Let $G$ be the direct product of two groups $G_1$ and $G_2, G:=G_1 \times G_2$. Then we can see that $G_1 \times 1$ is a normal subgroup of G and $G_1 \times 1$ is isomorphic to $G/(1\times G_2)$. In other words, $G_1$ is isomorphic to a (normal) subgroup and a quotient group of $G$.
Conversely, given two groups $G$ and $G_1$ such that $G_1$ is isomorphic to a (normal) subgroup and a quotient group of $G$, could we get another group $G_2$ such that $G \cong G_1 \times G_2$?