We discuss the problem of computing the density of sublattices L of \(Z^d\) which have the property that the quotient of \(Z^d\) by L has m invariant factors, for fixed m. We find that these densities follow a Cohen-Lenstra distribution.  Our main tool is a generalization of the subgroup growth zeta function of \(Z^d\) originally introduced by V. Petrogradsky.  This is a joint work with N. Kaplan and S. Koplewitz.