Normal Subgroups and Quotient Groups · Superchallenge problem

In $S_4$, let $H=\langle(1\,2\,3\,4),(1\,3)\rangle$, a subgroup of order $8$ containing the $4$-cycle $(1\,2\,3\,4)$ and the transposition $(1\,3)$. How many distinct subgroups appear as conjugates $gHg^{-1}$ for $g\in S_4$?