It is known that if a finite cyclic group $G$ has order divisible by $p^{3}$ for some prime $p$, then there are infinitely many non-isomorphic indecomposable $\mathbb{Z}G$-modules. One reference for this and related theorems is a 1963 paper of Alfredo Jones, which is available with Open Access on Project Euclid (https://projecteuclid.org/euclid.mmj/1028998908).

It is known that if a finite cyclic group $G$ has order divisible by $p^{3}$ for some prime $p$, then there are infinitely many non-isomorphic indecomposable $\mathbb{Z}G$-modules ( which are torsion free as $\mathbb{Z}$-modules). One reference for this and related theorems is a 1963 paper of Alfredo Jones, which is available with Open Access on Project Euclid (https://projecteuclid.org/euclid.mmj/1028998908).