There are $3{,}684{,}030{,}417$ different semigroups of order $8$. Early on in _"Nine Chapters on the Semigroup Art,"_ by Alan J. Cain it is claimed that > there are $3{,}684{,}030{,}417$ different (non-isomorphic) semigroups with $8$ elements. How was this number reached?

You already found yourself an important reference. You may also read the following articles:

[1] Andreas Distler and Tom Kelsey. The monoids of orders eight, nine & ten. _Ann. Math. Artif. Intell._ , **56** (1):3–21, 2009.

[2] Andreas Distler and Tom Kelsey. The semigroups of order 9 and their automorphism groups, _Semigroup Forum_ **88** (1):93–112, 2014.

[3] S. Satoh, K. Yama, and M. Tokizawa. Semigroups of order 8. _Semigroup Forum_ , **49** (1):7–29, 1994.