[1] With the exception of the group with three elements, each of these has one (or more) of the above two-element semigroups as subsemigroups.
[2] For example, the set {−1, 0, 1} under multiplication is a semigroup of order 3, and contains both {0, 1} and {−1, 1} as subsemigroups.
Algorithms and computer programs have been developed for determining nonisomorphic finite semigroups of a given order.
These have been applied to determine the nonisomorphic semigroups of small order.
OEIS: A001423 lists the number of non-equivalent semigroups, and OEIS: A023814 the number of associative binary operations, out of a total of nn2, determining a semigroup.