Defining templates of galaxy spectra is useful to quickly characterise new observations and organise databases from surveys. These templates are usually built from a pre-defined classification based on other criteria. Aims. We present an unsupervised classification of 702248 spectra of galaxies and quasars with redshifts smaller than 0.25 that were retrieved from the Sloan Digital Sky Survey (SDSS) database, release 7. The spectra were first corrected for redshift, then wavelet-filtered to reduce the noise, and finally binned to obtain about 1437 wavelengths per spectrum. The unsupervised clustering algorithm Fisher-EM, relying on a discriminative latent mixture model, was applied on these corrected spectra. The full set and several subsets of 100000 and 300000 spectra were analysed. The optimum number of classes given by a penalised likelihood criterion is 86 classes, of which the 37 most populated gather 99% of the sample. These classes are established from a subset of 302214 spectra. Using several cross-validation techniques we find that this classification agrees with the results obtained on the other subsets with an average misclassification error of about 15%. The large number of very small classes tends to increase this error rate. In this paper, we do an initial quick comparison of our classes with literature templates. This is the first time that an automatic, objective and robust unsupervised classification is established on such a large number of galaxy spectra. The mean spectra of the classes can be used as templates for a large majority of galaxies in our Universe.