Finite Group Databasebeta

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Subgroup lattices of abelian p-groups

A finite abelian group of prime-power order is a product of cyclic groups, so its shape is fixed by the exponents alone: C_p⁴ × C_p³ × C_p² × C_p is one shape, read at any prime. For the shapes below the entire subgroup lattice is one object valid at every prime — the number of subgroups of each kind is a polynomial in p, and substituting a prime gives the lattice of a concrete group. A shape marked with an exception leaves out the primes it names: there the polynomials say nothing about the group.

So C_p⁴ × C_p³ × C_p² × C_p describes C₁₆ × C₈ × C₄ × C₂ and C₈₁ × C₂₇ × C₉ × C₃ at once.

Order p¹

Order p²

Order p³

Order p⁴

Order p⁵

Order p⁶

Order p⁷

Order p⁸