Inequality involving the number of subgroups of a direct product of groups

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.

• What is $L(G)$? The number of subgroups of $G$? – Eric Wofsey Oct 23 '18 at 20:40
• Yes. It is the number of subgroups of a group. – Alchimist Oct 23 '18 at 20:45