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Classification system for crystals
In crystallography, a crystallographic point group is a three dimensional point group whose symmetry operations are compatible with a three dimensional crystallographic lattice. According to the crystallographic restriction it may only contain one , two , three , four and sixfold rotations or rotoinversions. This reduces the number of crystallographic point groups to 32 (from an infinity of general point groups). These 32 groups are one and the same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms. In the classification of crystals, to each space group is associated a crystallographic point group by "forgetting" the translational components of the symmetry operations. That is, by turning screw rotations into rotations, glide reflections into reflections and moving all symmetry elements into the origin. Each crystallographic point group defines the (geometric) crystal class of the crystal. The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including optical properties such as birefringency, or electro optical features such as the Pockels effect.

The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, mineralogists, and physicists. For the correspondence of the two systems below, see crystal system.

Leave out the Bravais lattice type. Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.) Axes of rotation, rotoinversion axes, and mirror planes remain unchanged.