A theorem of B.H. Neumann states that each subgroup of a group $G$ has finite index in a normal subgroup of $G$ if and only if the commutator subgroup $G'$ of $G$ is finite, i.e., $G$ is finite-by-abelian. As a group lattice version of this theorem for a periodic group $G$, it is proved that each subgroup of $G$ has finite index in a modular subgroup of $G$ if and only if $G$ is an extension of a finite group by a group with modular subgroup lattice.
"Periodic groups with nearly modular subgroup lattice." Illinois J. Math. 47 (1-2) 189 - 205, Spring/Summer 2003. https://doi.org/10.1215/ijm/1258488147