No, it isn't. Nazi Germany is known to the world as a "fascist," not a "socialist" regime. Robert Paxton's definition is well-known:

<<[fascism is] a form of political behavior marked by obsessive preoccupation with community decline, humiliation, or victimhood and by compensatory cults of unity, energy, and purity, in which a mass-based party of committed nationalist militants, working in uneasy but effective collaboration with traditional elites, abandons democratic liberties and pursues with redemptive violence and without ethical or legal restraints goals of internal cleansing and external expansion.>>

Kind of wordy to fit on a baseball cap, though.