Let $R$ be a commutative von Neumann regular ring. In this paper we study the lattice of subgroups of the general linear group $G=GL_n(R)$ containing the group $D=D_n(R)$ of diagonal matrices. We show that if $R$ satisfies some conditions (see Definition 2.4 in the text), then for every such subgroup $H$ there is a uniquely determined $D$-net $\sigma$ of ideals such that $G(\sigma)\leq H\leq N(\sigma)$, where $N(\sigma)$ is the normalizer of a $D$-net subgroup $G(\sigma)$.