= Small subgroups of tetrahedral symmetry
{title2=$25\text{ subgroups with }|H|\leq6$}
The full <symmetry group> of a <regular tetrahedron> is the <symmetric group> $S_4$ acting on its vertices. Its <subgroups> with at most six elements have eight types up to <conjugation>, with the following representative <subgroups> and counts:
$$
\begin{array}{c|l|r}
\text{abstract group}&\text{representative}&\text{count}\\\hline
C_1&\{e\}&1\\
C_2&\langle(12)\rangle&6\\
C_2&\langle(12)(34)\rangle&3\\
C_3&\langle(123)\rangle&4\\
C_4&\langle(1234)\rangle&3\\
C_2\times C_2&\{e,(12)(34),(13)(24),(14)(23)\}&1\\
C_2\times C_2&\langle(12),(34)\rangle&3\\
S_3&\{\sigma:\sigma(4)=4\}&4
\end{array}
$$
A <transposition> is a <reflection> through the plane containing the two fixed vertices and the midpoint of the exchanged edge. A double <transposition> is a half-turn about the axis joining the midpoints of the two exchanged edges. A three-cycle is a one-third-turn about a vertex-to-opposite-face axis. A four-cycle is a quarter-turn composed with a <reflection> perpendicular to the opposite-edge midpoint axis; it is an <improper orthogonal transformation>.
The unique <Klein four-group> containing only double <transpositions> consists of the three opposite-edge half-turns and the <identity element>. The other three <Klein four-groups> consist of two perpendicular mirror <reflections>, their half-turn product, and the <identity element>. Each $S_3$ is a <stabilizer subgroup> of a vertex, including the three <rotations> and three mirror <reflections> preserving that vertex.
For completeness, a <Klein four-group> containing $(12)$ must contain the commuting <involution> $(34)$ or $(12)(34)$, yielding the displayed <subgroup>; one containing no <transpositions> must contain all three double <transpositions>. An $S_3$ <subgroup> has a unique <cyclic subgroup> of <cardinality> three, whose unique fixed vertex must therefore be preserved by the entire $S_3$. These observations and the <disjoint cycle decomposition> exhaust the types allowed by <groups of order at most six>. The counts sum to $25$.
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