GroupSpaces has ability to manage multiple groups, mailing list, member management, wiki, file management and event management for groups.
You can use a free open source tool like http://dotproject.net
I really like so much of TeamBox, with some adapt you can get all this things. And the code is available on GitHub.
Collaborative Applications Spaces Social Applications Productivity and Interfaces Organization and Management Search, Navigation and Retrieval Customize and Extend Technology Platform
I know it's not open source, but It seems to fit your requirements. Features listed, are -
Member Management Shared Admin Access Payment Processing Subgroups Group Communication Calendar and Event RSVP Photo and File Sharing Reviews and Classifieds Public Group Page Private and Secure
Few more options are listed there at Wikipedia.
Have a look at BuddyPress
Social networking in a box. Build a social network for your company, school, sports team or niche community all based on the power and flexibility of WordPress.
BuddyPress is completely free and open source. Unlike hosted services, BuddyPress allows you to stay in control of your site and create a totally customized, unique experience
It looks like GroupServer might do what I want, it is a mailing list manager with online administration of members and lists.