In this paper, twofold triple systems of order v are enumerated for all v ≤ 19. The existence of TS(v , 2)'s (all terms are defined in §2) is completely settled; the condition v -0 or 1 (mod 3) is known to be both necessary and sufficient [4]. On the other hand, enumeration efforts have not enjoyed such success. In fact, the exact number of painvise nonisomorphic TS(v, 2)'s, denoted N(v) , has been determined only for v 5 10. In particular, we have N(3) = N(4) = 1 (trivial), N(6) = 1 [5], N(7) = 4 [13], N(9) = 36 [12, 81, and N(10) = 960 [l, 31. One reason for the unavailability of such enumeration results for higher values of v is the inherent computational complexity of the problem that leads to a combinatorial explosion effect. To curb this combinatorial explosion, extra conditions are often imposed to enumerate interesting classes of designs. One such condition involves specifying automorphisms that the desired designs must possess.