You can move a tag group containing several tags to another tag group by changing the "Parent tag group".
In the example below "Awareness of Human Rights" and "Barriers" are both tag groups that can be selected as parents to contain the "Human Rights Barriers" tag group (tag group that's currently being edited).
To reassign a parent group, click on tag group under Edit Tags in the Tags editor. In the pane, select a tag group you'd like the selected group to move to.