 Solution

The Klein four-group $V_4$ is normal in $S_4$, and $S_4/V_4\cong S_3$ through the action on the three partitions into pairs. A subgroup containing $V_4$ is the full preimage of its image in this quotient. Up to conjugacy the subgroups of $S_3$ have orders $1,2,3,6$. Their preimages give the complete listwith respective orders $4,8,12,24$, where $D_8$ denotes the dihedral group of order eight. The order-eight preimage is a Sylow two-subgroup of $S_4$, generated for example by a four-cycle and a reflection reversing it.