Article VI

Police officers aren't part of the executive branch. The executive branch in a state is the governor and the lieutenant governor.

jespah-
Thank you.

I am surprised though. I thought that all persons working in the courts are part of the Judicial Branch, all persons working for the legislature are employees of the Legislative Branch, and all other employees are employees of the Executive Branch. The employees of the Executive branch are not all executives, but they are part of the branch named, Executive Branch.

http://www.whitehouse.gov/our-government/state-and-local-government

It's possible that a Police Commissioner would be considered a member of the Executive branch of a state. But by your reasoning, people conducting driving tests would be members of the Executive branch, as well. And I've got my doubts about that.

It means, in part, you don't have to swear your oath to God, or put your hand upon a Bible, or such.
I believe the courts puts judges, and department heads as said judicial and executive officers, usually they are elected or need legislative confirmation to get their position.
As police officers and others who hold positions of public trust, they too must swear allegiance to support the Constitution. (That they had to say "..so help me God." is an Urban legend, added for dramatic purposes or proof that they weren't a godless communist.)