The current Imperial Dynasty, the Mizushima claims direct descendent from both gods and the Taiyo Family, with who's Clan they maintain close relations. Though largely uninvolved in the civil war, save for the occasional skirmish or act of espionage and/or courtship, the Mizushima maintain a tight control over those who would recognize them and those who would not. They are currently led by the Empress Mizushima Kazue.
The Imperial Families are, in a sense, clanless, though each acts with the authority of a clan in its own right. Few in number, but formidable, they act to serve the Mizushima and the current Emperor or Empress.
We haven't seen any yet.
The title of Shogun is granted by the current Emperor or Empress to the samurai which unifies the entirety of Nihon, ushering in an era of peace, no matter how brief. The last time the title was granted was over two-hundred years ago, to the Daimyo of a clan which was then known as the Shishi and is now known as the Ryoshun.
In times of war, it is the Shogunate's job to lead the forces of Nihon into battle at the command of the Imperial Ruler. Outside of that, they essentially act as the defacto leader, though only the most foolish of Shoguns would fail to heed the current Emperor or Empresses' wishes.