Why is the generalized quaternion group $Q_n$ not a semi-direct product? Why is the generalized quaternion group $Q_n$ not a semidirect product?

How many elements of order 2 does a generalized quaternion 2-group have?  How many elements of order 2 must each factor in the semi-direct product have?
Note that dicyclic groups (generalized quaternion groups that are not 2-groups) can be semi-direct products.  The dicyclic group of order 24 is a semi-direct product of a group of a quaternion group of order 8 acting on a cyclic group of order 3.
