  We give an asymptotic formula for the number of sublattices $\Lambda
\subseteq \mathbb{Z}^d$ of index at most $X$ for which $\mathbb{Z}^d/\Lambda$
has rank at most $m$, answering a question of Nguyen and Shparlinski. We
compare this result to recent work of Stanley and Wang on Smith Normal Forms of
random integral matrices and discuss connections to the Cohen-Lenstra
heuristics. Our arguments are based on Petrogradsky's formulas for the cotype
zeta function of $\mathbb{Z}^d$, a multivariable generalization of the subgroup
growth zeta function of $\mathbb{Z}^d$.
