When you say precinct most people think of patrol, in NY the precinct houses more than just patrol.
There are offices for Community Affairs, ,Community Policing, Crime Prevention, Domestic Violence,
Youth Officer, Auxiliary Coordinator and the precinct Detective Squad.
All are under the command of one person, often a Deputy Inspector sometimes a Captain.
Patrol is divided up into watches/platoons. !st watch is what normal people call 3rd shift (midnights) 2nd days and 3rd, the busiest watch evenings each watch has a boss, usually a captain some times a lieutenant. The community affairs and community policing office are usually run by a sergeant with patrol officers and civilians under him, the youth & domestic violence offices are usually bossed by one lieutenant with a sergeant running on each watch. The same goes for the detectives. I forgot the street crimes unit these are plain casual clothed officers who work the streets. They also have an Lt and sergeants for each watch.