Of course the "master race" has ethnic/mythological components also. The "overman" (I am not fond of the translation to "superman" -- I think it does not quite capture Nietzsche's sense) would be seen in Hitler and the Nazi elite. In Nietzsche's thought, the "overman" is morally compelled NOT to follow the moral rules which the weak follow. The "overman" is compelled by the "will to power." [Now it occurs to me, Nietzsche's "overman" explains Bill and Hill's approach to rules, morality, power, etc.!?]
There is other common ground between Nietzsche's thought and Nazi "philosophy". Both embraced the idea of eternal recurrence, for example. (The symbol of the swastika).
Without putting them on the same plane, there is also some similarity between Hitler's and Nietzsche's personalities: a type of meglomania --convinced of being right-- combined with not being understood, being rejected, etc. Mann's Dr. Faustus explores a main character in the end consumed by syphillitic madness. Given his other theses in the work, I have wondered whether this was a veiled reference to Hitler? Or to Nietzsche?