Noether's problem for dihedral 2-groups

Let K be any field and G be a finite group. Let G act on the rational function field K(x(g) : g is an element of G) by K-automorphisms defined by g (.) x(h) = x(gh) for any g, h is an element of G. Denote by K(G) the fixed field K(x(g) : g is an element of G)(G). Noether's problem asks whether K(G) is rational (= purely transcendental) over K. We shall prove that K(G) is rational over K if G is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.