CENTRAL ISLIP, N.Y. — The once popular and swaggering chief of the Suffolk County Police Department, James Burke, was sentenced to 46 months in federal prison on Wednesday for a series of misdeeds that began after a duffel bag belonging to him was stolen from his parked sport utility vehicle.

The duffel bag contained pornography and sex toys. Its disappearance in December 2012 set Mr. Burke off on a furious effort to find the thief, teach him a lesson, recover the bag and make sure the episode stayed quiet. It did not.

Mr. Burke’s efforts at a cover-up set in motion a scandal that reverberated through Long Island politics, making new enemies out of old allies and leading to an ever-widening inquiry by the Federal Bureau of Investigation that is now scrutinizing not only the police, but also prosecutors and the judiciary.

That larger investigation showed no signs of having ended on Wednesday, even as the man at its center was given a stiff sentence by a federal judge who was unimpressed by Mr. Burke’s apology, or the more than 80 letters he had received from Mr. Burke’s friends and family, pleading for leniency.