  We present a polynomial-time quantum algorithm for the Hidden Subgroup
Problem over $\mathbb{D}_{2^n}$. The usual approach to the Hidden Subgroup
Problem relies on harmonic analysis in the domain of the problem, and the best
known algorithm using this approach has time complexity in
$2^{\mathcal{O}(\sqrt{n})}$. By focusing on structure encoded in the codomain
of the problem, we develop a polynomial-time algorithm which uses this
structure to direct a "walk" down the subgroup lattice of $\mathbb{D}_{2^n}$
terminating at the hidden subgroup.
