36865

I think 6 different necklaces can be made with 8 beads.
I found all of them because I can't think of anything else.
If there were 9 beads, still 6 different necklaces can be made.
For 10 beads, no necklaces can be made.
For 11 beads, 10 necklaces can be made.

The more beads I have, the more different necklaces I can make, but if the number of beads is even and you divide it by 2, and each color beads is odd, you cannot make symmetrical necklace from odd number of color beads.