Abstract. Here we categorize all of the subloops of the unit octonions and in particular describe all of the finite maximal subloops. Lagrange's Theorem for Moufang loops then follows as a corollary.

AMS Subject Classification (1991): 20N05, 17A75

Received November 4, 2005, and in revised form May 29, 2006. (Registered under 7/2006.)