An essential step in understanding visual processing is to characterize the neuronal receptive fields (RFs) at each stage of the visual pathway. However, RF characterization beyond simple cells in the primary visual cortex (V1) remains a major challenge. Recent application of spike-triggered covariance (STC) analysis has greatly facilitated characterization of complex cell RFs in anesthetized animals. Here we apply STC to RF characterization in awake monkey V1. We found up to nine subunits for each cell, including one or two dominant excitatory subunits as described by the standard model, along with additional excitatory and suppressive subunits with weaker contributions. Compared with the dominant subunits, the nondominant excitatory subunits prefer similar orientations and spatial frequencies but have larger spatial envelopes. They contribute to response invariance to small changes in stimulus orientation, position, and spatial frequency. In contrast, the suppressive subunits are tuned to orientations 45 degrees -90 degrees different from the excitatory subunits, which may underlie cross-orientation suppression. Together, the excitatory and suppressive subunits form a compact description of RFs in awake monkey V1, allowing prediction of the responses to arbitrary visual stimuli.