We have a concrete floor. It was built as an auto shop for my dad and we remodeled it into an apartment. If your floor is not chipped or have cracks you could put down press on tiles. This is what I have. I want to change it because the floor is cold in winter but since my DH is in a wheelchair I can't use carpet. It is too hard for him and I don't want it because I have grandkids who like to paint and I don't want the mess.
But you might consider the carpet. No you don't have to lay a subflooring unless you want to level your floor. I and my DH come from families that are in construction and we have been debating this question for over a year now.
I want to lay hardwood flooring which means I would need a subfloor to even out our floor. But it is too expensive for now.
The only concern you might watch for is condensation coming up through the concrete.