Integration of molecular switching units into complex electronic circuits is considered to be the next step towards the realization of novel logic and memory devices. Here, we report on an ordered 2D network of neighboring ternary switching units represented by triazatruxene (TAT) molecules organized in a honeycomb lattice on a Ag(111) surface. Using low‐temperature scanning tunneling microscopy, we are able to control the bonding configurations of individual TAT molecules within the lattice, realizing up to 12 distinct states per molecule. The switching between those states shows a particularly strong bias dependence ranging from tens of millivolts to volts. Based on a single TAT molecule as a fundamental building block, we then explore the low‐bias switching behavior in units consisting of two and more interacting TAT molecules purposefully defined by the high‐bias switching within the honeycomb lattice. we demonstrate the possibility to realize up to 9 and 19 distinguishable states in a dyad and a tetrad of coupled switching units, respectively. The switching dynamics can be triggered and accessed by single‐point measurements on a single molecule. High experimental control over the desired state, owing to hierarchical switching and pronounced switching directionality, as well as the observed full reversibility, makes this system particularly appealing, paving the way to design complex molecule‐based memory systems.This article is protected by copyright. All rights reserved