As is the case with all three note chords, a quartal triad has six basic inversions:

The addition of a C flat to the specific form(s) above result in a tetrad for which the most common usage would include a C flat major chord (R-M7-M3-M6) or a first inversion A flat minor chord (m3-M9-P5-R):

The less obvious, but very useful, application for the chord voicing above would be as the first inversion altered G dominant (M3 - (+)9 - (+)5 - m9). One possible resolution to a C major (specifically, a G / C) is shown below:

Here’s hoping that you extrapolate on the core concept presented above.