You will need the Full Write Groups permission to set a group's parent, or the Create Groups permission when the group is new. A group admin without Full Write Groups can save the group's other fields only.
Setting and editing hierarchy of a group
Choose the group via Groups in the left-hand menu, open it, and choose Edit group from the Actions dropdown at the top right. When adding a group, there is a tab named Hierarchy to indicate a parent group as well as one or more child groups (a.k.a. subgroups).
Note: By setting a parent group the current group automatically becomes a subgroup.
After setting the parent group, subgroups from this Hierarchy tab for your group can also be added. This can be done by clicking on the add subgroups button.
Editing of the parent and subgroups can be done using the same steps outlined above.
Note: Group admins of a parent group have admin access to all of its subgroups, whatever their campus or organization limit.