The film gives the impression that the Gangster Squad were responsible for Mickey Cohen's arrest. In reality, Cohen was arrested in 1950 for the more mundane crime of tax evasion. He was also not sent to Alcatraz until 1961, over a decade after the film's time frame.
The movie suggests that Mickey Cohen had Jack Dragna murdered in his own house, but in real life Jack Dragna died of an heart attack years later. The movie is set in the late 40's/early 50's while Dragna died of the heart attack in '56.
When the two civilian men are playing Chinese Chess in Chinatown, the pieces clearly do not represent an actual game because some of the pieces are in positions that are not possible in a regular game.