True or False: Let $G, H$ be finite groups. Then any subgroup of $G × H$ is equal to $A × B$ for some subgroups $A
Let $G, H$ be finite groups. Then any subgroup of $G × H$ is equal to $A × B$ for some subgroups $A<G$ and $B<H$.
This question was asked in TIFR GS2018 exam.

The answer is FALSE.

I tried to think about such direct product of finite groups and their subgroups but couldn't find any counterexample for this statement.
I was thinking about a direct product of abelian groups(sometimes cyclic), but then every time I concluded that statement is true, then I tried for non-abelian groups but it was so difficult for me to think about a subgroup which will work as the counterexample for this statement.
Please help. Thanks in advance.

This is false: take for example $G=H$ and let $D=\{(g,g):g \in G \}$. Then $D$ is a subgroup of $G \times G$ and not a direct product of subgroups $A$ and $B$ of $G$. Why not? Since in $A \times B$ you will find elements of the form $(a,1_B)$ with $a \in A -\{1_A\}$. These do not exist in $D$.
