The number of divisors of $8!$  is

$\begin{array}{l}8!=1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\\ =2\times 3\times 2^2\times 5\times 2\times 3\times 7\times 2^3\\ =2^7\times 3^2\times 5\times 7\end{array}$

The number of divisors $=(7+1)(2+1)(1+1)(1+1)$

$=8\times 3\times 2\times 2$

$=96$