The term "socialist" has been thrown around by just about everyone in Europe of every political persuasion even before Karl Marx wrote his book. The tens of thousands of murdered and imprisoned German Marxists and Communists would be the first to raise an objection that the NAZI state was socialist.
The German system was all about control. Traditional socialism, as seen in Marxist or democratic socialist frameworks, emphasizes equality, worker control, and the abolition of class distinctions. Nazi Germany, by contrast, glorified hierarchy, racial supremacy, and the Führer principle, which centralized power in an authoritarian leader. Its social programs such as Strength Through Joy were designed to bolster regime loyalty, not to empower the working class.
Private property and businesses were preserved, and major industrialists—like those at Krupp, IG Farben, and Siemens—thrived under the regime, often profiting from state contracts and forced labor. The government did exert heavy control over the economy through policies like price controls, production quotas, and the suppression of labor unions, but it did not nationalize industries on a large scale as socialist states like the Soviet Union did. The Nazis dismantled independent trade unions and replaced them with the German Labour Front (DAF), which served to control workers rather than empower them.
The Nazis were vehemently anti-communist and anti-Marxist, viewing socialist movements like Bolshevism as existential threats. They persecuted and murdered communists, socialists, and other leftists, notably during the Night of the Long Knives in 1934, when Hitler purged the SA’s "left-leaning" elements (most famously Ernst Röhm), who had advocated for a "second revolution" with socialist-sounding reforms.
The inclusion of "socialist" in the party’s name was largely a propaganda tool to appeal to the working class and siphon support from leftist parties in the Weimar Republic. Early Nazi rhetoric, like the 1920 25-Point Program which
@SaintPanzer quotes, included vague promises of wealth redistribution and nationalization, but these were abandoned or reinterpreted once Hitler gained power.
As
@SaintPanzer also suggests, Goldberg's book, "Liberal Fascism: The Secret History of the American Left, From Mussolini to the Politics of Meaning," is worth reading if only because so many historians disagree with many of his conclusions (some of which I think have merit). I think there is much to be said for his notion that the extreme left uses very similar methods to what is traditionally considered those of the extreme right. That represents the circular nature of radicalism as it moves away from the center that I have noted elsewhere here. I would also agree that his position of viewing socialism and fascism through the lens of traditional American values, offers little to choose between the two systems if socialism is represented in its extreme forms. I, and quite a few historians far more schooled in the subject than I, believe his thesis falls apart when he tries to compare socialism to the Nazi fascists and essentially ignores what are crucial differences like those above.
And yes I know what the Etruscan, later Roman, "Fasces" was and its symbolism of a king's, or later, Caesar's centralized power and its role in the etymology of Fascism.