The correct option is B that he made a very big mistake - complement of a verb
A noun clause is a dependent clause that functions as a noun. It usually
begins with what, whatever, whom, that and other such relative
pronouns. A noun clause can act as a subject, object, predicate
nominative or object of a preposition, in a sentence.
A verb complement is a word, a phrase or a clause that is added to the verb to complete its meaning.
The
noun clause "that he made a very big mistake" is a verb
complement; the verb "admitted" makes no sense if it isn't followed by a
word that complements it. The noun clause in B functions as a verb
complement, hence it is correct.
Option A is incorrect because it
isn't a noun clause nor is it the subject; it contains the verb "admitted" and noun clauses do not contain the main verb.
Option C is
incorrect because it is a part of the noun clause with the verb "admitted".
Option D is incorrect because the answer lies in B.