This is joint work with Sue Sierra and Toby Stafford. One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative surfaces, and this work resolves a significant case of the problem. Namely, let S be the 3-dimensional Sklyanin algebra, and let T be its 3-veronese ring. Then we show how to describe all maximal orders which are contained in T, as certain kinds of blowups of T. The same techniques also apply to other noncommutative surfaces containing a smooth elliptic curve as a divisor.