A subgroup is a subset of group elements of a group that satisfies the four group requirements. It must therefore contain the identity element. " is a subgroup of " is written , or sometimes (e.g., Scott 1987, p. 16).

The order of any subgroup of a group of order must be a divisor of .

A subgroup of a group that does not include the entire group itself is known as a proper subgroup, denoted or .