  Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(D_4^\infty)$ 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
$\mathbb{Q}(D_4^\infty)$ is in fact the compositum of all $D_4$ extensions of
$\mathbb{Q}$ and then we prove that the torsion subgroup of
$E(\mathbb{Q}(D_4^\infty))$ 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.
