Number of subgroups of index n of the braid group on 3 strands <a,b|aba=bab>= <c,d|c^2=d^3>, the fundamental group of the complement of a trefoil knot.

The 4 subgroups of the braid group on 3 strands of index 3 are the 3 stabilizers of one strand and the subgroup of words of total weight divisible by 3.