This study presents applying recurrence quantification analysis (RQA) on EEG recordings and their subbands: delta, theta, alpha, beta, and gamma for epileptic seizure detection. RQA is adopted since it does not require assumptions about stationarity, length of signal, and noise. The decomposition of the original EEG into its five constituent subbands helps better identification of the dynamical system of EEG signal. This leads to better classification of the database into three groups: Healthy subjects, epileptic subjects during a seizure-free interval (Interictal) and epileptic subjects during a seizure course (Ictal). The proposed algorithm is applied to an epileptic EEG dataset provided by Dr. R. Andrzejak of the Epilepsy Center, University of Bonn, Bonn, Germany. Combination of RQA-based measures of the original signal and its subbands results in an overall accuracy of 98.67% that indicates high accuracy of the proposed method.