$H_1\times H_2 Let $H_1, H_2, G_1, G_2$ be groups. Clearly if $H_1<G_1$ and $H_2<G_2$, then $H_1\times H_2<G_1\times G_2$. 
I'm wondering if the converse statement is true. I'm quite sure it's not. Can you find a counterexample?

It depends even on the very construction of products and the notion of natural inclusion.
For example is $A\times (B\times C)=(A\times B)\times C$ or merely $A\times (B\times C)\cong(A\times B)\times C$? Is $A\times B<A\times (B\times C)$? If yes, this easily leads to a counterexample.
