So I am trying to figure out if this is possible or not:
Given $G$ a group where there exists a maximal subgroup $M$, is it possible for $G$ to have a subgroup $H$ where $H$ is not contained in any maximal subgroup?
The idea of this is that I have seen examples of groups where every subgroup is contained in a maximal subgroup, or groups where there are no maximal subgroups. But never have I seen, (or so I think) of an example where there is a maximal subgroup, but there are subgroups that are not contained in any maximal subgroup.
Any ideas would be largely appreciated.