Thats not exactly correct.
First off, the position of Cardinal is technically a Lay position, or at least it used to be. These days it is restricted to Bishops, but that did not used to be the case. Princes, Dukes, etc. were frequently made Cardinals. It was all part of renaissance politics.
So, understanding that, one understands that “Cardinal” is not so much a rank in the heirarchy above or below anything else. All Cardinals are bishops, and are of equal rank as all other bishops. Patriarchs can be cardinals. The Chaldean Patriarch of Babylon, [Mar Emmanuel III Delly ](Mar Emmanuel III Cardinal Delly) was made a cardinal by Pope Benedict very recently. Another good example is Armenian
Catholic Patriarch Krikor Bedros Aghajanian (Grégoire-Pierre Agagianian), who was nearly elected pope by the 1958 conclave whish eventually elected Pope John XXIII.
Patriarchs are also Bishops, althought hey are bishops with special powers to make laws, appoint other bishops, govern the litrugy, etc. of their Church. However, patriarchs are still bishops. The Pope is a good illustration of this. He is bishop of Rome, as well as the Patriarch of the Western Church, as well as the Pope of the Universal Church.
Hope this clears some things up