The currently available classification schemes of quadrilaterals do not include all possible types of quadrilaterals, but only an arbitrary subset. Additionally, students often have problems to understand these classifications. Therefore, this study aims at constructing a comprehensive and logically structured systematics of quadrilaterals. Herein the characteristics of its elements should determine its elements. The basis here are six quadrilaterals of first order, which are characterised by either two same angles, two sides of the same length (both either adjacent or opposed to each other), two parallel or orthogonally sides. Combination of these quadrilaterals with one of the other characters results in quadrilaterals of second order; further character combinations lead to quadrilaterals of third or fourth order, the latter including the square. This approach reveals that several, even highly ordered quadrilaterals, are yet unnamed. In addition to the square there is another quadrilateral of fourth order, the complex double-orthogonal three-sides-equal quadrilateral. The applicability of the newly found quadrilaterals still needs to be studied. The here presented systematics of quadrilaterals with its logical basis should probably be easier to teach than the classifications available to date.