The present work studies symmetry breaking for the subgraph isomorphism problem. This NP-Complete problem decides if a pattern graph is isomorphic to a subgraph of a target graph. The first part of the paper shows how to detect and break all variable and value global symmetries. The second part studies local symmetries, and shows that subgraphs of the initial instance allow to efficiently compute local variable and value symmetries. Experiments show that global symmetries are an efficient technique for subgraph isomorphism, and that limited local symmetries may be useful for difficult instances.