How to find subgroups of a product group containing the diagonal?

Question

Given a non-abelian group $G,$ I am curious about subgroups of $G\times G$ containing the diagonal $\{(a, a)\mid a\in G\}.$ As a starting point, I thought of looking at the simplest examples $S_3$ and $D_8$. Direct computations quickly become tedious as the product group is a bit large. Also, I think Goursat's lemma could be helpful, but don't know how to use it for an explicit calculation.

Hope someone with more experience in group theory can help me to find them for $S_3$ (and if possible $D_8$ and $Q_8$). At least if you can give me a list of all such groups and a hint to find them, that would be great.

Answer 1

The following observation seems to me useful.

Let $G$ be a group, $D$ be a diagonal in $G\times G$, $K$ be a normal subgroup of $G$. The set $$ H=\{(x,y)\mid x^{-1}y\in K\} \qquad(*) $$ is a subgroup of the group $G\times G$ and $D<H$.

Inversely, if $D<H<G\times G$ then there exists a normal subgroup $K$ in $G$ that $H$ constructs by formula (*).

The first statement is almost obvious. In the second one we have to specify the subgroup $K=\operatorname{gr}\{x\mid (1,x)\in H\}$.