Almost all Latin squares have trivial autoparatopy group

The main result is: almost all Latin squares of order $n$ have trivial autoparatopy group as $n\to\infty$. As a consequence we obtain the asymptotic number of main classes of Latin squares of order $n$:
$$ \frac{L_n}{6n!^3}(1+o(1)), $$
where $L_n$ – number of Latin squares of order $n$.