  The Chermak-Delgado lattice of a finite group $G$ is a self-dual sublattice
of the subgroup lattice of $G$. In this paper, we focus on finite groups whose
Chermak-Delgado lattice is a subgroup lattice of an elementary abelian
$p$-group. We prove that such groups are nilpotent of class $2$. We also prove
that, for any elementary abelian $p$-group $E$, there exists a finite group $G$
such that the Chermak-Delgado lattice of $G$ is a subgroup lattice of $E$.
