The officers in the military do not swear to obey the CIC or any orders from him. They are sworn to uphold the Constitution.
http://en.wikipedia.org/wiki/United_States_Uniformed_Services_Oath_of_Office
From your source...
Officers of the National Guard of the various States, however, take an additional oath:
I, [name], do solemly swear (or affirm) that I will support and defend the Constitution of the United States and the Constitution of the State (Commonwealth, District, Territory) of ___ against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I will obey the orders of the President of the United States and the Governor of the State (Commonwealth, District, Territory) of ___, that I make this obligation freely, without any mental reservations or purpose of evasion, and that I will well and faithfully discharge the duties of the Office of [grade] in the Army/Air National Guard of the State (Commonwealth, District, Territory) of ___ upon which I am about to enter, so help me God.[3]