gr.group theory - Subgroups of direct product of groups

I am interested in the following question on products of finite groups. Let $Gamma$ be a subgroup of $U_1times U_2$ such that the compositions with the canonical projections $Gamma subset U_1times U_2 rightarrow U_1$ and $Gamma subset U_1times U_2 rightarrow U_2$ are both surjective.

Does it follow that there is a group $G$ such that $Gamma$ is isomorphic to the fiber product $U_1 times_G U_2$? This means that there are surjections $pi_1:U_1rightarrow G$ and $pi_2:U_2rightarrow G$ such that $Gamma$ is the set of pairs $(u_1,u_2)$ with $pi_1(u_1)=pi_2(u_2)$.

Goursat's Lemma mentioned in this question proves the statement in the case $Gamma$ is a normal subgroup of $U_1times U_2$.

If the statement is not true without the normality assumption, then what would be a general characterization of these subgroups $Gamma$?