Geodesic
The enumeration of subgroups in finite $p$-groups
Matematičeskie zametki, Tome 13 (1973) no. 1, pp. 107-112

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We generalize the familiar principle of enumeration due to Hall and establish a new principle for the enumeration of subgroups of any $p$-group $G$ of order $p^m$, based on the following grouptheoretic relation found by the author: $\sum^m_{\lambda=0}(-1)^\lambda p^{\left(\lambda\atop2\right)}\mathscr E_\lambda(G)=0$, where $\mathscr E_\lambda(G)$ is the number of elementary Abelian subgroups of order $p^\lambda$ in $G$. 
@article{MZM_1973_13_1_a12,
     author = {V. N. Shokuev},
     title = {The enumeration of subgroups in finite $p$-groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {107--112},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_13_1_a12/}
}
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V. N. Shokuev. The enumeration of subgroups in finite $p$-groups. Matematičeskie zametki, Tome 13 (1973) no. 1, pp. 107-112. http://geodesic.mathdoc.fr/item/MZM_1973_13_1_a12/