subgroup vs group

subgroup and group both are nouns.

Word	Noun	Adjective	Verb	Adverb
subgroup	Yes	No	No	No
group	Yes	No	Yes	No

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

subgroup: (mathematics) a subset (that is not empty) of a mathematical group
group: a set that is closed, associative, has an identity element and every element has an inverse

Other hypernyms of subgroup include mathematical group.

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

Other hypernyms of subgroup include grouping.

subgroup (noun)	group (noun)
(mathematics) a subset (that is not empty) of a mathematical group	any number of entities (members) considered as a unit
a distinct and often subordinate group within a group	a set that is closed, associative, has an identity element and every element has an inverse
	(chemistry) two or more atoms bound together as a single unit and forming part of a molecule

subgroup (verb)	group (verb)
	arrange into a group or groups
	form a group or group together