The short answer is yes, you capitalize the word English regardless of whether you’re referring to the nationality, the school subject, or the language because all of these are proper nouns.
Normally English is a proper noun and all proper nouns in the English language are capitalized. Even when used as an adjective, such as in the “English language” or the “English curriculum,” English is capitalized because according to this Wikipedia article adjectives derived from proper nouns retain their capitalization:
In English, adjectives derived from proper nouns (except the names of characters in fictional works) usually retain their capitalization
– e.g. a Christian church, Canadian whisky, a Shakespearean sonnet, but not a quixotic mission, malapropism, holmesian nor pecksniffian.
Where the original capital is no longer at the beginning of the word, usage varies: anti-Christian, but Presocratic or Pre-Socratic or presocratic (not preSocratic).
If you want to know what other words should be capitalized, try our free title capitalization tool.