No subjobs, but jobs (advanced versions of classes that you can unlock at level 30) do have a secondary class requirement. For instance, I'm a BRD, which is the job version of ARC. To unlock it I needed a 30 ARC (with the level 30 class quest done) and a level 15 PGL. Then you can get the quest to unlock BRD. BRD is a support / DD job that deals ranged damage and sings songs.

Your duty list, on the right of your screen, will always show your current quests and the next objective. If there's nothing there it means you missed something somewhere.