2019 Cayley Problem 19

If x and y are positive integers with xy = 6, the sum of all of the possible values of \dfrac{2^{x+y}}{2^{x-y}} is

\textbf{(A)}\ 4180\quad \textbf{(B)}\ 4160\quad \textbf{(C)}\ 4164\quad \textbf{(D)}\ 4176\quad \textbf{(E)}\ 4128

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.