  In this paper we introduce a particular lattice of subgroups called a
"cyclic-diamond" and show that every finite non-cyclic group contains a
cyclic-diamond as a sublattice of its lattice of subgroups. Turning to the
infinite case, we show that an infinite abelian group does not contain a
cyclic-diamond in its subgroup lattice if and only if all of its finitely
generated subgroups are cyclic or isomorphic to $\mathbb{Z} \times
\mathbb{Z}_{2^N}$ for some $N$.
