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No, if you are referring to any country club it is NOT capitalized.
I went golfing at the country club.
If it is a specific country club, you capitalize it.
I went golfing at the Summertime Country Club.
Marlboro Country Music club
Tavistock Country Club was created in 1921.
Wykagyl Country Club was created in 1898.
The country that created Winx Club is Italy.