I'll tell you what 'grouping that anyone can conceive of' has killed more people in history. Fearful people. They've existed throughout human time. And that grouping transcends people, nations, ideologies, movements, or religions.
That last statement sounds like an extraordinarily extreme generalisation, if not an outright lie. And to answer your question, they oppressed the church because A. they didn't want competing grand narratives (Catholics and Christians more generally believe in God as the overarching supreme force of the world, and that Jesus, his messenger, will return and save his people, whereas Nazis believed that the 'Aryan Race' was supreme and that the ultimate goal was dominion over the Earth for their own gain) and B. because the Catholic Church was a non-political and powerful entity that existed outside the jurisdiction of the state, which made them a threat to Nazi-influence hegemony.
Besides these two main points, numerous prominent Catholics denounced Nazi ideology and i's 'false doctrines' (this use of quotations is not to paint the claim as ironic, but rather to show that those were the words used) before any real persecution of the Church began in Germany. Heck, church leaders were doing it since the 1920s, and Hitler came to power in 1933.
The most well known anti-Nazi Christian was the Lutheran pastor Dietrich Bonhöffer, and for his dignified resistance against the Nazis, he was hanged.
All of what I've said is factual, and none of it is motivated by a desire to defend Nazism, or deny the worst aspects of the Church.
I strongly disagree with your last statement. The ideological similarities between religion and Nazism are very, very, small. The first part of your response is confusing to me.
I have an extremely strong distaste for Nazism, as anyone with positive morality should. But I fear associating it, an ideology with nothing but disdain for most human life and for all good things in this world, with Christianity, a fundamentally humanist and positive belief system, does far more harm than good.
History teaches us that we should not go out of our way to alienate Christians, or Christianity. However much we may disagree with what the Church often stands for, they are not Nazis and never will be. It would help our movement to be more tolerant of Christianity.