Explained with orders:
In S4, x^4 is either identity or a 3-cycle (with order 3).
If a and b differ in order, but not by 3, there is no solution.
Explained with even and odd permutions:
(Do you know what even and odd permutations are?)
In S4, x^4 is always an even permutation.
If a is odd and b is even, then there is no solution.
Yes, you're missing 3 of them.
Did you already have them in your original solution?
As a challenge (when you find them), how should you count how many there are?