Size of maximal subgroups in the direct product of finite groups

Question

Let $\pi(G)$ be the set of all prime divisors of the order of a finite group $G$. Prove that: if $M$ is a maximal subgroup in $D=G \times G$ then $\pi(M)=\pi(G)$.
My attempt:
1) If $G$ is $p$-group then $\pi(M)=p$ for all maximal subgroups in $D$.
Now let $G$ be finite non $p$-group. One type of maximal subgroups in $D$ has the form $M\times G$. For this type of maximal subgroups its clear that they have the same set of prime divisors as $G$. What about other types of maximal subgroups in $D$!!
Any hint how to proceed.
Thank you.