From Star Trek: Theurgy Wiki
Notably, the civilian members, attachés, and diplomats didn't normally wear any uniform indicative of an actual department since they were not part of Starfleet.
Diplomats: As the name states, this group was composed of diplomats on starships and starbases.
Civilians: There were very few civilians aboard that don't actually perform some form of duty, be it as waiters or bartenders serving the officers in one of the lounges or a gardener maintaining the Arboretum.
Specialists: As part of the civilian crew, the Theurgy had people that were specialized in specific fields concerning the ship's missions before the ship had to flee Starfleet. These specialists were always graduated professionals that have earned themselves a reputation in and even outside of the Federation. (Example: Narik Cinsaj was tasked to analyse and retro-engineer the Reaver in the Fighter Assault Bay before she was killed in Episode 03, Part 3).