The processes that trigger active galactic nuclei (AGN) remain poorly understood. While lower luminosity AGN may be triggered by minor disturbances to the host galaxy, stronger disturbances are likely required to trigger luminous AGN. Major wet mergers of galaxies are ideal environments for AGN triggering since they provide large gas supplies and galaxy scale torques. There is however little observational evidence for a strong connection between AGN and major mergers. We analyse the morphological properties of AGN host galaxies as a function of AGN and host galaxy luminosity and compare them to a carefully matched sample of control galaxies. AGN are X-ray selected in the redshift range 0.5 < z < 0.8 and have luminosities 41 ≲ log (LX [erg s−1]) ≲ 44.5. ‘Fake AGN’ are simulated in the control galaxies by adding point sources with the magnitude of the matched AGN. We find that AGN host and control galaxies have comparable asymmetries, Sérsic indices and ellipticities at rest frame ∼950 nm. AGN host galaxies show neither higher average asymmetries nor higher fractions of very disturbed objects. There is no increase in the prevalence of merger signatures with AGN luminosity. At 95 per cent confidence we find that major mergers are responsible for <6 per cent of all AGN in our sample as well as <40 per cent of the highest luminosity AGN (log (LX [erg s−1]) ∼ 43.5). Major mergers therefore either play only a very minor role in the triggering of AGN in the luminosity range studied or time delays are too long for merger features to remain visible.