I have to prove that $D_4$ cannot be the internal direct product of two of its proper subgroups.Please suggest.
Since the order of the group is $8$. Internal direct is possible if there exists two normal subgroups $H$ and $K$ of $D_4$ such that $D_4 = H \times K$.
Then, by Lagranges Theorem we can have $|H| = 2$ and $|K| = 4$ or vice a versa. I can see that both $H$ and $K$ are abelian groups. How to proceed further in this ??