Let $G$ be a finite group such that there exists an epimorphism $\varphi: G \to \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3$. Show that $G$ has no cyclic Sylow $2$-subgroup.
Attempt: If I can prove that $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3$ has no cyclic Sylow $2$-subgroup then the result would follow immediately from the fact that the epimorphic image of a Sylow $p$-subgroup is a Sylow $p$-subgroup. The problem with this approach is (this was an exam problem) that it is a bit tedious (both the proof of the fact about the epimorphic image of Sylow subgroups as well as finding the subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3$). Is there an easier, perhaps more direct approach?
Any help would be appreciated!