Unfortunately, this is equivalent to the question of finding the densest lattice sphere packing in general. In particular, for any lattices $L$ and $L'$, there exists a sequence of sublattices $L_n$ of $L$ such that the density of $L_n$ approaches the density of $L'$ as $n \to \infty$.

To see this, simply note that if $L' \subset \mathbb{Q}^n$ is a rational lattice and $L = \mathbb{Z}^n$ is the integer lattice, then by clearing denominators, we find a single fixed sublattice of $L$ that is a scaling of $L'$. By applying a rational linear transformation, we see that the same holds for any two rational lattices (with full rank) $L, L' \subset \mathbb{Q}^n$. The full result for (full-rank) lattices $L,L' \subset \mathbb{Q}^n$ follows by taking appropriate sequences of rational lattices that converge to $L$ and $L'$.