Find the number of subgroups of a finite group
Mathematica, 62 48 bytes

Count[Subsets@First@#,x_/;Union@@#[[x,x]]==x]-1&amp;


Pure function expecting a 1-indexed 2D array. Counts the number of Subsets of the First row of the input array that are  closed under the group operation. Since this will include the empty set, we subtract 1. Note that a nonempty subset of a finite group that is closed under the group operation is in fact a subgroup, (and vice versa, by definition).

Strictly speaking I don't check that the subset x is closed under the group operation, I restrict the multiplication table to the subset x and check that it contains precisely the elements of x. Clearly this implies that x is closed with respect to the group operation. Conversely, any subgroup x will contain 1 and thus x will be a subset of the elements appearing in the restricted multiplication table, and since x is closed under the group operation, it must equal x.