Every 1-completed subgroup is contained in a maximal subgroup

Statement

Every 1-completed subgroup is contained in a maximal subgroup.

In symbols, suppose $G$ is a group and $H$ is a subgroup of $G$ such that there exists $x\in G,x\notin H$ such that $\langle H,x\rangle =G$. Then, $H$ is contained in a maximal subgroup.

Note that the statement is obviously true for finite groups; the interesting case is that where $G$ is an infinite group (so that it is not necessarily true that every proper subgroup is contained in a maximal subgroup).