= 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 list
$$
\boxed{V_4,\qquad D_8,\qquad A_4,\qquad S_4,}
$$
with 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.
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