The "socialism" of fascism was just wartime capitalism. Nothing more.
Did you even bother to read the info in the link??
Empty of content.
I notice that you are pegging people as "right" or "left" based on who they allied with and who they suppressed. Most people actually look at political policies when deciding if a person or party of "left" or "right". I will (again) help you out:
Euthanasia -- Nazi-supported/Democrats support
Gun control -- Nazi-supported/Democrats support
Government control over economy -- Nazi-supported/Democrats support
Atheists -- Nazi-supported/Democrats support
Big government welfare programs (Strength thru Joy, etc) -- Nazi-supported/Democrats support
Big publics works projects (Autobahn, etc.) -- Nazi-supported/Democrats support
Please be good enough to detail how they Conservative concept of Laissez-Faire was demonstrated by the Hitler regime.
This wartime capitalism of which you speak - would that be President Wilson's War Production Board, cited by many as the inspiration for both Lenin's Soviet central planning and the sundry futuristic and utopian schemes endemic in the post-WWI era?
Socialism comes in many guises - what distinguishes it from other belief systems is its ostensible elevation of the good of society as a whole over all subsidiary or individual goods.
With that characteristic, you have socialism, whatever the 'central committee' chooses to call its own particular Big Lie.
All in my not-so-humble opinion.