An octagon quadrangle is the graph consisting of an 8-cycle (x1; x2; :::; x8) with two additional chords: the edges fx1; x4g and fx5; x8g. An octagon quadrangle system of order v and index¸ [OQS] is a pair (X;H), where X is afinite set of v vertices and H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of ¸Kv defined on X. An octagon quadrangle system § = (X;H) of order v and index ¸ is said to be upper C4 - perfect if the collection of all of the upper 4- cycles contained in the octagon quadrangles form a ¹-fold 4-cycle system of order v; it is said to be upper strongly perfect, if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a ¹-fold 4-cycle system of order v and also the collection of all of the outside 8-cycles contained in the octagon quadrangles form a %-fold 8-cycle system of order v. In this paper, the authors determine the spectrum for these systems, in the case that it is the largest possible.