No, the subject of a sentence can belong to quite a few categories:
Ed likes drinking beer. (noun phrase) Under the mat isn't a good place to leave the key. (preposition phrase) That they refused didn't surprise us. (declarative content clause) What caused the delay remains a mystery. (interrogative content clause) To turn back now would be a mistake. (to-infinitival clause) Bringing your dad along was a good idea. (gerund-participial clause)