anaesthetic agent vs drug

anaesthetic agent and drug both are nouns.

Word	Noun	Adjective	Verb	Adverb
anaesthetic agent	Yes	No	No	No
drug	Yes	No	Yes	No

As nouns, drug is a hypernym of anaesthetic agent; that is, drug is a word with a broader meaning than anaesthetic agent:

anaesthetic agent (noun)	drug (noun)
a drug that causes temporary loss of bodily sensations	a substance that is used as a medicine or narcotic

anaesthetic agent (verb)	drug (verb)
	administer a drug to
	use recreational drugs