I don't have my books right in front of me right now so hopefully this answer will be considered acceptable...

You are right on the money that Chicago had a diverse criminal population among the varying key ethnic groups. During Capone's "tenure" it was indeed primarily controlled by the Italian-American dominated Capone group. I want to make that distinction because Al Capone is very similar to other rising gangsters of that time-frame such as Luciano, Costello, Genovese, etc. in that they all toed a very fine line between respect for old-world traditions and adaptation to new world realities.

For example, let's take Jake "Greasy Thumb" Guzik. Guzik was Capone's Secretary of the Treasury so-to-speak. He handed out the bribes (hence the nickname) and monitored the different funds for Capone's criminal empire. Of course Guzik is not an Italian-American but he carried more weight than most Italian-Americans in Capone's outfit.

I wanted to bring all this up because it answers one part of your question. Were there any other major racial gangs? I would yes but only one true major gang that rivaled or threatened Capone's hold at any point in time. Everyone else was more-or-less either rolled into Capone's criminal empire or sub-contractors to a degree. This trend carries through for quite some time across the US, so while the US "Mafia" is Italian-American dominated it is definitely an equal opportunity employer.

So now, who was the other rival group? None other than the North Side Gang which was originally headed up by Dion O'Banion. This was a primarily Irish-American gang but similar to Capone's group it had other ethnic groups represented. The North Side Gang was the primary rival to Capone's South Side focused group during Prohibition and are the victims of the St. Valentine's Day Massacre. O'Banion is eventually murdered and George "Bugs" Moran takes over the gang along with Hymie Weiss. Eventually though the North Side Gang dies out in the late 1930s.