Measurements of the subjet structure of quark and gluon jets in hadronic Z decays are presented. The analysis is based on one million hadronic events recorded by the ALEPH detector. Roughly symmetric three-jet events are selected with a coarse jet-resolution cut-off, y(1). Gluon jets are identified with a purity of 94.6% in those events where evidence of long-lived heavy-flavour hadrons in the other two jets is found. The jets are then analyzed using a smaller cut-off y(0) (< y(1)) so that subjets are resolved. The properties of the jets (subjet multiplicities [N-q], [N-g] and rates R(n)(g(q)) for n = 1,2,3,4) are determined and are found to be in good agreement with the expectations of perturbative QCD as long as the subjet resolution parameter y(0) is sufficiently large to keep non-perturbative effects small. In particular, the ratio [N-g - 1]/[N-q - 1], which to leading order in QCD is given by the ratio of colour factors C-A/C-F = 9/4, is measured to be 1.96 +/- 0.15 for y(0) = 2 . 10(-3), but falls to 1.29 +/- 0.03 for y(0) = 1.6 . 10(-5).