Classification of groups of prime-fourth order

Statement
Let $$p$$ be a prime number. Then, the groups of order $$p^4$$ can be classified as follows, with slightly different classifications for the cases $$p = 2$$, $$p = 3$$, and $$p \ge 5$$. The case $$p = 2$$ is different even in so far as the number of possible groups is concerned. The cases $$p = 3$$ and $$p \ge 5$$ have minor differences with each other.

The five abelian groups
The nature and classification of the five abelian groups of order $$p^4$$ is the same for both the $$p = 2$$ and odd $$p$$ cases; the abelian groups are classified by the set of unordered integer partitions of the number 4.