Upright orientation of a molecular adsorbate is one of the most practical keys for controlling surface functionalities by using self-assembled monolayers. However, lateral interactions between the upright molecules become more complex than planar molecules because they are not confined in a single plane. Here, a scanning tunneling microscope was used to study the intermolecular interactions among self-assembled 1,8,13-tris(mercaptomethyl)-triptycenes (TMMT) on Ag(111) with atomic spatial resolution. TMMT chemisorbs with a three-point contact by the S atoms leading to upright orientation, which has been verified by X-ray photoelectron spectroscopy. The resulting chiral stereoisomers self-assemble into highly ordered structures, suggesting stereochiral recognition because of intermolecular interactions. Theoretical simulations show weak C–H···π intermolecular interactions and a strong S–Ag molecule–substrate interaction, which suggests the former being responsible for the self-assembly formation with the latter being the stabilizing force. This provides fundamental insights into molecular assemblies with intermolecular interactions not confined within a single plane.