Is there another name for Goursat's Lemma on subgroups of a direct product of groups?

Goursat's Lemma is in several group theory textbooks, but not always by name.

It appears in Marshal Hall's classic The Theory of Groups. In the AMS Chelsea Publications edition, it is Theorem 5.5.1 on pages 63-64, and it occurs in the index under "subdirect product". It appears in W.R. Scott's classic Group Theory in Section 4.3, "Subdirect Products", as the statement labeled 4.3.1 (pp 71, Prentice Hall, 1964 printing). It shows up in Hungerford's Algebra, but in the section on Rings, discussing subdirect irreducibility, and there is usually a robust discussion of the concept in books on Universal Algebra when discussing subdirect representations; e.g., Gratzer's Universal Algebra.

Goursat's Lemma is not a special case of a more general theorem describing subgroups of a direct product: it is the general theorem that describes subgroups of a direct product. It may not look like it on first sight, but it really is.

To be explicit, here is Goursat's Lemma:

Goursat's Lemma. Let $G$ and $H$ be groups, and let $K$ be a subdirect product of $G$ and $H$; that is, $K\leq G\times H$, and $\pi_G(K)=G$, $\pi_H(K)=H$, where $\pi_G$ and $\pi_K$ are the projections onto the first and second factor, respectively from $G\times H$. Let $N_2=K\cap\mathrm{ker}(\pi_G)$ and $N_1=K\cap\mathrm{ker}(\pi_H)$. Then $N_2$ can be identified with a normal subgroup $N_G$ of $G$, $N_1$ can be identified with a normal subgroup $N_H$ of $H$, and the image of $K$ in $G/N_G\times H/N_H$ is the graph of an isomorphism $G/N_G \cong H/N_H$.

Another way to think about Goursat's Lemma is that we start with a quotient $G/N$ of $G$, and a quotient $H/M$ of $H$. If $\varphi\colon G/N\to H/M$ is an isomorphism, then $\varphi$ induces a subgroup of $G\times H$, by $$ K_{\varphi} = \{ (g,h)\in G\times H\mid \varphi(gN) = hM\}.$$ It is not hard to verify that $K_{\varphi}$ is a subdirect product of $G\times H$, and Goursat's Lemma is the statement that every subdirect product of $G\times H$ arises in this way:

Goursat's Lemma (restatement). Let $G$ and $H$ be groups, let $N\triangleleft G$, $M\triangleleft H$, and let $\varphi\colon G/N\to H/M$ be an isomorphism. Then $\varphi$ gives rise to a subgroup $$ K_{\varphi} = \{ (g,h)\in G\times H\mid \varphi(gN) = hM\}$$ with $\pi_G(K_{\varphi}) = G$ and $\pi_H(K_{\varphi}) = H$. Moreover, every subdirect product of $G\times H$ (every $K\leq G\times H$ with $\pi_G(K)=G$ and $\pi_H(K)=H$) arises in this way.

Now let $K$ be an arbitrary subgroup of $G\times H$, not necessarily a subdirect product. What can we say about $K$? Well, we can apply Goursat's Lemma, but not to $G\times H$, but rather to $\pi_G(K)\times\pi_H(K)$. That is, any subgroup of $G\times H$ is a subdirect product of a subgroup of $G\times H$ that is of the form $G_1\times H_1$, with $G_1\leq G$ and $H_1\leq H$. And so we can apply Goursat's Lemma to $K\leq G_1\times H_1$.

So Goursat's Lemma yields the following:

Goursat's Lemma for arbitrary sugroups of a direct product. Given groups $G$ and $H$, if $G_1\leq G$, $H_1\leq H$, $N\triangleleft G_1$, $M\triangleleft H_1$, and $\varphi\colon G_1/N \to H_1/M$ is an isomorphism, then $\varphi$ gives rise to a subgroup of $G\times H$, "the graph of $\varphi$", by $$ K_{\varphi} = \{ (g,h)\in G\times H\mid g\in G_1, h\in H_1, \varphi(gN)=hM\}.$$ Moreover, every subgroup of $G\times H$ arises in this way.