  The Cayley-Dickson algebras R (real numbers), C (complex numbers), H
(quaternions), O (octonions), S (sedenions), and T (trigintaduonions) have
attracted the attention of several mathematicians and physicists because of
their important applications in both pure mathematics and theoretical physics.
This paper deals with the determination of the subalgebra structure of the
algebra T by analyzing the loop T_L of order 64 generated by its 32 basis
elements. The analysis shows that T_L is a non-associative finite invertible
loop (NAFIL) with 373 non-trivial subloops of orders 32, 16, 8, 4, and 2 all of
which are normal. These subloops generate subalgebras of T of dimensions 16, 8,
4, 2, and 1 which form the elements of its structure.
