The correct answer is 18,816 different ways. The general formula for six fleurs-de-lys for all squares greater than 22 is simply this: Six times the square of the number of combinations of n things, taken three at a time, where n represents the number of fleurs-de-lys in the side of the square. Of course where n is even the remainders in rows and columns will be even, and where n is odd the remainders will be odd.
For further solution, see No. 358 in A. in M.