Cadmus was founded by Lex Luthor. He used the facility to seek the means to counter Superman, and named it for the mythical Cadmus. The facility was explicitly designed for secrecy; to the public eye, it was a two-story building, but an express elevator led to 52 sublevels.
In order to allow Project Kr to escape, Dubbilex started a fire on ground level. It got the attention of Aqualad, Robin and Kid Flash; Dubbilex coaxed them into exploring the facility by allowing himself to be seen. The three young heroes eventually found Project Kr and liberated him.
In the ensuing fight, Desmond tested the Blockbuster formula on himself, but was defeated. The ground levels collapsed, and "Blockbuster" was arrested.
The Light promoted Guardian to command the facility. He vowed to shut down all cloning and breeding experiments, and focused on medical research instead. The Light still had subtle means to control him though.
During the rebuild, Dubbilex set in motion a plan to liberate the genomorphs. He secreted them away one by one, and stored them in Genomorph City.
Lex Luthor sent Superboy back to Cadmus, claiming that Project Match was a recent project. He knew full well that it wasn't, but he hoped to find the missing genomorphs that way, as well as to make Superboy dependent on the Shields in his fight against Match. Both of Luthor's plans succeeded.
Roy and Jim Harper swept Cadmus, all 52 levels and the remains of Genomorph City, which they had done before several times. But even with the newest Wayne Tech ecogram unit, they could find no trace of Speedy.