In group theory, given a group *G* under a binary operation *, a subset *H* of *G* is called a **subgroup** of *G* if *H* also forms a group under the operation *. More precisely, *H* is a subgroup of *G* if the restriction of * to *H x H* is a group operation on *H*. This is usually represented notationally by *H* ≤ *G*, read as "*H* is a subgroup of *G*".

A **proper subgroup** of a group *G* is a subgroup *H* which is a proper subset of *G* (i.e. *H* ≠ *G*). The **trivial subgroup** of any group is the subgroup {*e*} consisting of just the identity element. If *H* is a subgroup of *G*, then *G* is sometimes called an *overgroup* of *H*.

The same definitions apply more generally when *G* is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group *G* is sometimes denoted by the ordered pair (*G*,*), usually to emphasize the operation * when *G* carries multiple algebraic or other structures.

This article will write *ab* for *a***b*, as is usual.

Read more about Subgroup: Basic Properties of Subgroups, Cosets and Lagrange's Theorem, Example: Subgroups of Z_{8}, Example: Subgroups of S_{4}