Let $G_1, G_2$ and $H$ be finite groups such that $|G_1|=|G_2|$. Assume that $|L(G_1)|<|L(G_2)|$. Is there a way to show that $|L(G_1\times H)|<|L(G_2\times H)|$?
I am especially interested if this would hold if all 3 groups ($G_1$, $G_2$ and $H$) are finite abelian $p$-groups. I know that the number of subgroups of a direct product of groups may be found via Goursat's Lemma. However, counting the number of subgroups in such a case seems difficult (even if we restrict the study to abelian $p$-groups). Any suggestion is appreciated.