You must be in a clan to make a social group.

If you are not in a clan, it's not difficult to be a part of one. Simply be a member for one month. After one month, you may take the clan test and get sorted into whatever clan the test says you belong in.

Once you are in a clan, you will have the option to create social groups. The button will appear on the Social Groups page, at the very bottom, and will say "Create New Social Group."

You can, however, still join as many social groups as you wish whether you're in a clan or not.