# Example of a group where a proper subgroup is not included in a maximal element

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.

• If a subgroup isn't contained in any larger, proper subgroup, doesn't that make it maximal, by definition? – G Tony Jacobs May 6 '18 at 21:30
• @GTonyJacobs It's contained in larger proper subgroups, but no maximal ones... – Robert Israel May 6 '18 at 21:31
• But that larger propper subgroup is not necessarily maximal. – Bajo Fondo May 6 '18 at 21:31

How about the direct product $G \times H$ of a group $G$ with no maximal subgroup and a cyclic group $H$ of order $2$? It has $G$ as maximal subgroup, but any subgroup containing $H$ is not contained in any maximal subgroup.