A dependent clause cannot stand alone, though they often contain both a subject and a verb. Where independent clauses express complete thoughts, dependent clauses do not, and left on their own, ...

The relative pronoun that always introduces essential information and can be omitted if it is the object of the clause. Relative pronouns CANNOT be omitted when they are the subject of the clause; ...

or a participle (doing), its subject is implied to be the subject of the clause, or sometimes the closest noun phrase. In such cases, construct your sentences carefully to avoid suggesting nonsense.