order Fagales vs plant order

order Fagales and plant order both are nouns.

Word	Noun	Adjective	Verb	Adverb
order Fagales	Yes	No	No	No
plant order	Yes	No	No	No

As nouns, plant order is a hypernym of order Fagales; that is, plant order is a word with a broader meaning than order Fagales:

order Fagales (noun)	plant order (noun)
an order of dicotyledonous trees of the subclass Hamamelidae	the order of plants