Why is the waitlist not updating correctly?

const Waitlist = () => {
  const [name, setName] = useState('');
  const [waitlist, setWaitlist] = useState([]);
  const onSubmit = (e) => {
    e.preventDefault();
    waitlist.push(name);
  };
  return (
    <div>
      <form onSubmit={onSubmit}>
        <label>
          Name: <input type="text" value={name} onChange={(e) => setName(e.target.value)} />
        </label>
        <button type="submit">Add to waitlist</button>
      </form>
      <ol>
        {waitlist.map((name) => (
          <li key={name}>{name}</li>
        ))}
      </ol>
    </div>
  );
};