Socialism and fascism are pretty much the same thing. Both have crony capitalism in them, under true socialism the businesses are mostly state run (but those who run them are treated like crony capitalists) and under social democracies there’s both, and the crony capitalism (basically inner party treatment) still occurs.
I think the more usually used terms are “international socialism” (i.e. Marxism/communism), “national socialism” (i.e. Adolph and company), and fascism. All are “socialist”.
Then there are “social democrats”, who like socialism but can’t admit it to others or to themselves.