Torsion subgroups of rational elliptic curves over the compositum of all $D_4$ extensions of the rational numbers

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(D_4^{\mathrm{\infty }})$ be the compositum of all extensions of $\mathbb{Q}$ whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of $D_4$. In this article we first show that $\mathbb{Q}(D_4^{\mathrm{\infty }})$ is in fact the compositum of all $D_4$ extensions of $\mathbb{Q}$ and then we prove that the torsion subgroup of $E(\mathbb{Q}(D_4^{\mathrm{\infty }}))$ 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.