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is there any finite p-group G that is subgroup or minimal/maximal subgroup of infinite p-group H? if yes what is the limits? can this happen with different p's? i'm more interested in being maximal subgroup of infinite p-group

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    $\begingroup$ Tarski Monsters have maximal (and minimal) subgroups of order $p$ and are infinite, and exist for all sufficiently large prime $p$. $\endgroup$
    – Geoff Robinson
    Commented Mar 8, 2015 at 11:34
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    $\begingroup$ So a question that springs to mind is whether, for example, a finite $2$-group can be a maximal subgroup of an infinite $2$-group. $\endgroup$
    – Derek Holt
    Commented Mar 8, 2015 at 12:35

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