I believe its 6 although I'm unsure. how I thought about it is if I take K_3 and give it three points a, b, c then the only isomorphic graph is trivial. A B and C are vertices in k_3 and - means there is an edge between them the subgraphs I can imagine are
A - B, A - C, B - C, A, B, C, a total of 6 am I correct?
The subgraphs of $K_3$ fall into 8 different isomorphism classes. The nonempty ones aredepicted in the following image.