Abstract Let G be a group of collineations of a finite thick generalised quadrangle Γ. Suppose that G acts primitively on the point set 𝒫 ${\mathcal{P}}$ of Γ, and transitively on the lines of Γ. We show that the primitive action of G on 𝒫 ${\mathcal{P}}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples Γ for which the action of G on 𝒫 ${\mathcal{P}}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup M and M is non-Abelian.