Sufficient dimension reduction (SDR) methods are a family of supervised methods for dimensionality reduction that seek to reduce dimensionality while preserving information about a target variable of interest. However, existing SDR methods typically require more observations than the number of dimensions ($N > p$). To overcome this limitation, we propose Class-conditional Factor Analytic Dimensions (CFAD), a model-based dimensionality reduction method for high-dimensional, small-sample data. We show that CFAD substantially outperforms existing SDR methods in the small-sample regime, and can be extended to incorporate prior information such as smoothness in the projection axes. We demonstrate the effectiveness of CFAD with an application to functional magnetic resonance imaging (fMRI) measurements during visual object recognition and working memory tasks, where it outperforms existing SDR and a variety of other dimensionality-reduction methods.