The subgraph isomorphism problem consists of two graphs, G and H and asks if there is some subgraph of G that is isomorphic to H. There are several families that H can be sampled from that preserve the NP-completeness of the problem. For example, if H is a cycle graph with the same number of vertices as G, this is equivalent to finding a Hamiltonian cycle. Similar constructions of H can be done to reduce the maximum clique problem to the subgraph isomorphism problem.
My question is if there are any constructions of G (and perhaps H as well) where the subgraph isomorphism problem on (G, H) is NP-complete.