Structure of finite abelian groups

Sylow decomposition. Since $G$ is abelian, $G\cong G_2\times G_3$ with $G_2$ the Sylow $2$-subgroup (order $4$) and $G_3$ the Sylow $3$-subgroup (order $3$).

$G_3$. Unique abelian group of order $3$: $\mathbb{Z}/3\mathbb{Z}$.

$G_2$. Abelian groups of order $4$: $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$.

Classification. The abelian groups of order $12$ are:
- $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\cong\mathbb{Z}/12\mathbb{Z}$;
- $(\mathbb{Z}/2\mathbb{Z})^2\times\mathbb{Z}/3\mathbb{Z}\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$.

Conclusion. Two abelian groups of order $12$ up to isomorphism.