# Relation between lattice theorem in groups and in rings

I was studying my abstract algebra notes, and couldn't help but notice a striking similarity between the following two statements:

1. Let $G$ be a group, and $H\triangleleft G$. The canonical projection $$\pi:G\to G/H$$ induces a bijection between subgroups of $G$ that contain $H$, and subgroups of $G/H$. As a corollary, $G/H$ is simple iff $H$ is maximal among proper normal subgroups of $G$.

2. Let $R$ be a ring, and $\mathfrak m$ an ideal. The canonical projection $$\pi:R\to R/\mathfrak m$$ induces a bijection between ideals in $R$ that contain $\mathfrak m$, and ideals of $R/\mathfrak m$. As a corollary, $R/\mathfrak m$ is a field iff $m$ is a maximal ideal.

There has to be a relationship between the two, right? I might even go so far as to say that $1. \Leftrightarrow 2.$ (and obviously $\text{true}\Leftrightarrow\text{true}$ always holds but, well, you know what I mean). Can someone shed some light?