Olympiad Geometry · Challenge problem

A cube has $8$ vertices, and any two of them determine a line, giving $28$ lines. Let $\overline{AC}$ be a diagonal of one face. Call a line orthogonal to line $AC$ if its direction is perpendicular to the direction of $AC$; the two lines may cross at a right angle or be skew. How many of the other $27$ lines are orthogonal to line $AC$?