Example of a $\mathbb{Z}$-module with exactly three proper submodules? What is an example of a $\mathbb{Z}$-module which has exactly three proper submodules?

A $\mathbf Z$-module is but an abelian group. For any prime $p$, $\mathbf Z/p^3\mathbf Z$ is an example of an abelian groups with three proper subgroups, which are linearly ordered. These subgroups are
$$0\subsetneq p^2\mathbf Z/p^3\mathbf Z\simeq \mathbf Z/p\mathbf Z\subsetneq p\mathbf Z/p^3\mathbf Z\simeq\mathbf Z/p^2\mathbf Z\subsetneq \mathbf Z/p^3\mathbf Z.$$
