WHIZZO“Is this a case in which the verb complain is a transitive verb?”
It's matter of terminology and depends on what one wants to call an object. In Scandinavian grammatical terminology a clause can be the object of a verb:
Students know [that] the teacher does not give enough homework.
What do the students know? That the teacher does not give enough homework.
Cf.
Students know that.
What do students know? That.
Similarly:
Students complain [that] the teacher does not give enough homework.
This is where we encounter a difficulty. We cannot say:
*What do the students complain?
This because complain requires a preposition:
What do the students complain about?
However, English grammar forbids a preposition before the cionjunction that:
*Students complain about that the teacher does not give enough homework.
To my mind the that clause in your sentence is the de facto object of complain, but since complain is [correctly] classified as an intransitive verb in dictionaries, Anglo-Saxon grammarians cannot call the subordinate clause the object of complain.
This is just one of innumerable grammatical anomalies in English grammar. You just have to live with them!
CB