What is the house of Cadmus in Oedipus Rex?
Cadmus is the original name for the city of Thebes. So, the House of Cadmus is really the House of Thebes. It is Oedipus and all his decedents as found in Sophocles' Oedipus Trilogy: Oedipus the King, Oedipus at Colonus, and Antigone.
It begins with Oedipus's parents, Laius and Jocasta. Then, as you know, Oedipus kills his father and marries his mother. They have four children: Eteocles, Polyneices, Antigone, and Ismene.
As you know, the House of Thebes is rather cursed, as is the city-state of Thebes accordingly. Not only is it cursed with murder, incest, plagues, and the Sphinx, but the dueling brothers Eteocles and Polyneices bring destruction on the city-state through a civil war. You can read about the Seven Armies Against Thebes in the links below.