No, in a socialist system property is privately owned, and can be bought and sold, but regardless of who owns it, it must be managed for the purposes of the state.
In a communist system it is communally owned, cannot be bought or sold, and is managed by the state.
Even Lenin had the NEP, which was an acknowledgment that market-based economies with some private property were better for the economy, especially in a pinch.
Hitler, by the same token, understood that if he wanted to build up his war machine quickly, needed to use the industrialists for his own ends.
But once they were no longer needed, the industrialists would go the same way as the Jews, as they were not a part of Hitler’s ideal New World Order.
That’s a “popular” definition of modern socialism but that’s not the way Marx defined it. In Marx’s socialism, property was “cooperatively owned.”