Point groups are symmetry operations with a common point that form a group. Point groups have an order number associated with them, with 1 being the least symmetric and then the number increases with increasing symmetry.

On this page, all highest order symmetry operations in each crystallographic point group are demonstrated on objects which have these symmetries. Crystallographic point groups are those 3-dimensional point groups which have translational symmetry, so the only permitted rotation symmetries are 1, 2, 3, 4, and 6-fold rotations. There are 32 crystallographic point groups in total with order numbers from 1 to 48. Point group names are given in Schönflies notation and Hermann-Mauguin notation.

Triclinic

Triclinic point groups contain only identity and inversion.

Only identity:
Schönflies: $C_1$
Hermann-Mauguin: $\mathbf{1}$
Point group order: 1

Inversion:
Schönflies: $C_i$
Hermann-Mauguin: $\mathbf{\bar{1}}$
Point group order: 2

Monoclinic

Monoclinic point groups have one 2-fold rotation axis.

One 2-fold axis of rotation or a mirror plane:
Schönflies: $C_2$
Hermann-Mauguin: $\mathbf{2}$
Point group order: 2

One 2-fold axis of rotoinversion:
Schönflies: $C_{1h}$ or $C_s$
Hermann-Mauguin: $\mathbf{\bar{2}}$ or $\mathbf{m}$
Point group order: 2

One 2-fold axis of rotation and a mirror plane perpendicular to it:
Schönflies: $C_{2h}$
Hermann-Mauguin: $\mathbf{\frac{2}{m}}$
Point group order: 4

Orthorhombic

Orthorhombic point groups have three 2-fold axes of rotation or two mirror planes and one 2-fold axis of rotation.

Three 2-fold axes of rotation:
Schönflies: $D_{2}$ or $V$
Hermann-Mauguin: $\mathbf{222}$
Point group order: 4

Two mirror planes and one 2-fold axis of rotation parallel to both of them:
Schönflies: $C_{2v}$
Hermann-Mauguin: $\mathbf{mm2}$
Point group order: 4

Three 2-fold axes of rotation with mirror planes perpendicular to each of them:
Schönflies: $D_{2h}$ or $V_{h}$
Hermann-Mauguin: $\mathbf{\frac{2}{m}\frac{2}{m}\frac{2}{m}}$ or $\mathbf{mmm}$
Point group order: 8

Trigonal

Trigonal point groups have one 3-fold axis of rotation.

One 3-fold axis of rotation:
Schönflies: $C_{3}$
Hermann-Mauguin: $\mathbf{3}$
Point group order: 3

One 3-fold and one 2-fold axis of rotation:
Schönflies: $D_{3}$
Hermann-Mauguin: $\mathbf{32}$
Point group order: 6

One 3-fold axis of rotation and a mirror plane parallel to it:
Schönflies: $C_{3v}$
Hermann-Mauguin: $\mathbf{3m}$
Point group order: 6

One 3-fold axis of rotoinversion:
Schönflies: $S_{6}$ or $C_{3i}$
Hermann-Mauguin: $\mathbf{\bar{3}}$
Point group order: 6

One 3-fold axis of rotation, inversion, and one 2-fold axis of rotation with a mirror plane perpendicular to it:
Schönflies: $D_{3d}$
Hermann-Mauguin: $\mathbf{\bar{3}\frac{2}{m}}$ or $\mathbf{\bar{3}m}$
Point group order: 12

Tetragonal

Tetragonal point groups have one 4-fold axis of rotation.

One 4-fold axis of rotation:
Schönflies: $C_{4}$
Hermann-Mauguin: $\mathbf{4}$
Point group order: 4

One 4-fold axis of rotoinversion:
Schönflies: $S_{4}$
Hermann-Mauguin: $\mathbf{\bar{4}}$
Point group order: 4

One 4-fold axis of rotation and two 2-fold axes of rotation:
Schönflies: $D_{4}$
Hermann-Mauguin: $\mathbf{422}$
Point group order: 8

One 4-fold axis of rotoinversion, one 2-fold axis of rotation and one mirror plane:
Schönflies: $D_{2d}$ or $V_{d}$
Hermann-Mauguin: $\mathbf{\bar{4}2m}$
Point group order: 8

One 4-fold axis of rotation with two mirror planes parallel to it:
Schönflies: $C_{4v}$
Hermann-Mauguin: $\mathbf{4mm}$
Point group order: 8

One 4-fold and two 2-fold axes of rotation with mirror planes perpendicular to each of them:
Schönflies: $D_{4h}$
Hermann-Mauguin: $\mathbf{\frac{4}{m}\frac{2}{m}\frac{2}{m}}$ or $\mathbf{\frac{4}{m}mm}$
Point group order: 8

One 4-fold axis of rotation with a mirror plane perpendicular to it:
Schönflies: $C_{4h}$
Hermann-Mauguin: $\mathbf{\frac{4}{m}}$
Point group order: 16

Hexagonal

Hexagonal point groups have one 6-fold axis of rotation.

One 6-fold axis of rotation:
Schönflies: $C_{6}$
Hermann-Mauguin: $\mathbf{6}$
Point group order: 6

One 6-fold axis of rotoinversion or one 3-fold axis of rotation with a mirror plane perpendicular to it:
Schönflies: $C_{3h}$
Hermann-Mauguin: $\mathbf{\bar{6}}$
Point group order: 6

One 6-fold and two 2-fold axes of rotation:
Schönflies: $D_{6}$
Hermann-Mauguin: $\mathbf{622}$
Point group order: 12

One 6-fold axis of rotoinversion, one mirror plane, and one 2-fold axis of rotation:
Schönflies: $D_{3h}$
Hermann-Mauguin: $\mathbf{\bar{6}m2}$
Point group order: 12

One 6-fold axis of rotation with two mirror planes parallel to it:
Schönflies: $C_{6v}$
Hermann-Mauguin: $\mathbf{\bar{6}mm}$
Point group order: 12

One 6-fold axis of rotation with a mirror plane perpendicular to it:
Schönflies: $C_{6h}$
Hermann-Mauguin: $\mathbf{\frac{6}{m}}$
Point group order: 12

One 6-fold and two 2-fold axes of rotation with mirror planes perpendicular to each of them:
Schönflies: $D_{6h}$
Hermann-Mauguin: $\mathbf{\frac{6}{m}mm}$ or $\mathbf{\frac{6}{m}\frac{2}{m}\frac{2}{m}}$
Point group order: 24

Cubic

Cubic point groups have four 3-fold axes of rotation.

One 2-fold and one 3-fold axis of rotation:
Schönflies: $T$
Hermann-Mauguin: $\mathbf{23}$
Point group order: 12

One 4-fold axis of rotoinversion, one 3-fold axis of rotation, and one mirror plane:
Schönflies: $T_{d}$
Hermann-Mauguin: $\mathbf{\bar{4}3m}$
Point group order: 24

One 2-fold axis of rotation with a mirror plane perpendicular to it and one 3-fold axis of rotation:
Schönflies: $T_{h}$
Hermann-Mauguin: $\mathbf{\frac{2}{m}\bar{3}}$ or $\mathbf{m\bar{3}}$
Point group order: 24

One 4-fold, one 2-fold, and one 3-fold axis of rotation and inversion:
Schönflies: $O$
Hermann-Mauguin: $\mathbf{\bar{432}}$
Point group order: 24

One 4-fold axis of rotation with a mirror plane perpendicular to it, one 3-fold axis of rotation, inversion, and one 2-fold axis of rotation with a mirror plane perpendicular to it:
Schönflies: $O_{h}$
Hermann-Mauguin: $\mathbf{\frac{4}{m}\bar{3}\frac{2}{m}}$ or $\mathbf{m\bar{3}m}$
Point group order: 48