Socialism was classically defined as total state control of the means of production. The more modern, 20th Century twist which became fascism was state via management of corporations, which divvied up the economy. As was famously stated, "Why own the cow when you can control the man who owns the cow." Mussolini called it "Corporatism" and scary enough many very prominent Americans loved it! In fact, Time Magazine named Hitler their Man of the Year for 1938, not because of his antisemitism but because of the "German miracle" brought about by their economic turnaround.
In every aspect of the German economy it was centrally controlled directly or indirectly. Your insistence in flavor as definition of socialism in consistent with about 70-years of whitewashing the socialist track record. Most socialists hate Nazism, but that doesn't make Nazis non-socialists.
"We are socialists, we are enemies of today's capitalistic economic system for the exploitation of the economically weak, with its unfair salaries, with its unseemly evaluation of a human being according to wealth and property instead of responsibility and performance, and we are all determined to destroy this system under all conditions."
Speech of May 1, 1927 as quoted in John Toland (1976),
Adolf Hitler: The Definitive Biography, p. 224
Many socialists fight the truth about Nazism. Historically, socialists have been so ashamed of Nazism they want it banished from the history books altogether. Interesting that while they hate Nazism they do their best to embrace Leninism and Maoism, mostly for the sake that those two radically ideologies attempted to spread their system of equality to other countries, international socialism. And since they cannot banish Nazism from the history books they attempt to re-brand it as 'right wing' and capitalism at its best. There is a similar, though much smaller mood about Mao and his 75 million murdered during his "Great Leaps Forward" during the Cultural Revolution. 99.9% of socialists oppose the whole 'murder 200 million of our own' during the 20th Century. It is a sad but undeniable part of the movement.
Good discussion, interesting, not the first time I've seen the refutation of the most obvious: Nazism was a rabid form of socialism, and I'll add, that most socialists hate.