The sentence doesn't have a subject. The sentence contains two subjects.
Even though "the subject of the sentence" is a common phrase, it's not the literal truth. Sentences typically contain clauses, and clauses have subjects. The sentence in question has two clauses, and each clause has its own subject.
[who/that/which] were involved in the march
This is a subordinate clause, specifically a relative clause in your model sentence. It has a subject -- or, it will as soon as we pick one of the relative pronoun options. Yes, the relative pronoun is required because it is the subject of its own clause.
The students that were involved in the march were arrested.
This is a matrix clause. The simple subject here is "students". The complete subject is "the students that were involved in the march" -- the subordinate clause that we already examined is just a part of this complete subject of this larger clause.
We have a subject inside another subject.
If you assume that a sentence can have only one subject, then you run into a problem. Here, there are two. You're left not knowing which one to call the subject or what you can call the other one.
Usually, we interpret "the subject of the sentence" to mean "the simple subject of the main clause of the sentence". So, if you ask me for the subject of that sentence, I would answer "students".
However, the sentence in question is complex. You could also ask me to identify every clause in the sentence. I would do that by matching each simple subject with its simple predicate: students / were arrested and [who/which/that] / were involved.
The sentence has two subjects, and neither of those subjects can be omitted.
