I said:
[I'm a little bit scared that maybe I've missed one out and two of the above are isomorphic, but I don't think so.]
I've now checked their abstract distinctness, so know they're OK, since I do remember the number 14. They are mostly distinguished by giving their centers and order spectrum, thus: number elts group center given order 16 16 1+1+2+4+8 8x2 8x2 1+3+4+8 4x4 4x4 1+3+12 4x2^2 4x2^2 1+7+8 2^4 2^4 1+15 D16 2 1+9+2+4 Q16 2 1+1+10+4 2xD8 2^2 1+11+4 2xQ8 2^2 1+3+12 2Q8 4 1+7+8 8 :3 2 2 1+5+6+4 8 :5 2 2^2 1+3+4+8 4 :3 4 2^2 1+3+12 4,4|2,2 2^2 1+7+8 The only candidate for an isomorphism that these invariants leave is between 2xQ8 and 4 :^3 4, which is prohibited by the unique factorisation theorem for direct products, since it's easy to see that the latter of these two isn't a non-trivial direct product. The argument is fairly robust, and in any case agrees with my memory, so these are indeed the groups. However, if anyone finds any errors in the above little table, I'd like to hear about them. JHC