Yes...there are also IRC (Internal Revenue Code) Sections 501(c)(4) through 501(c)(27) organizations that are considered tax-exempt, but not charitable. Examples include trade associations, social clubs and certain advocacy organizations involved in substantial political lobbying activity. The list is as follows:
501(c)(3) - Charitable, educational, religious, literary, testing for public safety, promotion of amateur athletics, prevention of cruelty to children or animals
501(c)(4) - Civic leagues and social welfare organizations
501(c)(5) - Labor, agricultural and horticultural organizations
501(c)(6) - Business leagues
501(c)(7) - Social and recreational clubs
501(c)(8) and 501(c)(10) - Fraternal beneficiary societies and domestic fraternal societies
501(c)(9) and 501(c)(17) - Employees' associations
501(c)(11) - Teachers' retirement fund associations
501(c)(12) - Local benevolent life insurance associations, mutual irrigation and telephone companies and like organizations
501(c)(13) - Cemetery companies
501(c)(14) - Credit unions and other mutual financial organizations
501(c)(15) - Mutual insurance companies
501(c)(16) - Corporations organized to finance crop operations
501(c)(18) - Employee pension fund trusts
501(c)(19) - Veterans' Organizations
501(c)(20) - Group legal services plan organizations
501(c)(21) - Black lung benefit trusts
501(c)(22) - Withdrawal liability payment fund
501(c)(23) - Veterans' organizations created before 1880
501(c)(24) - Surprise! There isn't one.
501(c)(25) - Title-holding companies or Trusts for multiple parents
501(c)(26) - State-sponsored high-risk health coverage organizations
501(c)(27) - State-sponsored workers' compensation reinsurance organizations
For more information, see our article The Other Nonprofits.