List of space groups#List

There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point group of the unit cell.

Symbols

In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.

These are the Bravais lattices in three dimensions:

P primitive
I body centered (from the German Innenzentriert)
F face centered (from the German Flächenzentriert)
A centered on A faces only
B centered on B faces only
C centered on C faces only
R rhombohedral

A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.

$a$ , $b$ , or $c$ : glide translation along half the lattice vector of this face
$n$ : glide translation along half the diagonal of this face
$d$ : glide planes with translation along a quarter of a face diagonal
$e$ : two glides with the same glide plane and translation along two (different) half-lattice vectors.{{refn|group=note|name=e|The symbol $e$ was introduced by the IUCR in 1992. Prior to this, the space groups Aem2 (No. 39), Aea2 (No. 41), Cmce (No. 64), Cmme (No. 67), and Ccce (No. 68) were known as Abm2 (No. 39), Aba2 (No. 41), Cmca (No. 64), Cmma (No. 67), and Ccca (No. 68) respectively. Historical literature may refer to the old names, but their meaning is unchanged.{{cite journal |last1=de Wolff |first1=P. M. |last2=Billiet |first2=Y. |last3=Donnay |first3=J. D. H. |last4=Fischer |first4=W. |last5=Galiulin |first5=R. B. |last6=Glazer |first6=A. M. |last7=Hahn |first7=T. |last8=Senechal |first8=M. |last9=Shoemaker |first9=D. P. |last10=Wondratschek |first10=H. |last11=Wilson |first11=A. J. C. |last12=Abrahams |first12=S. C. |title=Symbols for symmetry elements and symmetry operations. Final report of the IUCr Ad-Hoc Committee on the Nomenclature of Symmetry |journal=Acta Crystallographica Section A |volume=48 |issue=5 |date=1992-09-01 |issn=0108-7673 |pages=727–732 |doi=10.1107/s0108767392003428 |doi-access=free|bibcode=1992AcCrA..48..727D }}}}

A gyration point can be replaced by a screw axis denoted by a number, n, where the angle of rotation is $\color{Black}\tfrac{360^\circ}{n}$ . The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2₁ is a 180° (twofold) rotation followed by a translation of {{sfrac|1|2}} of the lattice vector. 3₁ is a 120° (threefold) rotation followed by a translation of {{sfrac|1|3}} of the lattice vector. The possible screw axes are: 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄, and 6₅.

Wherever there is both a rotation or screw axis n and a mirror or glide plane m along the same crystallographic direction, they are represented as a fraction $\frac{n}{m}$ or n/m. For example, 4₁/a means that the crystallographic axis in question contains both a 4₁ screw axis as well as a glide plane along a.

In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups. This is sometimes supplemented with a symbol of the form $\Gamma_x^y$ which specifies the Bravais lattice. Here $x \in \{t, m, o, q, rh, h, c\}$ is the lattice system, and $y \in \{\empty, b, v, f\}$ is the centering type.{{cite book |last1=Bradley |first1=C. J. |last2=Cracknell |first2=A. P. |title=The mathematical theory of symmetry in solids: representation theory for point groups and space groups |publisher=Clarendon Press |location=Oxford New York |year=2010 |isbn=978-0-19-958258-7 |oclc=859155300 |pages=127–134}}

In Fedorov symbol, the type of space group is denoted as s (symmorphic ), h (hemisymmorphic), or a (asymmorphic). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups.

=Symmorphic=

The 73 symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Example for point group 4/mmm ( $\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$ ): the symmorphic space groups are P4/mmm ( $P\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$ , 36s) and I4/mmm ( $I\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$ , 37s).

=Hemisymmorphic=

The 54 hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. Example for point group 4/mmm ( $\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$ ): hemisymmorphic space groups contain the axial combination 422, but at least one mirror plane m will be substituted with glide plane, for example P4/mcc ( $P\tfrac{4}{m}\tfrac{2}{c}\tfrac{2}{c}$ , 35h), P4/nbm ( $P\tfrac{4}{n}\tfrac{2}{b}\tfrac{2}{m}$ , 36h), P4/nnc ( $P\tfrac{4}{n}\tfrac{2}{n}\tfrac{2}{c}$ , 37h), and I4/mcm ( $I\tfrac{4}{m}\tfrac{2}{c}\tfrac{2}{m}$ , 38h).

=Asymmorphic=

The remaining 103 space groups are asymmorphic. Example for point group 4/mmm ( $\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$ ): P4/mbm ( $P\tfrac{4}{m}\tfrac{2_1}{b}\tfrac{2}{m}$ , 54a), P4₂/mmc ( $P\tfrac{4_2}{m}\tfrac{2}{m}\tfrac{2}{c}$ , 60a), I4₁/acd ( $I\tfrac{4_1}{a}\tfrac{2}{c}\tfrac{2}{d}$ , 58a) - none of these groups contains the axial combination 422.

List of triclinic

class="wikitable floatright" style="text-align:center;"

|+ Triclinic Bravais lattice

80px

class=wikitable

|+ Triclinic crystal system

! Number

! Point group

! Orbifold

! Short name

! Full name

! Schoenflies

! Fedorov

! Shubnikov

! Fibrifold

align=center

1

P 1

\Gamma_tC_1^1

(a/b/c)\cdot 1

(\circ)

align=center

\times

P{{overline|1}}

P {{overline|1}}

\Gamma_tC_i^1

(a/b/c)\cdot \tilde 2

(2222)

List of monoclinic

class="wikitable floatright" \|+ Monoclinic Bravais lattice
Simple (P) ! Base (C)
80px \| 80px

class="wikitable floatright"

|+ Monoclinic Bravais lattice

Simple (P)

! Base (C)

80px

| 80px

class=wikitable \|+ Monoclinic crystal system !Number ! Point group ! Orbifold ! Short name ! colspan=2\|Full name(s) ! Schoenflies ! Fedorov ! Shubnikov ! Fibrifold (primary) ! Fibrifold (secondary)
align=center \|3	rowspan=3\|2	rowspan=3\| $22$	P2	P 1 2 1	P 1 1 2	$\Gamma_mC_2^1$	3s	$(b:(c/a)):2$	$(2_02_02_02_0)$	$({}_0{}_0)$
align=center \|4	P2₁	P 1 2₁ 1	P 1 1 2₁	$\Gamma_mC_2^2$	1a	$(b:(c/a)):2_1$	$(2_12_12_12_1)$	$(\bar{\times}\bar{\times})$
align=center \|5	C2	C 1 2 1	B 1 1 2	$\Gamma_m^bC_2^3$	4s	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) :2$	$(2_02_02_12_1)$	$({}_1{}_1)$ , $({*}\bar{\times})$
align=center \|6	rowspan=4\|m	rowspan=4\| $*$	Pm	P 1 m 1	P 1 1 m	$\Gamma_mC_s^1$	5s	$(b:(c/a))\cdot m$	$[\circ_0]$	$({}{\cdot}{}{\cdot})$
align=center \|7	Pc	P 1 c 1	P 1 1 b	$\Gamma_mC_s^2$	1h	$(b:(c/a))\cdot \tilde c$	$(\bar\circ_0)$	$({}{:}{}{:})$ , $({\times}{\times}_0)$
align=center \|8	Cm	C 1 m 1	B 1 1 m	$\Gamma_m^bC_s^3$	6s	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot m$	$[\circ_1]$	$({}{\cdot}{}{:})$ , $({*}{\cdot}{\times})$
align=center \|9	Cc	C 1 c 1	B 1 1 b	$\Gamma_m^bC_s^4$	2h	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot \tilde c$	$(\bar\circ_1)$	$({*}{:}{\times})$ , $({\times}{\times}_1)$
align=center \|10	rowspan=6\|2/m	rowspan=6\| $2*$	P2/m	P 1 2/m 1	P 1 1 2/m	$\Gamma_mC_{2h}^1$	7s	$(b:(c/a))\cdot m:2$	$[2_02_02_02_0]$	$[*2{\cdot}22{\cdot}2)$
align=center \|11	P2₁/m	P 1 2₁/m 1	P 1 1 2₁/m	$\Gamma_mC_{2h}^2$	2a	$(b:(c/a))\cdot m:2_1$	$[2_12_12_12_1]$	$(22{*}{\cdot})$
align=center \|12	C2/m	C 1 2/m 1	B 1 1 2/m	$\Gamma_m^bC_{2h}^3$	8s	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot m:2$	$[2_02_02_12_1]$	$(2{\cdot}22{:}2)$ , $(2\bar{}2{\cdot}2)$
align=center \|13	P2/c	P 1 2/c 1	P 1 1 2/b	$\Gamma_mC_{2h}^4$	3h	$(b:(c/a))\cdot \tilde c:2$	$(2_02_022)$	$(2{:}22{:}2)$ , $(22{}_0)$
align=center \|14	P2₁/c	P 1 2₁/c 1	P 1 1 2₁/b	$\Gamma_mC_{2h}^5$	3a	$(b:(c/a))\cdot \tilde c:2_1$	$(2_12_122)$	$(22{*}{:})$ , $(22{\times})$
align=center \|15	C2/c	C 1 2/c 1	B 1 1 2/b	$\Gamma_m^bC_{2h}^6$	4h	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot \tilde c:2$	$(2_02_122)$	$(2\bar{}2{:}2)$ , $(22{}_1)$

List of orthorhombic

class=wikitable style="text-align:center;" \|+ Orthorhombic Bravais lattice
Simple (P) ! Body (I) ! Face (F) ! Base (A or C)
80px \| 80px \| 80px \| 80px

class=wikitable style="text-align:center;"

|+ Orthorhombic Bravais lattice

Simple (P)

! Body (I)

! Face (F)

! Base (A or C)

80px

| 80px

class=wikitable \|+ Orthorhombic crystal system !Number ! Point group ! Orbifold ! Short name ! Full name ! Schoenflies ! Fedorov ! Shubnikov ! Fibrifold (primary) ! Fibrifold (secondary)
align=center \|16	rowspan=9\|222	rowspan=9\| $222$	P222	P 2 2 2	$\Gamma_oD_2^1$	9s	$(c:a:b):2:2$	$(*2_02_02_02_0)$
align=center \|17	P222₁	P 2 2 2₁	$\Gamma_oD_2^2$	4a	$(c:a:b):2_1:2$	$(*2_12_12_12_1)$	$(2_02_0{*})$
align=center \|18	P2₁2₁2	P 2₁ 2₁ 2	$\Gamma_oD_2^3$	7a	$(c:a:b):2$ 16px $2_1$	$(2_02_0\bar{\times})$	$(2_12_1{*})$
align=center \|19	P2₁2₁2₁	P 2₁ 2₁ 2₁	$\Gamma_oD_2^4$	8a	$(c:a:b):2_1$ 16px $2_1$	$(2_12_1\bar{\times})$
align=center \|20	C222₁	C 2 2 2₁	$\Gamma_o^bD_2^5$	5a	$\left ( \tfrac{a+b}{2}:c:a:b\right ) :2_1:2$	$(2_1{*}2_12_1)$	$(2_02_1{*})$
align=center \|21	C222	C 2 2 2	$\Gamma_o^bD_2^6$	10s	$\left ( \tfrac{a+b}{2}:c:a:b\right ) :2:2$	$(2_0{*}2_02_0)$	$(*2_02_02_12_1)$
align=center \|22	F222	F 2 2 2	$\Gamma_o^fD_2^7$	12s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) :2:2$	$(*2_02_12_02_1)$
align=center \|23	I222	I 2 2 2	$\Gamma_o^vD_2^8$	11s	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) :2:2$	$(2_1{*}2_02_0)$
align=center \|24	I2₁2₁2₁	I 2₁ 2₁ 2₁	$\Gamma_o^vD_2^9$	6a	$\left ( \tfrac{a+b+c}{2}/c:a:b \right ) :2:2_1$	$(2_0{*}2_12_1)$
align=center \|25	rowspan=22\|mm2	rowspan=22\| $*22$	Pmm2	P m m 2	$\Gamma_oC_{2v}^1$	13s	$(c:a:b):m \cdot 2$	$(*{\cdot}2{\cdot}2{\cdot}2{\cdot}2)$	$[{}_0{\cdot}{}_0{\cdot}]$
align=center \|26	Pmc2₁	P m c 2₁	$\Gamma_oC_{2v}^2$	9a	$(c:a:b): \tilde c \cdot 2_1$	$(*{\cdot}2{:}2{\cdot}2{:}2)$	$(\bar{}{\cdot}\bar{}{\cdot})$ , $[{\times_0}{\times_0}]$
align=center \|27	Pcc2	P c c 2	$\Gamma_oC_{2v}^3$	5h	$(c:a:b): \tilde c \cdot 2$	$(*{:}2{:}2{:}2{:}2)$	$(\bar{}_0\bar{}_0)$
align=center \|28	Pma2	P m a 2	$\Gamma_oC_{2v}^4$	6h	$(c:a:b): \tilde a \cdot 2$	$(2_02_0{*}{\cdot})$	$[{}_0{:}{}_0{:}]$ , $({\cdot}{}_0)$
align=center \|29	Pca2₁	P c a 2₁	$\Gamma_oC_{2v}^5$	11a	$(c:a:b): \tilde a \cdot 2_1$	$(2_12_1{*}{:})$	$(\bar{}{:}\bar{}{:})$
align=center \|30	Pnc2	P n c 2	$\Gamma_oC_{2v}^6$	7h	$(c:a:b): \tilde c \odot 2$	$(2_02_0{*}{:})$	$(\bar{}_1\bar{}_1)$ , $({*}_0{\times}_0)$
align=center \|31	Pmn2₁	P m n 2₁	$\Gamma_oC_{2v}^7$	10a	$(c:a:b): \widetilde{ac} \cdot 2_1$	$(2_12_1{*}{\cdot})$	$(*{\cdot}\bar{\times})$ , $[{\times}_0{\times}_1]$
align=center \|32	Pba2	P b a 2	$\Gamma_oC_{2v}^8$	9h	$(c:a:b): \tilde a \odot 2$	$(2_02_0{\times}_0)$	$({:}{}_0)$
align=center \|33	Pna2₁	P n a 2₁	$\Gamma_oC_{2v}^9$	12a	$(c:a:b): \tilde a \odot 2_1$	$(2_12_1{\times})$	$(*{:}{\times})$ , $({\times}{\times}_1)$
align=center \|34	Pnn2	P n n 2	$\Gamma_oC_{2v}^{10}$	8h	$(c:a:b): \widetilde{ac} \odot 2$	$(2_02_0{\times}_1)$	$(*_0{\times}_1)$
align=center \|35	Cmm2	C m m 2	$\Gamma_o^bC_{2v}^{11}$	14s	$\left ( \tfrac{a+b}{2}:c:a:b\right ) :m \cdot 2$	$(2_0{*}{\cdot}2{\cdot}2)$	$[_0{\cdot}{}_0{:}]$
align=center \|36	Cmc2₁	C m c 2₁	$\Gamma_o^bC_{2v}^{12}$	13a	$\left ( \tfrac{a+b}{2}:c:a:b\right ) :\tilde c \cdot 2_1$	$(2_1{*}{\cdot}2{:}2)$	$(\bar{}{\cdot}\bar{}{:})$ , $[{\times}_1{\times}_1]$
align=center \|37	Ccc2	C c c 2	$\Gamma_o^bC_{2v}^{13}$	10h	$\left ( \tfrac{a+b}{2}:c:a:b\right ) : \tilde c \cdot 2$	$(2_0{*}{:}2{:}2)$	$(\bar{}_0\bar{}_1)$
align=center \|38	Amm2	A m m 2	$\Gamma_o^bC_{2v}^{14}$	15s	$\left ( \tfrac{b+c}{2}/c:a:b\right ):m \cdot 2$	$(*{\cdot}2{\cdot}2{\cdot}2{:}2)$	$[{}_1{\cdot}{}_1{\cdot}]$ , $[*{\cdot}{\times}_0]$
align=center \|39	Aem2	A b m 2	$\Gamma_o^bC_{2v}^{15}$	11h	$\left ( \tfrac{b+c}{2}/c:a:b\right ) :m \cdot 2_1$	$(*{\cdot}2{:}2{:}2{:}2)$	$[{}_1{:}{}_1{:}]$ , $(\bar{}{\cdot}\bar{}_0)$
align=center \|40	Ama2	A m a 2	$\Gamma_o^bC_{2v}^{16}$	12h	$\left ( \tfrac{b+c}{2}/c:a:b\right ) : \tilde a \cdot 2$	$(2_02_1{*}{\cdot})$	$({\cdot}{}_1)$ , $[*{:}{\times}_1]$
align=center \|41	Aea2	A b a 2	$\Gamma_o^bC_{2v}^{17}$	13h	$\left ( \tfrac{b+c}{2}/c:a:b\right ) : \tilde a \cdot 2_1$	$(2_02_1{*}{:})$	$({:}{}_1)$ , $(\bar{}{:}\bar{}_1)$
align=center \|42	Fmm2	F m m 2	$\Gamma_o^fC_{2v}^{18}$	17s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) :m \cdot 2$	$(*{\cdot}2{\cdot}2{:}2{:}2)$	$[{}_1{\cdot}{}_1{:}]$
align=center \|43	Fdd2	F d d 2	$\Gamma_o^fC_{2v}^{19}$	16h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b \right ) : \tfrac{1}{2} \widetilde{ac} \odot 2$	$(2_02_1{\times})$	$({*}_1{\times})$
align=center \|44	Imm2	I m m 2	$\Gamma_o^vC_{2v}^{20}$	16s	$\left ( \tfrac{a+b+c}{2}/c:a:b \right ) :m \cdot 2$	$(2_1{*}{\cdot}2{\cdot}2)$	$[*{\cdot}{\times}_1]$
align=center \|45	Iba2	I b a 2	$\Gamma_o^vC_{2v}^{21}$	15h	$\left ( \tfrac{a+b+c}{2}/c:a:b \right ) : \tilde c \cdot 2$	$(2_1{*}{:}2{:}2)$	$(\bar{}{:}\bar{}_0)$
align=center \|46	Ima2	I m a 2	$\Gamma_o^vC_{2v}^{22}$	14h	$\left ( \tfrac{a+b+c}{2}/c:a:b \right ) : \tilde a \cdot 2$	$(2_0{*}{\cdot}2{:}2)$	$(\bar{}{\cdot}\bar{}_1)$ , $[*{:}{\times}_0]$
align=center \|47	rowspan=28\| $\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	rowspan=28\| $*222$	Pmmm	P 2/m 2/m 2/m	$\Gamma_oD_{2h}^1$	18s	$\left ( c:a:b \right ) \cdot m:2 \cdot m$	$[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]$
align=center \|48	Pnnn	P 2/n 2/n 2/n	$\Gamma_oD_{2h}^2$	19h	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \odot \widetilde{ac}$	$(2\bar{*}_12_02_0)$
align=center \|49	Pccm	P 2/c 2/c 2/m	$\Gamma_oD_{2h}^3$	17h	$\left ( c:a:b \right ) \cdot m:2 \cdot \tilde c$	$[*{:}2{:}2{:}2{:}2]$	$(*2_02_02{\cdot}2)$
align=center \|50	Pban	P 2/b 2/a 2/n	$\Gamma_oD_{2h}^4$	18h	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \odot \tilde a$	$(2\bar{*}_02_02_0)$	$(*2_02_02{:}2)$
align=center \|51	Pmma	P 2₁/m 2/m 2/a	$\Gamma_oD_{2h}^5$	14a	$\left ( c:a:b \right ) \cdot \tilde a :2 \cdot m$	$[2_02_0{*}{\cdot}]$	$[{\cdot}2{:}2{\cdot}2{:}2]$ , $[2{\cdot}2{\cdot}2{\cdot}2]$
align=center \|52	Pnna	P 2/n 2₁/n 2/a	$\Gamma_oD_{2h}^6$	17a	$\left ( c:a:b \right ) \cdot \tilde a:2 \odot \widetilde{ac}$	$(2_02\bar{*}_1)$	$(2_0{}2{:}2)$ , $(2\bar{}2_12_1)$
align=center \|53	Pmna	P 2/m 2/n 2₁/a	$\Gamma_oD_{2h}^7$	15a	$\left ( c:a:b \right ) \cdot \tilde a:2_1 \cdot \widetilde{ac}$	$[2_02_0{*}{:}]$	$(2_12_12{\cdot}2)$ , $(2_0{}2{\cdot}2)$
align=center \|54	Pcca	P 2₁/c 2/c 2/a	$\Gamma_oD_{2h}^8$	16a	$\left ( c:a:b \right ) \cdot \tilde a:2 \cdot \tilde c$	$(2_02\bar{*}_0)$	$(2{:}2{:}2{:}2)$ , $(2_12_12{:}2)$
align=center \|55	Pbam	P 2₁/b 2₁/a 2/m	$\Gamma_oD_{2h}^9$	22a	$\left ( c:a:b \right ) \cdot m:2 \odot \tilde a$	$[2_02_0{\times}_0]$	$(*2{\cdot}2{:}2{\cdot}2)$
align=center \|56	Pccn	P 2₁/c 2₁/c 2/n	$\Gamma_oD_{2h}^{10}$	27a	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \cdot \tilde c$	$(2\bar{*}{:}2{:}2)$	$(2_12\bar{*}_0)$
align=center \|57	Pbcm	P 2/b 2₁/c 2₁/m	$\Gamma_oD_{2h}^{11}$	23a	$\left ( c:a:b \right ) \cdot m:2_1 \odot \tilde c$	$(2_02\bar{*}{\cdot})$	$(2{:}2{\cdot}2{:}2)$ , $[2_12_1{}{:}]$
align=center \|58	Pnnm	P 2₁/n 2₁/n 2/m	$\Gamma_oD_{2h}^{12}$	25a	$\left ( c:a:b \right ) \cdot m:2 \odot \widetilde{ac}$	$[2_02_0{\times}_1]$	$(2_1{*}2{\cdot}2)$
align=center \|59	Pmmn	P 2₁/m 2₁/m 2/n	$\Gamma_oD_{2h}^{13}$	24a	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \cdot m$	$(2\bar{*}{\cdot}2{\cdot}2)$	$[2_12_1{*}{\cdot}]$
align=center \|60	Pbcn	P 2₁/b 2/c 2₁/n	$\Gamma_oD_{2h}^{14}$	26a	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2_1 \odot \tilde c$	$(2_02\bar{*}{:})$	$(2_1{}2{:}2)$ , $(2_12\bar{}_1)$
align=center \|61	Pbca	P 2₁/b 2₁/c 2₁/a	$\Gamma_oD_{2h}^{15}$	29a	$\left ( c:a:b \right ) \cdot \tilde a:2_1 \odot \tilde c$	$(2_12\bar{*}{:})$
align=center \|62	Pnma	P 2₁/n 2₁/m 2₁/a	$\Gamma_oD_{2h}^{16}$	28a	$\left ( c:a:b \right ) \cdot \tilde a:2_1 \odot m$	$(2_12\bar{*}{\cdot})$	$(2\bar{*}{\cdot}2{:}2)$ , $[2_12_1{\times}]$
align=center \|63	Cmcm	C 2/m 2/c 2₁/m	$\Gamma_o^bD_{2h}^{17}$	18a	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2_1 \cdot \tilde c$	$[2_02_1{*}{\cdot}]$	$(2{\cdot}2{\cdot}2{:}2)$ , $[2_1{}{\cdot}2{:}2]$
align=center \|64	Cmce	C 2/m 2/c 2₁/a	$\Gamma_o^bD_{2h}^{18}$	19a	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2_1 \cdot \tilde c$	$[2_02_1{*}{:}]$	$(2{\cdot}2{:}2{:}2)$ , $(2_12{\cdot}2{:}2)$
align=center \|65	Cmmm	C 2/m 2/m 2/m	$\Gamma_o^bD_{2h}^{19}$	19s	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot m$	$[2_0{*}{\cdot}2{\cdot}2]$	$[*{\cdot}2{\cdot}2{\cdot}2{:}2]$
align=center \|66	Cccm	C 2/c 2/c 2/m	$\Gamma_o^bD_{2h}^{20}$	20h	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot \tilde c$	$[2_0{*}{:}2{:}2]$	$(*2_02_12{\cdot}2)$
align=center \|67	Cmme	C 2/m 2/m 2/e	$\Gamma_o^bD_{2h}^{21}$	21h	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2 \cdot m$	$(*2_02{\cdot}2{\cdot}2)$	$[*{\cdot}2{:}2{:}2{:}2]$
align=center \|68	Ccce	C 2/c 2/c 2/e	$\Gamma_o^bD_{2h}^{22}$	22h	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2 \cdot \tilde c$	$(*2_02{:}2{:}2)$	$(*2_02_12{:}2)$
align=center \|69	Fmmm	F 2/m 2/m 2/m	$\Gamma_o^fD_{2h}^{23}$	21s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot m$	$[*{\cdot}2{\cdot}2{:}2{:}2]$
align=center \|70	Fddd	F 2/d 2/d 2/d	$\Gamma_o^fD_{2h}^{24}$	24h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) \cdot \tfrac{1}{2}\widetilde{ab}:2 \odot \tfrac{1}{2}\widetilde{ac}$	$(2\bar{*}2_02_1)$
align=center \|71	Immm	I 2/m 2/m 2/m	$\Gamma_o^vD_{2h}^{25}$	20s	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot m:2 \cdot m$	$[2_1{*}{\cdot}2{\cdot}2]$
align=center \|72	Ibam	I 2/b 2/a 2/m	$\Gamma_o^vD_{2h}^{26}$	23h	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot m:2 \cdot \tilde c$	$[2_1{*}{:}2{:}2]$	$(*2_02{\cdot}2{:}2)$
align=center \|73	Ibca	I 2/b 2/c 2/a	$\Gamma_o^vD_{2h}^{27}$	21a	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot \tilde a :2 \cdot \tilde c$	$(*2_12{:}2{:}2)$
align=center \|74	Imma	I 2/m 2/m 2/a	$\Gamma_o^vD_{2h}^{28}$	20a	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot \tilde a :2 \cdot m$	$(*2_12{\cdot}2{\cdot}2)$	$[2_0{*}{\cdot}2{:}2]$

List of tetragonal

class="wikitable floatright" \|+ Tetragonal Bravais lattice
Simple (P) ! Body (I)
80px \| 80px

class="wikitable floatright"

|+ Tetragonal Bravais lattice

Simple (P)

! Body (I)

80px

| 80px

class=wikitable \|+ Tetragonal crystal system !Number ! Point group ! Orbifold ! Short name ! Full name ! Schoenflies ! Fedorov ! Shubnikov ! Fibrifold
align=center \|75	rowspan=6\|4	rowspan=6\| $44$	P4	P 4	$\Gamma_qC_4^1$	22s	$(c:a:a):4$	$(4_04_02_0)$
align=center \|76	P4₁	P 4₁	$\Gamma_qC_4^2$	30a	$(c:a:a) :4_1$	$(4_14_12_1)$
align=center \|77	P4₂	P 4₂	$\Gamma_qC_4^3$	33a	$(c:a:a) :4_2$	$(4_24_22_0)$
align=center \|78	P4₃	P 4₃	$\Gamma_qC_4^4$	31a	$(c:a:a) :4_3$	$(4_14_12_1)$
align=center \|79	I4	I 4	$\Gamma_q^vC_4^5$	23s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4$	$(4_24_02_1)$
align=center \|80	I4₁	I 4₁	$\Gamma_q^vC_4^6$	32a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1$	$(4_34_12_0)$
align=center \|81	rowspan=2\|{{overline\|4}}	rowspan=2\| $2\times$	P{{overline\|4}}	P {{overline\|4}}	$\Gamma_qS_4^1$	26s	$(c:a:a):\tilde 4$	$(442_0)$
align=center \|82	I{{overline\|4}}	I {{overline\|4}}	$\Gamma_q^vS_4^2$	27s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4$	$(442_1)$
align=center \|83	rowspan=6\|4/m	rowspan=6\| $4*$	P4/m	P 4/m	$\Gamma_qC_{4h}^1$	28s	$(c:a:a)\cdot m:4$	$[4_04_02_0]$
align=center \|84	P4₂/m	P 4₂/m	$\Gamma_qC_{4h}^2$	41a	$(c:a:a)\cdot m:4_2$	$[4_24_22_0]$
align=center \|85	P4/n	P 4/n	$\Gamma_qC_{4h}^3$	29h	$(c:a:a)\cdot \widetilde{ab}:4$	$(44_02)$
align=center \|86	P4₂/n	P 4₂/n	$\Gamma_qC_{4h}^4$	42a	$(c:a:a)\cdot \widetilde{ab}:4_2$	$(44_22)$
align=center \|87	I4/m	I 4/m	$\Gamma_q^vC_{4h}^5$	29s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4$	$[4_24_02_1]$
align=center \|88	I4₁/a	I 4₁/a	$\Gamma_q^vC_{4h}^6$	40a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1$	$(44_12)$
align=center \|89	rowspan=10\|422	rowspan=10\| $224$	P422	P 4 2 2	$\Gamma_qD_4^1$	30s	$(c:a:a):4:2$	$(*4_04_02_0)$
align=center \|90	P42₁2	P42₁2	$\Gamma_qD_4^2$	43a	$(c:a:a):4$ 16px $2_1$	$(4_0{*}2_0)$
align=center \|91	P4₁22	P 4₁ 2 2	$\Gamma_qD_4^3$	44a	$(c:a:a):4_1:2$	$(*4_14_12_1)$
align=center \|92	P4₁2₁2	P 4₁ 2₁ 2	$\Gamma_qD_4^4$	48a	$(c:a:a):4_1$ 16px $2_1$	$(4_1{*}2_1)$
align=center \|93	P4₂22	P 4₂ 2 2	$\Gamma_qD_4^5$	47a	$(c:a:a):4_2:2$	$(*4_24_22_0)$
align=center \|94	P4₂2₁2	P 4₂ 2₁ 2	$\Gamma_qD_4^6$	50a	$(c:a:a):4_2$ 16px $2_1$	$(4_2{*}2_0)$
align=center \|95	P4₃22	P 4₃ 2 2	$\Gamma_qD_4^7$	45a	$(c:a:a):4_3:2$	$(*4_14_12_1)$
align=center \|96	P4₃2₁2	P 4₃ 2₁ 2	$\Gamma_qD_4^8$	49a	$(c:a:a):4_3$ 16px $2_1$	$(4_1{*}2_1)$
align=center \|97	I422	I 4 2 2	$\Gamma_q^vD_4^9$	31s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4:2$	$(*4_24_02_1)$
align=center \|98	I4₁22	I 4₁ 2 2	$\Gamma_q^vD_4^{10}$	46a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4:2_1$	$(*4_34_12_0)$
align=center \|99	rowspan=12\|4mm	rowspan=12\| $*44$	P4mm	P 4 m m	$\Gamma_qC_{4v}^1$	24s	$(c:a:a):4\cdot m$	$(*{\cdot}4{\cdot}4{\cdot}2)$
align=center \|100	P4bm	P 4 b m	$\Gamma_qC_{4v}^2$	26h	$(c:a:a):4\odot \tilde a$	$(4_0{*}{\cdot}2)$
align=center \|101	P4₂cm	P 4₂ c m	$\Gamma_qC_{4v}^3$	37a	$(c:a:a):4_2\cdot \tilde c$	$(*{:}4{\cdot}4{:}2)$
align=center \|102	P4₂nm	P 4₂ n m	$\Gamma_qC_{4v}^4$	38a	$(c:a:a):4_2\odot \widetilde{ac}$	$(4_2{*}{\cdot}2)$
align=center \|103	P4cc	P 4 c c	$\Gamma_qC_{4v}^5$	25h	$(c:a:a):4\cdot \tilde c$	$(*{:}4{:}4{:}2)$
align=center \|104	P4nc	P 4 n c	$\Gamma_qC_{4v}^6$	27h	$(c:a:a):4\odot \widetilde{ac}$	$(4_0{*}{:}2)$
align=center \|105	P4₂mc	P 4₂ m c	$\Gamma_qC_{4v}^7$	36a	$(c:a:a):4_2\cdot m$	$(*{\cdot}4{:}4{\cdot}2)$
align=center \|106	P4₂bc	P 4₂ b c	$\Gamma_qC_{4v}^8$	39a	$(c:a:a):4\odot \tilde a$	$(4_2{*}{:}2)$
align=center \|107	I4mm	I 4 m m	$\Gamma_q^vC_{4v}^9$	25s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4\cdot m$	$(*{\cdot}4{\cdot}4{:}2)$
align=center \|108	I4cm	I 4 c m	$\Gamma_q^vC_{4v}^{10}$	28h	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4\cdot \tilde c$	$(*{\cdot}4{:}4{:}2)$
align=center \|109	I4₁md	I 4₁ m d	$\Gamma_q^vC_{4v}^{11}$	34a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1\odot m$	$(4_1{*}{\cdot}2)$
align=center \|110	I4₁cd	I 4₁ c d	$\Gamma_q^vC_{4v}^{12}$	35a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1\odot \tilde c$	$(4_1{*}{:}2)$
align=center \|111	rowspan=12\|{{overline\|4}}2m	rowspan=12\| $2{*}2$	P{{overline\|4}}2m	P {{overline\|4}} 2 m	$\Gamma_qD_{2d}^1$	32s	$(c:a:a):\tilde 4 :2$	$(*4{\cdot}42_0)$
align=center \|112	P{{overline\|4}}2c	P {{overline\|4}} 2 c	$\Gamma_qD_{2d}^2$	30h	$(c:a:a):\tilde 4$ 16px $2$	$(*4{:}42_0)$
align=center \|113	P{{overline\|4}}2₁m	P {{overline\|4}} 2₁ m	$\Gamma_qD_{2d}^3$	52a	$(c:a:a):\tilde 4 \cdot \widetilde{ab}$	$(4\bar{*}{\cdot}2)$
align=center \|114	P{{overline\|4}}2₁c	P {{overline\|4}} 2₁ c	$\Gamma_qD_{2d}^4$	53a	$(c:a:a):\tilde 4 \cdot \widetilde{abc}$	$(4\bar{*}{:}2)$
align=center \|115	P{{overline\|4}}m2	P {{overline\|4}} m 2	$\Gamma_qD_{2d}^5$	33s	$(c:a:a):\tilde 4 \cdot m$	$(*{\cdot}44{\cdot}2)$
align=center \|116	P{{overline\|4}}c2	P {{overline\|4}} c 2	$\Gamma_qD_{2d}^6$	31h	$(c:a:a):\tilde 4 \cdot \tilde c$	$(*{:}44{:}2)$
align=center \|117	P{{overline\|4}}b2	P {{overline\|4}} b 2	$\Gamma_qD_{2d}^7$	32h	$(c:a:a):\tilde 4 \odot \tilde a$	$(4\bar{*}_02_0)$
align=center \|118	P{{overline\|4}}n2	P {{overline\|4}} n 2	$\Gamma_qD_{2d}^8$	33h	$(c:a:a):\tilde 4 \cdot \widetilde{ac}$	$(4\bar{*}_12_0)$
align=center \|119	I{{overline\|4}}m2	I {{overline\|4}} m 2	$\Gamma_q^vD_{2d}^9$	35s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \cdot m$	$(*4{\cdot}42_1)$
align=center \|120	I{{overline\|4}}c2	I {{overline\|4}} c 2	$\Gamma_q^vD_{2d}^{10}$	34h	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \cdot \tilde c$	$(*4{:}42_1)$
align=center \|121	I{{overline\|4}}2m	I {{overline\|4}} 2 m	$\Gamma_q^vD_{2d}^{11}$	34s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 :2$	$(*{\cdot}44{:}2)$
align=center \|122	I{{overline\|4}}2d	I {{overline\|4}} 2 d	$\Gamma_q^vD_{2d}^{12}$	51a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \odot \tfrac{1}{2}\widetilde{abc}$	$(4\bar{*}2_1)$
align=center \|123	rowspan=20\|4/m 2/m 2/m	rowspan=20\| $*224$	P4/mmm	P 4/m 2/m 2/m	$\Gamma_qD_{4h}^1$	36s	$(c:a:a)\cdot m:4\cdot m$	$[*{\cdot}4{\cdot}4{\cdot}2]$
align=center \|124	P4/mcc	P 4/m 2/c 2/c	$\Gamma_qD_{4h}^2$	35h	$(c:a:a)\cdot m:4\cdot \tilde c$	$[*{:}4{:}4{:}2]$
align=center \|125	P4/nbm	P 4/n 2/b 2/m	$\Gamma_qD_{4h}^3$	36h	$(c:a:a)\cdot \widetilde{ab}:4\odot \tilde a$	$(*4_04{\cdot}2)$
align=center \|126	P4/nnc	P 4/n 2/n 2/c	$\Gamma_qD_{4h}^4$	37h	$(c:a:a)\cdot \widetilde{ab}:4\odot \widetilde{ac}$	$(*4_04{:}2)$
align=center \|127	P4/mbm	P 4/m 2₁/b 2/m	$\Gamma_qD_{4h}^5$	54a	$(c:a:a)\cdot m:4\odot \tilde a$	$[4_0{*}{\cdot}2]$
align=center \|128	P4/mnc	P 4/m 2₁/n 2/c	$\Gamma_qD_{4h}^6$	56a	$(c:a:a)\cdot m:4\odot \widetilde{ac}$	$[4_0{*}{:}2]$
align=center \|129	P4/nmm	P 4/n 2₁/m 2/m	$\Gamma_qD_{4h}^7$	55a	$(c:a:a)\cdot \widetilde{ab}:4\cdot m$	$(*4{\cdot}4{\cdot}2)$
align=center \|130	P4/ncc	P 4/n 2₁/c 2/c	$\Gamma_qD_{4h}^8$	57a	$(c:a:a)\cdot \widetilde{ab}:4\cdot \tilde c$	$(*4{:}4{:}2)$
align=center \|131	P4₂/mmc	P 4₂/m 2/m 2/c	$\Gamma_qD_{4h}^9$	60a	$(c:a:a)\cdot m:4_2\cdot m$	$[*{\cdot}4{:}4{\cdot}2]$
align=center \|132	P4₂/mcm	P 4₂/m 2/c 2/m	$\Gamma_qD_{4h}^{10}$	61a	$(c:a:a)\cdot m:4_2\cdot \tilde c$	$[*{:}4{\cdot}4{:}2]$
align=center \|133	P4₂/nbc	P 4₂/n 2/b 2/c	$\Gamma_qD_{4h}^{11}$	63a	$(c:a:a)\cdot \widetilde{ab}:4_2\odot \tilde a$	$(*4_24{:}2)$
align=center \|134	P4₂/nnm	P 4₂/n 2/n 2/m	$\Gamma_qD_{4h}^{12}$	62a	$(c:a:a)\cdot \widetilde{ab}:4_2\odot \widetilde{ac}$	$(*4_24{\cdot}2)$
align=center \|135	P4₂/mbc	P 4₂/m 2₁/b 2/c	$\Gamma_qD_{4h}^{13}$	66a	$(c:a:a)\cdot m:4_2\odot \tilde a$	$[4_2{*}{:}2]$
align=center \|136	P4₂/mnm	P 4₂/m 2₁/n 2/m	$\Gamma_qD_{4h}^{14}$	65a	$(c:a:a)\cdot m:4_2\odot \widetilde{ac}$	$[4_2{*}{\cdot}2]$
align=center \|137	P4₂/nmc	P 4₂/n 2₁/m 2/c	$\Gamma_qD_{4h}^{15}$	67a	$(c:a:a)\cdot \widetilde{ab}:4_2\cdot m$	$(*4{\cdot}4{:}2)$
align=center \|138	P4₂/ncm	P 4₂/n 2₁/c 2/m	$\Gamma_qD_{4h}^{16}$	65a	$(c:a:a)\cdot \widetilde{ab}:4_2\cdot \tilde c$	$(*4{:}4{\cdot}2)$
align=center \|139	I4/mmm	I 4/m 2/m 2/m	$\Gamma_q^vD_{4h}^{17}$	37s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4\cdot m$	$[*{\cdot}4{\cdot}4{:}2]$
align=center \|140	I4/mcm	I 4/m 2/c 2/m	$\Gamma_q^vD_{4h}^{18}$	38h	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4\cdot \tilde c$	$[*{\cdot}4{:}4{:}2]$
align=center \|141	I4₁/amd	I 4₁/a 2/m 2/d	$\Gamma_q^vD_{4h}^{19}$	59a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1\odot m$	$(*4_14{\cdot}2)$
align=center \|142	I4₁/acd	I 4₁/a 2/c 2/d	$\Gamma_q^vD_{4h}^{20}$	58a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1\odot \tilde c$	$(*4_14{:}2)$

List of trigonal

class="wikitable floatright" \|+ Trigonal Bravais lattice
Rhombohedral (R) ! Hexagonal (P)
style="vertical-align:top;" \| 100px \| 100px

class="wikitable floatright"

|+ Trigonal Bravais lattice

Rhombohedral (R)

! Hexagonal (P)

style="vertical-align:top;"

| 100px

class=wikitable \|+ Trigonal crystal system !Number ! Point group ! Orbifold ! Short name ! Full name ! Schoenflies ! Fedorov ! Shubnikov ! Fibrifold
align=center \|143	rowspan=4\|3	rowspan=4\| $33$	P3	P 3	$\Gamma_hC_3^1$	38s	$(c:(a/a)):3$	$(3_03_03_0)$
align=center \|144	P3₁	P 3₁	$\Gamma_hC_3^2$	68a	$(c:(a/a)):3_1$	$(3_13_13_1)$
align=center \|145	P3₂	P 3₂	$\Gamma_hC_3^3$	69a	$(c:(a/a)):3_2$	$(3_13_13_1)$
align=center \|146	R3	R 3	$\Gamma_{rh}C_3^4$	39s	$(a/a/a)/3$	$(3_03_13_2)$
align=center \|147	rowspan=2\|{{overline\|3}}	rowspan=2\| $3\times$	P{{overline\|3}}	P {{overline\|3}}	$\Gamma_hC_{3i}^1$	51s	$(c:(a/a)):\tilde 6$	$(63_02)$
align=center \|148	R{{overline\|3}}	R {{overline\|3}}	$\Gamma_{rh}C_{3i}^2$	52s	$(a/a/a)/\tilde 6$	$(63_12)$
align=center \|149	rowspan=7\|32	rowspan=7\| $223$	P312	P 3 1 2	$\Gamma_hD_3^1$	45s	$(c:(a/a)):2:3$	$(*3_03_03_0)$
align=center \|150	P321	P 3 2 1	$\Gamma_hD_3^2$	44s	$(c:(a/a))\cdot 2:3$	$(3_0{*}3_0)$
align=center \|151	P3₁12	P 3₁ 1 2	$\Gamma_hD_3^3$	72a	$(c:(a/a)):2:3_1$	$(*3_13_13_1)$
align=center \|152	P3₁21	P 3₁ 2 1	$\Gamma_hD_3^4$	70a	$(c:(a/a))\cdot 2:3_1$	$(3_1{*}3_1)$
align=center \|153	P3₂12	P 3₂ 1 2	$\Gamma_hD_3^5$	73a	$(c:(a/a)):2:3_2$	$(*3_13_13_1)$
align=center \|154	P3₂21	P 3₂ 2 1	$\Gamma_hD_3^6$	71a	$(c:(a/a))\cdot 2:3_2$	$(3_1{*}3_1)$
align=center \|155	R32	R 3 2	$\Gamma_{rh}D_3^7$	46s	$(a/a/a)/3:2$	$(*3_03_13_2)$
align=center \|156	rowspan=6\|3m	rowspan=6\| $*33$	P3m1	P 3 m 1	$\Gamma_hC_{3v}^1$	40s	$(c:(a/a)):m\cdot 3$	$(*{\cdot}3{\cdot}3{\cdot}3)$
align=center \|157	P31m	P 3 1 m	$\Gamma_hC_{3v}^2$	41s	$(c:(a/a))\cdot m\cdot 3$	$(3_0{*}{\cdot}3)$
align=center \|158	P3c1	P 3 c 1	$\Gamma_hC_{3v}^3$	39h	$(c:(a/a)):\tilde c:3$	$(*{:}3{:}3{:}3)$
align=center \|159	P31c	P 3 1 c	$\Gamma_hC_{3v}^4$	40h	$(c:(a/a))\cdot\tilde c :3$	$(3_0{*}{:}3)$
align=center \|160	R3m	R 3 m	$\Gamma_{rh}C_{3v}^5$	42s	$(a/a/a)/3\cdot m$	$(3_1{*}{\cdot}3)$
align=center \|161	R3c	R 3 c	$\Gamma_{rh}C_{3v}^6$	41h	$(a/a/a)/3\cdot\tilde c$	$(3_1{*}{:}3)$
align=center \|162	rowspan=6\|{{overline\|3}} 2/m	rowspan=6\| $2{*}3$	P{{overline\|3}}1m	P {{overline\|3}} 1 2/m	$\Gamma_hD_{3d}^1$	56s	$(c:(a/a))\cdot m\cdot\tilde 6$	$(*{\cdot}63_02)$
align=center \|163	P{{overline\|3}}1c	P {{overline\|3}} 1 2/c	$\Gamma_hD_{3d}^2$	46h	$(c:(a/a))\cdot\tilde c \cdot\tilde 6$	$(*{:}63_02)$
align=center \|164	P{{overline\|3}}m1	P {{overline\|3}} 2/m 1	$\Gamma_hD_{3d}^3$	55s	$(c:(a/a)):m\cdot\tilde 6$	$(*6{\cdot}3{\cdot}2)$
align=center \|165	P{{overline\|3}}c1	P {{overline\|3}} 2/c 1	$\Gamma_hD_{3d}^4$	45h	$(c:(a/a)):\tilde c \cdot\tilde 6$	$(*6{:}3{:}2)$
align=center \|166	R{{overline\|3}}m	R {{overline\|3}} 2/m	$\Gamma_{rh}D_{3d}^5$	57s	$(a/a/a)/\tilde 6 \cdot m$	$(*{\cdot}63_12)$
align=center \|167	R{{overline\|3}}c	R {{overline\|3}} 2/c	$\Gamma_{rh}D_{3d}^6$	47h	$(a/a/a)/\tilde 6 \cdot\tilde c$	$(*{:}63_12)$

List of hexagonal

class="wikitable floatright"

|+ Hexagonal Bravais lattice

80px

class=wikitable \|+ Hexagonal crystal system !Number ! Point group ! Orbifold ! Short name ! Full name ! Schoenflies ! Fedorov ! Shubnikov ! Fibrifold
align=center \|168	rowspan=6\|6	rowspan=6\| $66$	P6	P 6	$\Gamma_hC_6^1$	49s	$(c:(a/a)):6$	$(6_03_02_0)$
align=center \|169	P6₁	P 6₁	$\Gamma_hC_6^2$	74a	$(c:(a/a)):6_1$	$(6_13_12_1)$
align=center \|170	P6₅	P 6₅	$\Gamma_hC_6^3$	75a	$(c:(a/a)):6_5$	$(6_13_12_1)$
align=center \|171	P6₂	P 6₂	$\Gamma_hC_6^4$	76a	$(c:(a/a)):6_2$	$(6_23_22_0)$
align=center \|172	P6₄	P 6₄	$\Gamma_hC_6^5$	77a	$(c:(a/a)):6_4$	$(6_23_22_0)$
align=center \|173	P6₃	P 6₃	$\Gamma_hC_6^6$	78a	$(c:(a/a)):6_3$	$(6_33_02_1)$
align=center \|174	{{overline\|6}}	$3*$	P{{overline\|6}}	P {{overline\|6}}	$\Gamma_hC_{3h}^1$	43s	$(c:(a/a)):3:m$	$[3_03_03_0]$
align=center \|175	rowspan=2\|6/m	rowspan=2\| $6*$	P6/m	P 6/m	$\Gamma_hC_{6h}^1$	53s	$(c:(a/a))\cdot m :6$	$[6_03_02_0]$
align=center \|176	P6₃/m	P 6₃/m	$\Gamma_hC_{6h}^2$	81a	$(c:(a/a))\cdot m :6_3$	$[6_33_02_1]$
align=center \|177	rowspan=6\|622	rowspan=6\| $226$	P622	P 6 2 2	$\Gamma_hD_6^1$	54s	$(c:(a/a))\cdot 2 :6$	$(*6_03_02_0)$
align=center \|178	P6₁22	P 6₁ 2 2	$\Gamma_hD_6^2$	82a	$(c:(a/a))\cdot 2 :6_1$	$(*6_13_12_1)$
align=center \|179	P6₅22	P 6₅ 2 2	$\Gamma_hD_6^3$	83a	$(c:(a/a))\cdot 2 :6_5$	$(*6_13_12_1)$
align=center \|180	P6₂22	P 6₂ 2 2	$\Gamma_hD_6^4$	84a	$(c:(a/a))\cdot 2 :6_2$	$(*6_23_22_0)$
align=center \|181	P6₄22	P 6₄ 2 2	$\Gamma_hD_6^5$	85a	$(c:(a/a))\cdot 2 :6_4$	$(*6_23_22_0)$
align=center \|182	P6₃22	P 6₃ 2 2	$\Gamma_hD_6^6$	86a	$(c:(a/a))\cdot 2 :6_3$	$(*6_33_02_1)$
align=center \|183	rowspan=4\|6mm	rowspan=4\| $*66$	P6mm	P 6 m m	$\Gamma_hC_{6v}^1$	50s	$(c:(a/a)):m\cdot 6$	$(*{\cdot}6{\cdot}3{\cdot}2)$
align=center \|184	P6cc	P 6 c c	$\Gamma_hC_{6v}^2$	44h	$(c:(a/a)):\tilde c \cdot 6$	$(*{:}6{:}3{:}2)$
align=center \|185	P6₃cm	P 6₃ c m	$\Gamma_hC_{6v}^3$	80a	$(c:(a/a)):\tilde c \cdot 6_3$	$(*{\cdot}6{:}3{:}2)$
align=center \|186	P6₃mc	P 6₃ m c	$\Gamma_hC_{6v}^4$	79a	$(c:(a/a)):m\cdot 6_3$	$(*{:}6{\cdot}3{\cdot}2)$
align=center \|187	rowspan=4\|{{overline\|6}}m2	rowspan=4\| $*223$	P{{overline\|6}}m2	P {{overline\|6}} m 2	$\Gamma_hD_{3h}^1$	48s	$(c:(a/a)):m\cdot 3:m$	$[*{\cdot}3{\cdot}3{\cdot}3]$
align=center \|188	P{{overline\|6}}c2	P {{overline\|6}} c 2	$\Gamma_hD_{3h}^2$	43h	$(c:(a/a)):\tilde c \cdot 3:m$	$[*{:}3{:}3{:}3]$
align=center \|189	P{{overline\|6}}2m	P {{overline\|6}} 2 m	$\Gamma_hD_{3h}^3$	47s	$(c:(a/a))\cdot m:3\cdot m$	$[3_0{*}{\cdot}3]$
align=center \|190	P{{overline\|6}}2c	P {{overline\|6}} 2 c	$\Gamma_hD_{3h}^4$	42h	$(c:(a/a))\cdot m:3\cdot \tilde c$	$[3_0{*}{:}3]$
align=center \|191	rowspan=4\|6/m 2/m 2/m	rowspan=4\| $*226$	P6/mmm	P 6/m 2/m 2/m	$\Gamma_hD_{6h}^1$	58s	$(c:(a/a))\cdot m:6\cdot m$	$[*{\cdot}6{\cdot}3{\cdot}2]$
align=center \|192	P6/mcc	P 6/m 2/c 2/c	$\Gamma_hD_{6h}^2$	48h	$(c:(a/a))\cdot m:6\cdot\tilde c$	$[*{:}6{:}3{:}2]$
align=center \|193	P6₃/mcm	P 6₃/m 2/c 2/m	$\Gamma_hD_{6h}^3$	87a	$(c:(a/a))\cdot m:6_3\cdot\tilde c$	$[*{\cdot}6{:}3{:}2]$
align=center \|194	P6₃/mmc	P 6₃/m 2/m 2/c	$\Gamma_hD_{6h}^4$	88a	$(c:(a/a))\cdot m:6_3\cdot m$	$[*{:}6{\cdot}3{\cdot}2]$

List of cubic

class="wikitable" style="text-align:center;" \|+ Cubic Bravais lattice
Simple (P) ! Body centered (I) ! Face centered (F)
100px \| 100px \| 100px

class="wikitable" style="text-align:center;"

|+ Cubic Bravais lattice

Simple (P)

! Body centered (I)

! Face centered (F)

100px

| 100px

{{Gallery

| title=Example cubic structures

| width=160 |height=170

| File:CsCl crystal.svg

| (221) Caesium chloride. Different colors for the two atom types.

| File:Sphalerite-unit-cell-depth-fade-3D-balls.png|(216) Sphalerite

| File:12-14-hedral honeycomb.png

| (223) Weaire–Phelan structure

}}

class=wikitable \|+ Cubic crystal system !Number ! Point group ! Orbifold ! Short name ! Full name ! Schoenflies ! Fedorov ! Shubnikov ! Conway ! Fibrifold (preserving $z$ ) ! Fibrifold (preserving $x$ , $y$ , $z$ )
align=center \|195	rowspan=5\|23	rowspan=5\| $332$	P23	P 2 3	$\Gamma_cT^1$	59s	$\left ( a:a:a\right ) :2/3$	$2^\circ$	$(*2_02_02_02_0){:}3$	$(*2_02_02_02_0){:}3$
align=center \|196	F23	F 2 3	$\Gamma_c^fT^2$	61s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :2/3$	$1^\circ$	$(*2_02_12_02_1){:}3$	$(*2_02_12_02_1){:}3$
align=center \|197	I23	I 2 3	$\Gamma_c^vT^3$	60s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :2/3$	$4^{\circ\circ}$	$(2_1{*}2_02_0){:}3$	$(2_1{*}2_02_0){:}3$
align=center \|198	P2₁3	P 2₁ 3	$\Gamma_cT^4$	89a	$\left ( a:a:a\right ) :2_1/3$	$1^\circ/4$	$(2_12_1\bar{\times}){:}3$	$(2_12_1\bar{\times}){:}3$
align=center \|199	I2₁3	I 2₁ 3	$\Gamma_c^vT^5$	90a	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :2_1/3$	$2^\circ/4$	$(2_0{*}2_12_1){:}3$	$(2_0{*}2_12_1){:}3$
align=center \|200	rowspan=7\|2/m {{overline\|3}}	rowspan=7\| $3{*}2$	Pm{{overline\|3}}	P 2/m {{overline\|3}}	$\Gamma_cT_h^1$	62s	$\left ( a:a:a\right ) \cdot m/ \tilde 6$	$4^-$	$[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}3$	$[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}3$
align=center \|201	Pn{{overline\|3}}	P 2/n {{overline\|3}}	$\Gamma_cT_h^2$	49h	$\left ( a:a:a\right ) \cdot \widetilde{ab} / \tilde 6$	$4^{\circ+}$	$(2\bar{*}_12_02_0){:}3$	$(2\bar{*}_12_02_0){:}3$
align=center \|202	Fm{{overline\|3}}	F 2/m {{overline\|3}}	$\Gamma_c^fT_h^3$	64s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) \cdot m/ \tilde 6$	$2^-$	$[*{\cdot}2{\cdot}2{:}2{:}2]{:}3$	$[*{\cdot}2{\cdot}2{:}2{:}2]{:}3$
align=center \|203	Fd{{overline\|3}}	F 2/d {{overline\|3}}	$\Gamma_c^fT_h^4$	50h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) \cdot \tfrac{1}{2}\widetilde{ab} / \tilde 6$	$2^{\circ+}$	$(2\bar{*}2_02_1){:}3$	$(2\bar{*}2_02_1){:}3$
align=center \|204	Im{{overline\|3}}	I 2/m {{overline\|3}}	$\Gamma_c^vT_h^5$	63s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) \cdot m/\tilde 6$	$8^{-\circ}$	$[2_1{*}{\cdot}2{\cdot}2]{:}3$	$[2_1{*}{\cdot}2{\cdot}2]{:}3$
align=center \|205	Pa{{overline\|3}}	P 2₁/a {{overline\|3}}	$\Gamma_cT_h^6$	91a	$\left ( a:a:a\right ) \cdot \tilde a /\tilde 6$	$2^-/4$	$(2_12\bar{*}{:}){:}3$	$(2_12\bar{*}{:}){:}3$
align=center \|206	Ia{{overline\|3}}	I 2₁/a {{overline\|3}}	$\Gamma_c^vT_h^7$	92a	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) \cdot \tilde a /\tilde 6$	$4^-/4$	$(*2_12{:}2{:}2){:}3$	$(*2_12{:}2{:}2){:}3$
align=center \|207	rowspan=8\|432	rowspan=8\| $432$	P432	P 4 3 2	$\Gamma_cO^1$	68s	$\left ( a:a:a\right ) :4/3$	$4^{\circ-}$	$(*4_04_02_0){:}3$	$(*2_02_02_02_0){:}6$
align=center \|208	P4₂32	P 4₂ 3 2	$\Gamma_cO^2$	98a	$\left ( a:a:a\right ) :4_2//3$	$4^+$	$(*4_24_22_0){:}3$	$(*2_02_02_02_0){:}6$
align=center \|209	F432	F 4 3 2	$\Gamma_c^fO^3$	70s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/3$	$2^{\circ-}$	$(*4_24_02_1){:}3$	$(*2_02_12_02_1){:}6$
align=center \|210	F4₁32	F 4₁ 3 2	$\Gamma_c^fO^4$	97a	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//3$	$2^+$	$(*4_34_12_0){:}3$	$(*2_02_12_02_1){:}6$
align=center \|211	I432	I 4 3 2	$\Gamma_c^vO^5$	69s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4/3$	$8^{+\circ}$	$(4_24_02_1){:}3$	$(2_1{*}2_02_0){:}6$
align=center \|212	P4₃32	P 4₃ 3 2	$\Gamma_cO^6$	94a	$\left ( a:a:a\right ) :4_3//3$	$2^+/4$	$(4_1{*}2_1){:}3$	$(2_12_1\bar{\times}){:}6$
align=center \|213	P4₁32	P 4₁ 3 2	$\Gamma_cO^7$	95a	$\left ( a:a:a\right ) :4_1//3$	$2^+/4$	$(4_1{*}2_1){:}3$	$(2_12_1\bar{\times}){:}6$
align=center \|214	I4₁32	I 4₁ 3 2	$\Gamma_c^vO^8$	96a	$\left ( \tfrac{a+b+c}{2}/:a:a:a\right ) :4_1//3$	$4^+/4$	$(*4_34_12_0){:}3$	$(2_0{*}2_12_1){:}6$
align=center \|215	rowspan=6\|{{overline\|4}}3m	rowspan=6\| $*332$	P{{overline\|4}}3m	P {{overline\|4}} 3 m	$\Gamma_cT_d^1$	65s	$\left ( a:a:a\right ) :\tilde 4 /3$	$2^\circ{:}2$	$(*4{\cdot}42_0){:}3$	$(*2_02_02_02_0){:}6$
align=center \|216	F{{overline\|4}}3m	F {{overline\|4}} 3 m	$\Gamma_c^fT_d^2$	67s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :\tilde 4 /3$	$1^\circ{:}2$	$(*4{\cdot}42_1){:}3$	$(*2_02_12_02_1){:}6$
align=center \|217	I{{overline\|4}}3m	I {{overline\|4}} 3 m	$\Gamma_c^vT_d^3$	66s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :\tilde 4 /3$	$4^\circ{:}2$	$(*{\cdot}44{:}2){:}3$	$(2_1{*}2_02_0){:}6$
align=center \|218	P{{overline\|4}}3n	P {{overline\|4}} 3 n	$\Gamma_cT_d^4$	51h	$\left ( a:a:a\right ) :\tilde 4 //3$	$4^\circ$	$(*4{:}42_0){:}3$	$(*2_02_02_02_0){:}6$
align=center \|219	F{{overline\|4}}3c	F {{overline\|4}} 3 c	$\Gamma_c^fT_d^5$	52h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :\tilde 4 //3$	$2^{\circ\circ}$	$(*4{:}42_1){:}3$	$(*2_02_12_02_1){:}6$
align=center \|220	I{{overline\|4}}3d	I {{overline\|4}} 3 d	$\Gamma_c^vT_d^6$	93a	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :\tilde 4 //3$	$4^\circ/4$	$(4\bar{*}2_1){:}3$	$(2_0{*}2_12_1){:}6$
align=center \|221	rowspan=10\|4/m {{overline\|3}} 2/m	rowspan=10\| $*432$	Pm{{overline\|3}}m	P 4/m {{overline\|3}} 2/m	$\Gamma_cO_h^1$	71s	$\left ( a:a:a\right ) :4/\tilde 6 \cdot m$	$4^-{:}2$	$[*{\cdot}4{\cdot}4{\cdot}2]{:}3$	$[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}6$
align=center \|222	Pn{{overline\|3}}n	P 4/n {{overline\|3}} 2/n	$\Gamma_cO_h^2$	53h	$\left ( a:a:a\right ) :4/\tilde 6 \cdot \widetilde{abc}$	$8^{\circ\circ}$	$(*4_04{:}2){:}3$	$(2\bar{*}_12_02_0){:}6$
align=center \|223	Pm{{overline\|3}}n	P 4₂/m {{overline\|3}} 2/n	$\Gamma_cO_h^3$	102a	$\left ( a:a:a\right ) :4_2//\tilde 6 \cdot \widetilde{abc}$	$8^\circ$	$[*{\cdot}4{:}4{\cdot}2]{:}3$	$[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}6$
align=center \|224	Pn{{overline\|3}}m	P 4₂/n {{overline\|3}} 2/m	$\Gamma_cO_h^4$	103a	$\left ( a:a:a\right ) :4_2//\tilde 6 \cdot m$	$4^+{:}2$	$(*4_24{\cdot}2){:}3$	$(2\bar{*}_12_02_0){:}6$
align=center \|225	Fm{{overline\|3}}m	F 4/m {{overline\|3}} 2/m	$\Gamma_c^fO_h^5$	73s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/\tilde 6 \cdot m$	$2^-{:}2$	$[*{\cdot}4{\cdot}4{:}2]{:}3$	$[*{\cdot}2{\cdot}2{:}2{:}2]{:}6$
align=center \|226	Fm{{overline\|3}}c	F 4/m {{overline\|3}} 2/c	$\Gamma_c^fO_h^6$	54h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/\tilde 6 \cdot \tilde c$	$4^{--}$	$[*{\cdot}4{:}4{:}2]{:}3$	$[*{\cdot}2{\cdot}2{:}2{:}2]{:}6$
align=center \|227	Fd{{overline\|3}}m	F 4₁/d {{overline\|3}} 2/m	$\Gamma_c^fO_h^7$	100a	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//\tilde 6 \cdot m$	$2^+{:}2$	$(*4_14{\cdot}2){:}3$	$(2\bar{*}2_02_1){:}6$
align=center \|228	Fd{{overline\|3}}c	F 4₁/d {{overline\|3}} 2/c	$\Gamma_c^fO_h^8$	101a	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//\tilde 6 \cdot \tilde c$	$4^{++}$	$(*4_14{:}2){:}3$	$(2\bar{*}2_02_1){:}6$
align=center \|229	Im{{overline\|3}}m	I 4/m {{overline\|3}} 2/m	$\Gamma_c^vO_h^9$	72s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4/\tilde 6 \cdot m$	$8^\circ{:}2$	$[*{\cdot}4{\cdot}4{:}2]{:}3$	$[2_1{*}{\cdot}2{\cdot}2]{:}6$
align=center \|230	Ia{{overline\|3}}d	I 4₁/a {{overline\|3}} 2/d	$\Gamma_c^vO_h^{10}$	99a	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4_1//\tilde 6 \cdot \tfrac{1}{2}\widetilde{abc}$	$8^\circ/4$	$(*4_14{:}2){:}3$	$(*2_12{:}2{:}2){:}6$

Notes

References

External links

[https://www.iucr.org/ International Union of Crystallography]
[http://neon.mems.cmu.edu/degraef/pointgroups/ Point Groups and Bravais Lattices]
[http://img.chem.ucl.ac.uk/sgp/mainmenu.htm Full list of 230 crystallographic space groups]
[https://www.emis.de/journals/BAG/vol.42/no.2/b42h2con.pdf Conway et al. on fibrifold notation]

Category:Symmetry

Category:Crystallography