Multidrug treatments are increasingly important in medicine and for probing biological systems. Although many studies have focused on interactions between specific drugs, little is known about the system properties of a full drug interaction network. Like their genetic counterparts, two drugs may have no interaction, or they may interact synergistically or antagonistically to increase or suppress their individual effects. Here we use a sensitive bioluminescence technique to provide quantitative measurements of pairwise interactions among 21 antibiotics that affect growth rate in Escherichia coli. We find that the drug interaction network possesses a special property: it can be separated into classes of drugs such that any two classes interact either purely synergistically or purely antagonistically. These classes correspond directly to the cellular functions affected by the drugs. This network approach provides a new conceptual framework for understanding the functional mechanisms of drugs and their cellular targets and can be applied in systems intractable to mutant screening, biochemistry or microscopy.