Determining histological subtypes, such as invasive ductal and invasive lobular carcinomas (IDCs and ILCs) and immunohistochemical markers, such as estrogen response (ER), progesterone response (PR), and the HER2 protein status is important in planning breast cancer treatment. MRI-based radiomic analysis is emerging as a non-invasive substitute for biopsy to determine these signatures. We explore the effectiveness of radiomics-based and CNN (convolutional neural network)-based classification models to this end. T1-weighted dynamic contrast-enhanced, contrast-subtracted T1, and T2-weighted MR images of 429 breast cancer tumors from 323 patients are used. Various combinations of input data and classification schemes are applied for ER+ vs. ER−, PR+ vs. PR−, HER2+ vs. HER2−, and IDC vs. ILC classification tasks. The best results were obtained for the ER+ vs. ER− and IDC vs. ILC classification tasks, with their respective AUCs reaching 0.78 and 0.73 on test data. The results with multi-contrast input data were generally better than the mono-contrast alone. The radiomics and CNN-based approaches generally exhibited comparable results. ER and IDC/ILC classification results were promising. PR and HER2 classifications need further investigation through a larger dataset. Better results by using multi-contrast data might indicate that multi-parametric quantitative MRI could be used to achieve more reliable classifiers.