At the top is the Director of the United Fire Defense Agency, Reiji Osakabe (AKA R0).
Next up are the UFDA's Branches. The main one we see is the Japan Branch, where Rescue Fire resides. Rescue Force used to operate here but it was since moved to the Europe Branch for unknown reasons. The UFDA also operates a Branch in the United States, it has been mentioned but never seen.
Next are the Force's themsleves. Rescue force was headed by Supreme Commander Natsuno Nanbu (and my still be unless she retired).
It appear's Rescue fire has no supreme commander, as one has never been seen. But one may exist.
Next up are the Capitan's, the one's who organize and head up that arm's squad. Eiji Ishiguro and Riku Taiga head up the squads for Force and Fire respectively.
Then we have the squad's, who in themselves have a leader to lead them into disaster's. Hikaru Todoroki and Tatsuya Homura respectively.
Outside the braches we have the R&D (who research new tech) and the Maintenance (Who build and maintain the tech) Departments. They most likely have facilities at every UFDA Base and Campus.
Needless to say, there are likely other "Force's" or Branches not yet seen.