Example

Let G be the symmetric group S₃. The subgroup H = {e, (1,2,3), (1,3,2)} of order 3 is a normal subgroup.

It has index 2. In fact, more generally, whenever H is a subgroup of G of index 2, it is a normal subgroup. For then, for g G, either g H and so gH = H = Hg or or not, in which case gH = G\H = Hg.