Predictive models that satisfy group fairness criteria in aggregate for members of a protected class, but do not guarantee subgroup fairness, could produce biased predictions for individuals at the intersection of two or more protected classes. To address this risk, we propose Conditional Bias Scan (CBS), a flexible auditing framework for detecting intersectional biases in classification models. CBS identifies the subgroup for which there is the most significant bias against the protected class, as compared to the equivalent subgroup in the non-protected class, and can incorporate multiple commonly used fairness definitions for both probabilistic and binarized predictions. We show that this methodology can detect previously unidentified intersectional and contextual biases in the COMPAS pre-trial risk assessment tool and has higher bias detection power compared to similar methods that audit for subgroup fairness.