Subgroups of Semidirect Product of the elementary abelian group of order 8 by $S_3$

Question

What are the subgroups of the semidirect product of the elementary abelian group of order 8 by $S_3$?

This is the group $(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2)\rtimes S_3$ of order 48; $S_3$ acts on the elementary abelian $2$-subgroup by permuting three copies of $\mathbb{Z}_2$.

(i.e. if $\langle a_1\rangle \times \langle a_2\rangle \times \langle a_3\rangle$ is the elementary abelian $2$-group, and $\sigma\in S_3$, the action of $\sigma$ on the elementary abelian $2$-group is $\sigma (a_1^{\epsilon_1} a_2^{\epsilon_2}a_3^{\epsilon_3})\sigma^{-1}=a_{\sigma(1)}^{\epsilon_1} a_{\sigma(2)}^{\epsilon_2}a_{\sigma(3)}^{\epsilon_3}$ and $\epsilon_i\in\{0,1\}$.

[Unfortunately, I don't have GAP.]

Answer 1

The comments specify that the question is specifically for subgroups of order 24. An explicit calculation shows that there are three such subgroups:

1. $ℤ_2^3 ⋊ A_3$
2. A subgroup, isomorphic to $S_4$ generated by $a_1a_2, a_2a_3, \sigma, \tau$, where $\sigma$ (order 3) and $\tau$ (order 2) are the generators of the original $S_3$ complement. It corresponds to the permutation action of $S_4$ on the cosets of $\langle(1,2),(3,4)\rangle$ (intransitive $C_2\times C_2$).
3. A subgroup, also isomorphic to $S_4$, generated by $a_1a_2, a_2a_3, \sigma, a_1\cdot\tau$. It corresponds to the permutation action of $S_4$ on the cosets of $\langle(1,2,3,4)\rangle$ (subgroup $C_4$).

In the classification of transitive groups of degree 3 (your group is a wreath product $C_2\wr S_3$) from http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6560180&fileId=S1461157000000115

they are groups number 6,7 and 8, respectively.