The Sylow Theorems · Hard problem

$D_{44}$, the symmetries of a regular $44$-gon, has order $88=2^3\cdot11$. A Sylow $2$-subgroup has order $8$, built from the rotations through multiples of $4$ steps (there are $44/4=11$ such rotations) together with the matching reflections. How many Sylow $2$-subgroups does $D_{44}$ have?