"Let's start with Augustus John (1878-1961), best known as a portraitist who sired children by his wife and other women. His second son (by his wife) was Caspar John (1903-1984), who went on to become First Sea Lord (1960-63), attaining the rank of Admiral of the Fleet in 1962. In the Royal Navy, First Sea Lord [was] the highest position that an officer can attain."
Caspar John's Wikipedia entry is here.
Recently I became aware that the grandson of Sir John Everett Millais, 1st Baronet (1829-1896) also became a Royal Navy Admiral. Millais was one of the original members of the Pre-Raphaelite Brotherhood and later went on the become a successful portrait painter and, not long before his death, president of the Royal Academy.
"Bubbles," a 1886 painting of a young boy, became famous because it was controversially for Millais used for many years in advertising material by England's Pears Soap company (more information about it here).
The boy in the painting was Millais' grandson William Milbourne James (1881-1973) who later rose to the rank of Admiral in the Royal Navy.
Sir William had to bear the cross of the painting in the form of having the nickname "Bubbles" during his naval career. He was a prolific author during and after his time in the navy. In the early years of the Great War he was executive officer of the battlecruiser Queen Mary, serving under Sir William Reginald "Blinker" Hall who later was in charge of the famous Room 40 decoding center where James also served. Both Hall and James transferred from Queen Mary before the Battle of Jutland where the ship was destroyed when a magazine exploded: only 20 men survived of a complement of 1,286.
When he was First Sea Lord.