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We present a computational study of the M3QX7 family of two-dimensional compounds, focusing specifically on their flat-band properties. We use a high-throughput search methodology, accelerated by machine learning, to explore the vast chemical space spawned by this family. In this way, we identify numerous stable 2D compounds within the M3QX7 family. We investigate how the chemical composition can be manipulated to modulate the position and dispersion of the flat bands. By employing a tight-binding model we explain the formation of flat bands as a result of a relatively loose connection between triangular M3QX3 clusters via bridges of X atoms. The model provides an understanding of the residual interactions that can impact the band dispersion. The same loose connection between clusters not only leads to strongly localized electronic states and thus flat electronic bands but also leads to localized phonon modes and flat bands in the phonon dispersion.