Label the vertices of a regular tetrahedron , and include all its Euclidean isometries, both rotations and reflections. The following list describes every subgroup of cardinality at most six, allowing arbitrary relabelling of the vertices. The numbers count actual subgroups, rather than just their group isomorphism types.
- One trivial subgroup contains only the identity element.
- Six reflection subgroups are generated by a transposition, for example . Its mirror plane contains vertices and the midpoint of the opposite edge .
- Three half-turn subgroups are generated by a product of two disjoint transpositions, for example . The rotation axis joins the midpoints of the opposite edges and .
- Four cyclic subgroups consist of the three rotations about an axis joining a vertex to the centre of its opposite face. For example, fixes vertex .
- Three cyclic subgroups are generated by an improper orthogonal transformation: a quarter-turn about an axis through opposite edge midpoints, composed with a reflection in the perpendicular plane through the centre. For , the axis joins the midpoints of edges and . Its square is the half-turn .
- One rotational Klein four-group consists of the identity element and the three half-turns about opposite-edge midpoint axes:
- Three further Klein four-groups arise from a pair of opposite edges. For edges , the subgroup isIt contains two reflections in perpendicular planes and their product, the half-turn about the line where those planes meet. It preserves each of the two opposite edges as a set.
- Four subgroups are the stabilizer subgroups of individual vertices. Each contains three rotations about the vertex-to-opposite-face axis and three reflections in planes containing that axis. For the fixed vertex , this is every permutation of .
Here is why this is exhaustive. Part b leaves only the trivial group, . The disjoint cycle decomposition gives exactly the two listed kinds of involutions, the three-cycles, and the four-cycles, accounting for all the cyclic subgroups. In a Klein four-group, the nonidentity elements commute. If it contains , any other nonidentity element must be a commuting involution, hence or ; either choice yields the displayed opposite-edge subgroup. If it has no transposition, its three nonidentity elements must be all three double transpositions, yielding the rotational subgroup.
Finally, an subgroup has a unique cyclic subgroup of cardinality three. Its two three-cycles fix the same unique vertex. Every element of the subgroup preserves this vertex, since conjugation preserves that unique cyclic subgroup. Thus it is the full stabilizer subgroup of that vertex. The counts in the list are consequently exhaustive, givingThe two kinds of and the two kinds of Klein four-groups have different geometric behavior despite having the same abstract group isomorphism type within each pair. This is the distinction recorded by the small subgroups of tetrahedral symmetry.
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