Well... nobody's asked for this, but I did it... and it's not doing anyone any good just sitting on my hard drive. So here it is.
Most of the ranks and the characters they belong to are obvious to anyone who's seen the film: Commander, Ship's Surgeon and Lieutenant belong to the characters of Adams (Leslie Nielsen), Ostrow (Warren Stevens) and Farman (Jack Kelly). The Crewman 1st Class patch is worn by the extras and the few "crewman" speaking parts in the film; the moon-shaped 2nd Class patch is worn by Cookie (Earl Holliman), the ship's cook, and no one else as far as I can see.
The remaining two noncom ranks don't come from the script, but are purely conjectural based on clues. The Bosun (George Wallace) is in charge of all the crewmembers, so I decided he was probably a CPO. (Bosun, or boatswain, isn't a formal rank as such, being a holdover from early merchant marine vessels, so it's being used in FP as a traditional, informal title, much the same way as first officers on Star Trek are referred to as Number One). Chief Quinn (Richard Anderson) is the head engineer, and noncommissioned technical leads in the Navy are usually warrant officers, so that's where that came from.
Enough jawing. Here's the graphic: