This paper describes the measurements of flow harmonics v(2)-v(6) in 3 mu b(-1) of Xe Xe collisions at root S-NN = 5.44 TeV performed using the ATLAS detector at the Large Hadron Collider (LHC). Measurements of the centrality, multiplicity, and p(T) dependence of the v(n) obtained using two-particle correlations and the scalar product technique are presented. The measurements are also performed using a template-fit procedure, which was developed to remove nonflow correlations in small collision systems. This nonflow removal is shown to have a significant influence on the measured v(n) at high p(T), especially in peripheral events. Comparisons of the measured v(n) with measurements in Pb + Pb collisions and p + Pb collisions at root S-NN = 5.02 TeV are also presented. The v(n) values in Xe + Xe collisions are observed to be larger than those in Pb + Pb collisions for n = 2, 3, and 4 in the most central events. However, with decreasing centrality or increasing harmonic order n, the v(n) values in Xe + Xe collisions become smaller than those in Pb + Pb collisions. The v(n) in Xe + Xe and Pb + Pb collisions are also compared as a function of the mean number of participating nucleons, < N-part >, and the measured charged-particle multiplicity in the detector. The v(3) values in Xe + Xe and Pb + Pb collisions are observed to be similar at the same < N-part > or multiplicity, but the other harmonics are significantly different. The ratios of the measured v(n) in Xe + Xe and Pb + Pb collisions, as a function of centrality, are also compared to theoretical calculations.