Can a fascist government also be socialist?

From what I understand, true communism is stateless and doesn't allow the private ownership of property, so it doesn't go with fascism. However, in countries like Cuba and Venezuala, this is not the case. So is there anything else that makes this kind of socialism incompatible with fascism? From what I understand, fascism as characterized by authoritarian control by the government, extreme nationalism and militarism. They can be pro-capitalism, but that isn't a requirement is it?