The answer to your question would be that the necesssary element to complete the sentence above would be the following one: Predicate. "My father's 1969 Volkswagen beetle" is a sentence fragment.
A sentence fragment is a group of words that looks like a sentence but isn't. To be a sentence, group of words must have at least one independent clause. An independent clause is any group of words that contains both a subject and a verb and can stand on its own. In this case, the fragment does have the subject but it lacks the predicate.
