Under the assumption that the incidence matrix of a 2-(45, 12, 3) design has a certain block structure, we determine completely the number of nonisomorphic designs involved. We discover 1136 such designs with trzvial automorphism group. In addition we analyze all 2-(45, 12, 3) designs having an automorphism of order 5 or 11. Altogether, the total number of nonisomorphic 2-(45,12,3) designs found in 3752. Many of these designs are self-dual and each of these self-dual designs possess a polarity. Some have polarities with no absolute points, giving rise to strongly regular (45,12,3,3) graphs. In total we discovered 58 pairwise nonisomorphic strongly regular graphs, one of which has a trivial automorphism group. Further, we analyzed completely all the designs for subdesigns with parameters 2-(12,4,3), 2-(9,3,3), and 2-(5,4,3). In the first case, the number of 2-(12,4,3) subdesigns that a design possessed, if non-zero, turned out to be a multiple of 3, whereas 2-(9,3,3) subdesigns were so abundant it was more unusual to find a design without them. Finally, in the case of 2-(5, 4, 3) subdesigns there is a design, unique amongst the ones discovered, that has precisely 9 such subdesigns and these form a partition of the point set of the design. This design has a transitive group of automorphisms of order 360.