You will face the problem how to assign new instances to their clusters.
The more common combination is to run cluster analysis to check if any class consists maybe of multiple clusters. Then use this information to train multiple classifiers for such classes (i.e. Class1A, Class1B, Class1C), and in the end strip the cluster information from the output (i.e. Class1A -> Class1).
If you are running cluster analysis first, then split your data, you have a data snooping bias problem. Don't do this. Running clusterig on the train and test set independently will usually not work, as clusters will often be very different.
In some rare cases (e.g. k-means) you can of course assign new instances to the nearest mean, with the means optimized on the training set only. But this is in fact no longer k-means, but NN-classification to a simplified data set (consisting of the means obtained by k-means). This approach is however only possible for the particular case of k-means with squared Euclidean distance.