rugby vs football game

rugby and football game both are nouns.

Word	Noun	Adjective	Verb	Adverb
rugby	Yes	No	No	No
football game	Yes	No	No	No

As nouns, football game is a hypernym of rugby; that is, football game is a word with a broader meaning than rugby:

rugby: a form of football played with an oval ball
football game: any of various games played with a ball (round or oval) in which two teams try to kick or carry or propel the ball into each other's goal

Other hypernyms of rugby include football.

rugby (noun)	football game (noun)
a form of football played with an oval ball	any of various games played with a ball (round or oval) in which two teams try to kick or carry or propel the ball into each other's goal