How show that not exist two subgroups $H,K \leqslant S_{4} $ such that $H \times K\approx \operatorname{Dih}_4 $

Question

I know that $S_4$ have 6 abelian subgroups of order 4 and 12 abelian subgroups of order 2, if i choose a subgroup $H$ of order 4 in $S_4$ and a subgroup $K$ of order 2 in $S_4$, when i do the direct product $H \times K$,this will be abelian (theorem). Then not exist two subgroups such that the direct product be isomorph to $\operatorname{Dih}_4$.

But, i've needed see the subgroups of $S_4$ for show that. Exist another way to show the same thing?.

I appreciate your help. thanks.