In the sea of data generated daily, unlabeled samples greatly outnumber labeled ones. This is due to the fact that, in many application areas, labels are scarce or hard to obtain. In addition, unlabeled samples might belong to new classes that are not available in the label set associated with data. In this context, we propose A3SOM, an abstained explainable semi-supervised neural network that associates a self-organizing map to dense layers in order to classify samples. Abstained classification enables the detection of new classes and class overlaps. The use of a self-organizing map in A3SOM allows integrated visualization and makes the model explainable. Along with describing our approach, this paper shows that the method is competitive with other classifiers and demonstrates the benefits of including abstention rules. A use case is presented on breast cancer subtype classification and discovery to show the relevance of our method in real-world medical problems.