Departments on a Ship

Each US Navy ship is divided into at least seven departments: Engineering, Operations, Deck, Administration, Medical, Communications, and Supply Departments. Each department has its own chain of command based on rank and specific knowledge. (You go to this person to figure out a problem in one system but another person or division to work on another system.) Each department helps each other and the culture on a ship is often one where the right hand washes the left, so to say. When it comes to covering up problems, not so much but favoritism/friendships and favors are a huge part of ship life as it tends to get things done faster.
Engineering Department is where all the engine and power plant would get maintained on your fictional starship as well as all the piping and fuel needs on ship. On Navy ships they also take care of the balance of a ship underway and are the most highly trained to fix any damage created by hitting something or a battle, or even common wear and tear. It’s called Damage Control and their drills are another major part of ship life.
Operations Department runs the ship. They steer the ship and keep an eye on where everything else nearby (other ships, land, reefs, planes, all that) is so they don’t hit anything. Operations is also the department in control of weapons of attack or defense onboard the ship.
Deck Department is the miscellaneous area of the ship. Their main job seems to be making sure the ship always looks its best and to stand look out for what computers may miss.
Administration is the department in charge or all the paperwork involved in updating records, dealing with schooling or college, and all the paperwork involved in weekly ship and system maintenances. You will need to decide how such paperwork is submitted and recorded for inspections on ship so you may want to keep the question at the back of your brain to think about.
Medical handles everything from simple colds to emergency surgery or stitches as well as the basics of specialized medicine like the typical female issues, references for outside surgery, dental issues and exams, vaccinations, and possibly things like pregnancies or life threatening illnesses, depending on how long you plan to be gone.
Communications mainly handles the traffic between ships of the fleet and the wires on the ship as well as many of the antennas aboard.
The job of the Supply Department is to make sure everyone is fed, has all the supplies they need, and has enough sugar, caffeine, and microwavable dinners to be in a mild mood. They run the vending machines, barbershop, post office, the laundry services for the officers, and are well trained for the security of the ship using handheld weapons.