Sure, there is the 501(c)3 organization structure a non-profit can file as. But, it turns out there are 30 other types of 501 organizations. Surprised and intrigued? Take a look at the list below, as detailed in IRS publication 557.
Once you've made it through the options, check out the Related Resources for more information on each. Nobody said the IRS doesn't give non-profits a few filing choices.
501(c)(1) -- Corporations Organized Under Act of Congress (including Federal Credit Unions)
501(c)(2) -- Title Holding Corporation for Exempt Organization
501(c)(3) -- Religious, Educational, Charitable, Scientific, Literary, Testing for Public Safety, to Foster National or International Amateur Sports Competition, or Prevention of Cruelty to Children or Animals Organizations
501(c)(4) -- Civic Leagues, Social Welfare Organizations, and Local Associations of Employees
501(c)(5) -- Labor, Agricultural, and Horticultural Organizations
501(c)(6) -- Business Leagues, Chambers of Commerce, Real Estate Boards, etc.
501(c)(7) -- Social and Recreational Clubs
501(c)(8) -- Fraternal Beneficiary Societies and Associations