The core definition of socialism is that the means of production are held in common.

I think Nazism's desire to take care of all it's "true" citizens (barf!) with social welfare is characteristic of Socialism.

However, the owners of businesses colluded with the govt. and accepted it's authority, but retained ownership in exchange for a suite of benefits from the govt. That aspect of Nazism is a move away from popular control of industry. Furthermore, since the people had no control over government decision-making, it was impossible for them to own the means of production even if all industries were nationalized. Finally, the Nazis have a history of brutally repressing popular steps toward common ownership of industry e.g. outlawing unions and killing or abusing all domestic groups historically associated with Marxism.