Solvable group

The basic example of solvable groups are abelian groups. They are trivially solvable since a subnormal series being given by just the group itself and the trivial group. But non-abelian groups may or may not be solvable.
In particular, the quaternion group is a solvable group given by the group extension
The celebrated Feit–Thompson theorem states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.
If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups: