Subgroup need not be isomorphic to any quotient group
- Subgroup lattice and quotient lattice of finite abelian group are isomorphic and in particular, this implies that for a finite abelian group, every subgroup is isomorphic to some quotient group, and vice versa.
Example of symmetric group of degree three
Let be the symmetric group of degree three (order six) acting on the set and be the unique subgroup of order three, i.e., . Then, there is no quotient group of isomorphic to : the only normal subgroups of are the trivial subgroup, , and , and in none of these cases is the quotient of order three.
Example of non-abelian groups of order eight
Example of arbitrary degree symmetric groups
If we take to be the symmetric group of degree , , and to be any subgroup other than the trivial subgroup, the whole subgroup, and a cyclic subgroup of order two, then is not isomorphic to any quotient of .