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I wanted to recast this question, it was asked at least once for example here: Known bounds for the number of groups of a given order. The reason I ask again is, that in the answers is not much fundamental explanations contained and that I would like to get an opinion on the following (maybe very naive or even wrong) ideas: A group is defined by the group table. In each line of the table every element must occur exactly one time (say the first element contains the multiplication with 1 (from the correct side)). From that the upper bond $\prod^n_{i=0} n!-i$ for groups of order $n+1$ would follow. Then associativity must hold. As far as I understand it, associativity could be formulated as a certain symmetry of a "three-dimensional group table" (for example here: http://www.jstor.org/stable/3613856). Could this symmetry be used to get a tighter approximation of number of possible groups (I expect this is somehow much too simple an approach, since it apparently is quite a tricky problem, however maybe someone can help me to understand what I misunderstand).

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Raphael J.F. Berger
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    Another item of note is that product is an upper bound for the number of group structures on a set of size $n$, which is itself bigger than the number of isomorphism classes of groups of size $n$. – anon Dec 06 '15 at 17:47
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    you could also ask for the number of embeddings of a group into another one. Such as the different finite subgroups of the rotation group in $SO(3)$ or in higher dimensions such as $SO(n)$. – cactus314 Dec 06 '15 at 19:37

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