What is the direct object in this entence/ Socrates desired that his house be full of true friends.
That introduces a clause, and a clause can function as a the object of a verb. Moreover, the direct object is the answer to the question, What did Socrates desire?
Ex: Socrates desired <verb's object here> .
Ex: What did Socrates desire?
Does that help?
You may want to have a look here, The five basic sentence sentence types