For a finite abelian group $(G,+, 0)$ the Harborth constant $\mathsf{g}(G)$ is the smallest integer $k$ such that each squarefree sequence over $G$ of length $k$, equivalently each subset of $G$ of cardinality at least $k$, has a subsequence of length $\exp(G)$ whose sum is $0$. In this paper, it is established that $\mathsf{g}(G)= 3n + 3$ for prime $n \neq 3$ and $\mathsf{g}(C_3 \oplus C_9)= 13$.