How to approximate the number of groups? 
How can I find a reasonable approximation for the number of groups upto isomorphism of some higher order $\ n\ $ with relatively large exponents in the prime factorization without excessive calculations ?

This number is usually called $\ gnu(n)\ $.
For example, how many groups upto isomorphism of order $\ 86\ 400\ $ (The number of seconds in a day) approximately exist ?
The supermultiplicativity gives $$gnu (86400)\ge gnu(128)\cdot gnu(675)=39\ 576$$ but his bound is even worse than $gnu(1920)=241\ 004$. Since the prime factorization of $\ 86\ 400\ $  is $\ 2^7\cdot 3^3\cdot 5^2\ $ , I think, that $gnu(1920)$ is still much too low.
Any advices ?
 A: I follow here the book "Enumeration of finite groups".
You can estimate the number $f_b(n)$ of binary systems $S$ with $n$ elements (sets $S$ with map from $S\times S \to S$), and that's also what was already addressed in the comments as
$$ f_b(n) \le n^{n^{2}}.$$ It means you can have one of $n$ entries in each of the $n^2$ fields of the multiplication table. The error you make here is to neglect the isomorphism bit. One can "over-correct" that by simply eliminating all possible permutations of the $n$-elements in the former estimate:
$$ \frac{n^{n^{2}}}{n!} \le f_b(n) \le n^{n^{2}}.$$
In a next step one can include the occurrence of a unit element (and using some approximation) which reduces the number of possibilities for binary sets with unit elements to $f_{b1}$ to
$$ n^{n^{2}-3n+O(n)} \le f_{b1}(n) \le n^{(n-1)^{2}}.$$
In a further step one can try to estimate what happens when we include associativity (that's actually the really hard bit for this approach, obviously associativity is no "simple symmetry property" of a multiplication table), that would yield us the number of semigroups $f_{s}$, some consideration of the multiplications table shows that one can get
$$ n^{{(1-\varepsilon)}n^2} \le f_{s}(n) \le n^{n^2} $$.
Next one can consider the number of latin squares  $f_l(n)$, those are mult. tables where each element occurs once and only once (those are actually the group tables with and without associativity). For that one can get
$$ \frac{(n!)^{2n}}{n^{n^2}}\le f_l(n) \le n^{n^2} $$
Finally for groups one can estimate the minimal number of generators $d(n)$ to $\le \log(n)$  and then using the Lagrange and Cayley theorem one can arrive at the substantially smaller upper bound for the the number of groups with $n$-elements $f(n)$
$$ f(n) \le n^{n\log(n)}$$.
The estimate on $d(n)$ can be improved a little to arrive at lower
$$ f(n) \le n^{n d(n)} \le n^{n\log(n)}$$.
This is about where the current knowledge on the general problem ends and the book starts. Essentially different classes of finite groups are then considered. For example for groups of order $p^m$ with prime $p$ one can get
$$ n^{\frac{2}{27} m^2 (m-6)} \le f(p^m) \le n^{ 
       \frac{2}{27} m^3 + O(m^{\frac{5}{2}})
     } $$
Then the book continues to regard special groups like soluble subgroups of the symmetric groups or Abelian groups and so on.
