Number of groups of given order

Definition
Let $$n$$ be a natural number. The number of groups of order $$n$$ is defined as the number of isomorphism classes of groups whose order is $$n$$.

This is a finite number and is bounded by $$n^{n^2}$$ for obvious reasons. The function is not strictly increasing in $$n$$ and depends heavily on the nature of the prime factorization of $$n$$.

Numbers up till 100
Orders 10 to 36. We omit the prime numbers since there is only one group of each such order.

Orders greater than 36. We omit prime numbers, squares of primes, and numbers of the form $$pq$$ where $$p,q$$ both primes, since these are covered by standard cases.

Small powers of small primes
(For general formulas, see the next section).

Powers of 2: Powers of 3: Powers of 5:

Powers of 7:

Asymptotic facts and conjectures

 * Higman-Sims asymptotic formula on number of groups of prime power order: This states that the number of groups of order $$p^n$$ is about $$p^{(2n^3/27) + O(n^{8/3})}$$.
 * Conjecture that most finite groups are nilpotent
 * Pyber's theorem on logarithmic quotient of number of nilpotent groups to number of groups approaching unity
 * Higman's PORC conjecture states that the number of groups of order $$p^n$$ is a PORC function in $$p$$ for fixed $$n$$.

Supermultiplicativity
If $$n = ab$$ with $$a$$ and $$b$$ relatively prime, the number of groups of order $$n$$ is bounded from below by the product of the number of groups of orders $$a$$ and $$b$$ respectively. This is because we can take direct products for every pair of a group of order $$a$$ and a group of order $$b$$.