One study (Zhao et. al., 2010), investigating the sleep architecture of two bat species (one nocturnal, and one mixed), notes the following:
C. sphinx was found to sleep predominantly throughout the day (60% of total sleep quota) during which time it spent significantly longer
time in NREM and REM sleep. Compared to E. spelaea, C. sphinx had
significantly less REM and NREM sleep episodes but their duration was
significantly longer. E. spelaea appears to distribute its time at
wake and in REM and NREM sleep equally throughout the light and dark
Top three rows are for the predominantly nocturnal bat C. sphinx. Note the "Wake" rows for the two species.
Zhao, X., Sun, H., Tang, Z., Flanders, J., Zhang, S., & Ma, Y. (2010). Characterization of the sleep architecture in two species of fruit bat. Behavioural brain research, 208(2), 497-501.