hunting dog vs domestic dog

hunting dog and domestic dog both are nouns.

Word	Noun	Adjective	Verb	Adverb
hunting dog	Yes	No	No	No
domestic dog	Yes	No	No	No

As nouns, domestic dog is a hypernym of hunting dog; that is, domestic dog is a word with a broader meaning than hunting dog:

hunting dog: a dog used in hunting game
domestic dog: a member of the genus Canis (probably descended from the common wolf) that has been domesticated by man since prehistoric times; occurs in many breeds

Other hypernyms of hunting dog include Canis familiaris, dog.

hunting dog (noun)	domestic dog (noun)
a dog used in hunting game	a member of the genus Canis (probably descended from the common wolf) that has been domesticated by man since prehistoric times; occurs in many breeds