I am not sure the Church (as opposed to the Conciliar sect) ever defined precisely what crimes for which the death penalty could legitimately be used.
The Catechism of Saint Pius X says:
3 Q. Are there cases in which it is lawful to kill?
A. It is lawful to kill when fighting in a just war; when carrying out by order of the Supreme Authority a sentence of death in punishment of a crime; and, finally, in cases of necessary and lawful defence of one's own life against an unjust aggressor.
But it doesn't say which crime legitimately use this penalty.
Based on my reading of history, I would suggest that any crime that is contrary to the good order of a moral society could legitimately resort to the death penalty in a Christian nation. This does not require, of course, the State to execute each and every perpetrator of those crimes, but the Church would certainly not rule out execution for them.
Thus, treason against one's nation certainly qualifies. So does heresy when it becomes clear it is unrepentant and pertinacious and the heretic attempts to spread the heresy. Other crimes that have often carried the death penalty to which the Church never objected would include: murder, kidnapping, rape, sodomy, and theft (especially from another's substance as opposed to his excess).
This is not, of course, an exhaustive list but merely a few of the more commonly considered crimes for which execution is a legitimate punishment. The Church has traditionally given the State great latitude in determining the punishment due crimes. The criteria given by the Conciliarists are, undeniably, ridiculously narrow.