The Nazis were clearly evil, but were they really fascists?

The Nazis were National Socialists. Fascists are not. Beyond that, i understand fascism as the ideology that the only duty is towards the state along with an attachment to the idea of race. Nazis seem to me to have both of these, and fascists clearly do believe in a strong state, which to me sounds rather like one version of socialism, so what's the fundamental difference between Nazism and fascism?