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What are the Crystallographic Point Groups

An infinite number of point groups exist. For example, if a point group contains rotation operations as its elements, one can create arbitrarily many new ones by changing the rotation angle. However, when restricting the subject to crystals, the rotation angle cannot be set freely. Since crystals must fill space through unit cell translations, the allowed rotation (rotoinversion) angles are restricted to 0°, 60°, 90°, 120°, 180°, 240°, 270°, and 300°. This requirement is equivalent to saying that “the order of rotation or rotoinversion is limited to 1, 2, 3, 4, and 6.” When rotations and rotoinversions¹ of order 1, 2, 3, 4, and 6 are combined in various orientations to form groups, only 32 types can be produced in total. These 32 types are called crystallographic point groups².

Below, we present the notation for the 32 crystallographic point groups, organized by crystal system.

• Sch.: Point group or space group notation following Schoenflies notation
• HM (short): Abbreviated point group or space group notation following Hermann–Mauguin notation
• HM (full): Full point group or space group notation following Hermann–Mauguin notation

Subgroups of Crystallographic Point Groups

Below, we show the subgroup/supergroup relationships for all 32 crystallographic point groups.

Among all crystallographic point groups, the point group \(1\) sits at the bottom, with an order of 1. It is a point group consisting only of the identity matrix. On the other hand, the point group \(m\bar{3}m\), belonging to the cubic crystal system, is the point group with the highest symmetry, with an order of 48. However, the point group \(m\bar{3}m\) is not a supergroup of all point groups. The point groups of the hexagonal crystal system, which have point group \(6/mmm\) as their local apex, cannot reach the cubic point groups by adding or removing any symmetry operations. There is no supergroup/subgroup relationship between point groups belonging to the cubic crystal system and those belonging to the hexagonal crystal system. On the other hand, the trigonal crystal system can be a subgroup of both the cubic and hexagonal systems. For details on this, please refer to the “6.1. Topics in Trigonal/Hexagonal Crystal Systems” page. Below is an explanation of the terms used in the diagram.

• order k of groups: The order, i.e., the number of elements (operations) that the group contains. It can also be thought of as the number of positions to which a general point is mapped.
• normal subgroup: This refers to a normal subgroup. When there are 2 or 3 lines, it means that 2 or 3 subgroups exist in different orientations. For example, from point group \(222\), three subgroups in different orientations (all denoted as point group \(2\)) can be extracted.
• conjugate subgroup: This refers to a conjugate subgroup. As the term “conjugate” implies, there naturally exist multiple subgroups in a conjugate relationship (though they share the same notation). For example, the conjugate subgroups of point group \(32\) are three point groups \(2\) in different orientations.
• Maximal subgroup: This is called a maximal subgroup. It is a subgroup, excluding the group itself, that has the maximal number of elements. Multiple maximal subgroups may exist. For example, the maximal subgroups of point group \(6\) are point group \(3\) and point group \(2\). For a detailed explanation, see “2.4. Subgroups and Cosets“.

Example: Subgroups of \(4/m\)

This is a group whose elements are the 8 symmetry operations of point group \(4/m\). When the symmetry operations are expressed in Seitz notation and the multiplication table is written out, it appears as follows.

From the multiplication table above, let us try to select as many elements as possible while satisfying the group requirements. First, the group formed by selecting the operations (elements) enclosed in red below is point group \(4\).

The group formed by selecting the operations (elements) enclosed in red below is point group \(\bar{4}\).

The group formed by selecting the operations (elements) enclosed in red below is point group \(2/m\).

Note that for \(4/m\), \(4, \bar{4}, 2/m\) are all normal subgroups, so quotient groups can be formed. For \(4/m\), the quotient group modulo \(4\) is \(m\), the quotient group modulo \(\bar{4}\) is \(\bar{1}\), and the quotient group modulo \(2/m\) is \(2\).

Example: Subgroups of Point Group \(2\ 2\ 2\)

The structure of point group \(2\ 2\ 2\) can be expressed by the following multiplication table.

There are the following three ways to select subgroups from the multiplication table above that satisfy the group requirements.

All of these are “point group \(2\)” in terms of notation, but the directions of the 2-fold rotation axes differ. In other words, the maximal subgroups of \(222\) are three \(2\) subgroups in different orientations, and all of these are normal subgroups. In the diagram shown at the beginning of this page, this situation is represented by three lines.

Footnotes

1. Inversion centers and mirrors are types of rotoinversion operations (an inversion center is a 1-fold rotoinversion, and a mirror is a 2-fold rotoinversion), so they are not listed separately here. ↩︎
2. On the other hand, space groups always contain a translation group as a subgroup, and since possessing translational symmetry operations is part of the definition of a crystal, we do not bother saying “crystallographic space groups” but simply call them “space groups.” ↩︎