well, first we have to have a way of describing the elements of order 4. let's tackle S8 first.
clearly any 4-cycle is of order 4. since 4 = 2*2, the disjoint cycle decomposition of an element of S8 of order 4, can only contain 2-cycles and 4-cycles.
that gives the following possibilities:
(a b c d) a 4-cycle
(a b c d)(e f) a disjoint 4-cycle and 2-cycle
(a b c d)(e f)(g h) a 4-cycle and 2 disjoint 2-cycles
(a b c d)(e f g h) 2 disjoint 4-cycles.
in Sn, (disjoint) cycle types determine the conjugacy classes. that is two permutations with the same disjoint cycle types are conjugate, and vice-versa. can you tackle S12, now?