In fact almost all FWP's and ALL paid hosts will permit you to make sub directories. If you are using a FTP client, you will find a "MkDir" button on the remote server pane and click on it to give a name for the new (sub-)directory on the root directory and say "Ok".
Creating a sub directory is easy enough, if your ISP will allow it. Not all do. You simply add a folder to your current web directory and use FTP to upload the files to that folder, in the same way as you would upload to your main web.