# Estimating Number Of normal subgroups of a p-Group

Does someone have an idea about a possible way to count the number of normal subgroup that a group of order $p ^n$ has ($n \in \mathbb{N}$ )? Is there anyway we can count the maximal subgroups it has (i.e.- the groups of order $p^{n-1}$ ? ) ?

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Counting maximal subgroups is equivalent to counting p-cycles in the elementary abelian group $P/\phi(P)$. – peoplepower Oct 1 '12 at 23:38
It's going to depend a lot on the group, isn't it? Just among groups of order 8, the answer varies from 4 to 16. – Gerry Myerson Oct 1 '12 at 23:56
In case you aren't the same person, there is a thread about this on MO. mathoverflow.net/questions/108581/108584#108584 – Alexander Gruber Oct 2 '12 at 4:09