What is sub graph isomorphism?
No, that is not the correct statement.

Every graph contains the empty graph as a subgraph, the problem as you define it would return TRUE for all pairs of graphs, which isn't very interesting.

Finding the largest common subgraph of two graphs would be an interesting problem, and it is at least as hard as the actual subgraph isomorphism problem (since $G$ contains a subgraph isomorphic to $H$ if and only if the largest common subgraph is isomorphic to $H$.)