Most two-level fractional factorial designs used in practice involve independent or fully confounded effects (so-called regular designs). For example, for 16 runs and 6 factors, the classical resolution IV design with defining relation I = ABCE = BCDF = ADEF has become the de facto gold standard. Recent work has indicated that non-regular orthogonal designs could be preferable in some circumstances. Inhibiting a wider usage of these non-regular designs seems to be a combination of inertia/status quo and perhaps the general resistance and suspicion to designs that are computer generated to achieve ‘X–Y–Z’ optimality. In this paper each of the orthogonal non-isomorphic two-level, 16 run designs with 6, 7, or 8 factors (both regular and non-regular) are shown to have a classical-type construction with a 24 or a replicated 23 starting point. Additional factor columns are defined either using the familiar one-term column generators or generators using weighted sums of effects. Copyright © 2010 John Wiley & Sons, Ltd.