If $G$ is any group, then by $\text{Sub}(G)$ we denote the collection of all subgroups, ordered by $\subseteq$. If $(P,\leq)$ is a [partially ordered set](https://en.wikipedia.org/wiki/Partially_ordered_set) we let $\text{Max}(P)$ and the set of maximal elements. 


Let $\frak{S}$ be the group of all bijections $f:\omega\to\omega$ together with composition. 


Is there a member of $\text{Sub}(\frak{S}) \setminus \{\frak{S}\}$ that is not contained in some member of $\text{Max}\big(\text{Sub}(\frak{S})\setminus \{\frak{S}\}\big)$?



$\text{Max}\big(\text{Sub}(\text{Sym}(\omega))\setminus \{\text{Sym}(\omega)\}\big)$
As was written by Andreas Thom in his comment, Google gives an extensive literature on this subject. For example, look at the papers
<https://link.springer.com/chapter/10.1007/978-94-011-2080-7_18>
or <https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms/s2-42.1.85>
of Macpherson and his coauthors.


The question posed by Dominic van der Zypen was considered by [Baumgartner, Shelah and Thomas](https://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093634559) who constructed a consistent example of a proper subgroup of $Sym(\omega)$ that is not contained in a proper maximal subgroup of $Sym(\omega)$.


