Nazism, fascism and the far right?

Currently I am watching a doentary on Auschwitz with my brother and we are debating about the political nature of the Nazi movement. As a committed socialist/liberal, he has postulated that fascism is equal to Nazism, which I disagree with. Fascism, as I see it, is just a name for the extreme right which should be contrasted with Marxism. My opinion is that the further left you go, the more equality and power is distributed amongst the people and the further right you go, the more a hierarchical system is established. I am a proponent of the Nolan Chart, which I believe simplifies the whole contradictory nature of having both authoritarian and liberal extreme politics. The Nazis were without a doubt extremely authoritarian, but under my logic, the Nazis had quite contradictory politics. They believed in the Volksgemeinschaft, which sought to bring down the cl divisions however they retained an authoritarian regime. Likewise, they encouraged collectivism within the greater community but encouraged rugged individualism. Personally, I am of the opinion that the Nazis had both fascistic and socialistic policies and that you cannot just group them with the far right; therefore, we should just call Nazis autocratic authoritarians rather than fascist, because anyone who identifies themselves as fascists automatically ume the connotations of the Nazi movement and is not conductive to rational discourse nor objectivity. So, am I completely wrong? Should fascism define Nazism or the reverse? Is my brother correct in erting that Nazis are fascists and therefore fascists Nazis? Thanks in advance.