society vs social group

society and social group both are nouns.

Word	Noun	Adjective	Verb	Adverb
society	Yes	No	No	No
social group	Yes	No	No	No

As nouns, social group is a hypernym of society; that is, social group is a word with a broader meaning than society:

society: an extended social group having a distinctive cultural and economic organization
social group: people sharing some social relation

society (noun)	social group (noun)
an extended social group having a distinctive cultural and economic organization	people sharing some social relation
a formal association of people with similar interests
the fashionable elite
the state of being with someone