I have started reading about subgroup growth and, to my surprise, I haven't found a reference to whether direct products preserve subgroup growth. 

What I have found is the following: Let $G$ be a finitely generated group, let $N$ be a normal subgroup and let $Q=G/N$. Then Proposition 1.3.2 in *Subgroup growth* by A. Lubotzky and D. Segal states
\begin{align}
s_n(G)&\leq s_n(Q)s_n(N)n^{\text{rk}(Q)},\\
s_n(G)&\leq s_n(Q)s_n(N)c^n,\qquad \text{where}\ c=3^{d(Q)/3}.
\end{align}
Of course, these inequalities can be applied to a product $G\times G$ by taking $N=G\times 1$.

This seems to suggest that there should exist a group $G$ of intermediate growth and infinite rank such that $G\times G$ has strictly faster subgroup growth than $G$. Does such an example exist?