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Is it true that a pro-2 group $G$ with $G/\Phi(G)\cong C_2\times C_2$ may have a maximal subgroup which is not open?

Motivation: The Frattini subgroup of a profinite group by definition, is the intersection of all maximal proper open subgroups of the group. So I am asking of existence of maximal subgroups (between all subgroups) which are not open. Some of properties of such maximal subgroups are mentioned below in the comments:

1) They are not normal.

2) They are not closed. They are dense in the group.

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  • $\begingroup$ If $H$ is a closed subgroup then it's the intersection of $HN$ when $N$ ranges open normal subgroups of $G$; in particular if $H$ is maximal and closed then it's open. So you're asking about dense subgroups, right? $\endgroup$
    – YCor
    Commented Jun 9, 2015 at 16:01
  • $\begingroup$ I think you are right. But I thought to find a non-normal maximal subgroup. $\endgroup$
    – Alireza Abdollahi
    Commented Jun 9, 2015 at 16:45
  • $\begingroup$ I was speaking of subgroups, not necessarily normal subgroups. $\endgroup$
    – YCor
    Commented Jun 9, 2015 at 16:52
  • $\begingroup$ What I ask is equivalent to find a non- normal maximal subgroup $\endgroup$
    – Alireza Abdollahi
    Commented Jun 9, 2015 at 18:13
  • $\begingroup$ I was asking for clarification: you're asking about subgroups that are not necessarily closed? $\endgroup$
    – YCor
    Commented Jun 9, 2015 at 20:11

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