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For a group G and 
$m\ge 1$, let 
$G^m$ denote the subgroup generated by the elements 
$g^m$, where g runs through G. The subgroups not of the form 
$G^m$ are the nonpower subgroups of G. We classify the groups with at most nine nonpower subgroups.
No group has exactly one or two nonpower subgroups. We classify groups containing exactly three nonpower subgroups and show that there is a unique finite group with exactly four nonpower subgroups. Finally, we show that given any integer k greater than
$4$
, there are infinitely many groups with exactly k nonpower subgroups.
