Torsion subgroups of rational elliptic curves over the compositum of all D4 extensions of the rational numbers

Let E/Q be an elliptic curve and let Q(D4∞) be the compositum of all extensions of Q whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of D4. In this article we first show that Q(D4∞) is in fact the compositum of all D4 extensions of Q and then we prove that the torsion subgroup of E(Q(D4∞)) is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have each possible torsion structure in terms of their j-invariants.