Yes there are several important subdivisions of Trinitarian theology. If we start with the Nicene Creed as the baseline definition of the Trinity, then there were three early positions concerning Christology and the incarnation: Nestorianism (two persons in Christ), monophysitism/miaphysitism (one joined nature of Christ), and dyophysitism (two natures in the one person of Christ.) Dyophysitism is the position of the great majority of Christians today, but Nestorianism survives into the present in the Church of the East, and miaphysitism in the Oriental Orthodox Churches. A few centuries later, the debate developed into the debate over monothelitism, whether Jesus had one will or two. This may all seem like a debate over Christology, not the Trinity, but with monothelitism we saw that our Christology is dependent on questions of the Trinity: whether the faculties of God such as his will are possessed by the divine nature or by the persons of the Trinity, although at the time of Chalcedon they would not have phrased the debate in that way.
A second aspect in which Trinitarian theology can be distinguished is through the immanent and economic views of God. The immanent Trinity focuses on how God knows himself, while the economic Trinity is how God shows himself to us in his actions. Can we know anything of God as he truly is? If the economic Trinity is entirely arbitrary and the Spirit or the Father could have equally become incarnate then that implies the names or titles of God reflect only his roles and not his true identities. But if the economic reflects the immanent then the roles reveal the immanent to us, with questions of the nature by which the Father is the source of the Son or the Son is submissive to the Father. In this the categories are less well defined and debate continues.
One modern development is called Social Trinitarianism. I don't really understand it properly, so I'll quote Wikipedia to explain how it reinterprets the single divine essence of classical Trinitarianism: "this one essence can be thought of as the loving relationship between Father, Son, and Spirit." Debate continues on how strong a model and explanation this is and how compatible it is with the Nicene Creed.