A3 in S4

This article is about the subgroup $$H$$ in the group $$G$$, where $$G$$ is the symmetric group of degree four, acting on the set $$\{ 1,2,3,4 \}$$, and $$H$$ is the three-element subgroup:

$$\! H =\{, (1,2,3), (1,3,2) \}$$

In other words, $$H$$ is the alternating group on $$\{ 1,2,3 \}$$ viewed naturally as a subgroup of $$G$$.

There are three other conjugate subgroups of $$H$$ in $$G$$ (so a total of four), with each conjugate characterized as the alternating group on some subset of size three. Indexing these by the fixed point, we get:

$$H_1 = \{, (2,3,4), (2,4,3 \}, \qquad H_2 = \{ , (1,3,4), (1,4,3) \}, \qquad H_3 = \{ , (1,2,4), (1,4,2) \}, H_4 = H = \{ , (1,2,3), (1,3,2) \}$$

Complements
The permutable complements to $$H$$ (and also to each of its conjugates) are precisely the 2-Sylow subgroups of $$G$$, which are the D8 in S4s, namely, copies of dihedral group:D8 sitting inside the whole group. One such complement is:

$$\! \{, (1,2,3,4), (1,3)(2,4), (1,4,3,2), (1,2)(3,4), (1,4)(2,3), (1,3), (2,4) \}$$

There are also other lattice complements that are not permutable complements -- for instance, any Z4 in S4 is a lattice complement that is not a permutable complement.

Subgroup properties
The subgroup is a satisfies property::Sylow subgroup for the prime 3. Many properties follow from this fact.