The latest version of the free Tag Groups plugin comes with a new feature that makes it easier to assign tags to groups when you have a long list of groups: On the drop-down menu you now find a filter to shrink that list. Just start typing in the search field.
Get the new release through your WordPress admin backend.
A Note For Developers
The WordPress options for Tag Groups tag_group_labels, tag_group_ids and max_tag_group_id are now deprecated and will be retired soon. Currently they are not in use, but their values are still kept up-to-date for backwards compatibility.
If you directly access them, you should start using the replacements term_groups, term_group_positions and term_group_labels. term_groups is an array of all existing term group IDs, term_group_positions is an array of term group ID => position (in the sequence of tag groups) pairs and, similarly, term_group_labels an array of term group ID => label (name) pairs.
The original structure stemmed from the first beginnings of the plugin when there was no clear idea about developing it much further. Over the years, however, it turned out to be unnecessarily confusing and error-prone.