Order of ${\rm Aut}(G)$ given $G$ is abelian group of order $30$

Question

Since $G$ is abelian of order $30$ we have that $$G \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z}$$ $$\text{Aut}(\mathbb{Z}/p\mathbb{Z}) = p-1, \text{ p prime} $$ does that mean $$|\text{Aut}(G)| = 1\cdot2\cdot4 = 8$$ or is it false that $$\text{Aut}(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z}) \cong \text{Aut}(\mathbb{Z}/2\mathbb{Z}) \times \text{Aut}(\mathbb{Z}/3\mathbb{Z}) \times \text{Aut}(\mathbb{Z}/5\mathbb{Z})$$