In Classical Athens, there was no distinction between commanders of land and naval forces. Every year the Assembly elected a board of ten strategoi (which we tend to translate as "generals") who would then be assigned to particular military missions. These missions could involve the command of garrison troops, land armies, fleets, or combined (often amphibious) expeditionary forces. The mission given to (or claimed by) a particular strategos did not affect his title or his potential authority over other troop types. In the Hellenistic period, there were particular strategoi in charge of the city walls, the defence of the harbour and so on, but initially there was nothing to formally distinguish one strategos from another. Indeed, over time, individual strategoi were often given a range of different commands, so that we can't even informally pigeonhole some of them as "generals" and others as "admirals".

Take the career of the strategos Iphikrates (lived c.418-353 BC) for example. He initially came to prominence as an ordinary marine, fighting with the Athenian fleet at the battle of Knidos (394 BC). Next we hear of him, he commands a unit of mercenary light troops garissoning Corinth and fighting Sparta and her allies on land (392-390 BC). He is then put in command of a fleet task force sent to the Hellespont, where he lands his troops and ambushes the Spartans on land (389 BC). Around 376 BC, he is sent to Egypt to command Greek mercenary infantry in Persian service; in 373 BC, he is back in Athenian service, leading a fleet of triremes around the Peloponnese and fighting a naval engagement near Corcyra. In 369 BC he commands the entire Athenian army, marching into the Peloponnese in support of Sparta. After some troubles in Thrace, he receives a final commission as co-commander of the Athenian fleet during the Social War (357-355 BC).

As you can see, the Athenians did not regard the duties of generals and admirals as mutually exclusive; rather, they tended to combine them and assign both to the same commander. Even when command is held by a board of generals, they do not specialise.

Interestingly, Athens' perennial rivals the Spartans did create a separate office of admiral (called nauarchos, literally "ship-leader") at some point late in the 5th century BC. Supreme command over the Spartan fleet gave the nauarchos so much influence that he could often operate almost as the equal of the kings, who traditionally only commanded land forces. In particular, the nauarchos Lysander, who defeated the Athenian fleet in 405 BC and demolished the Athenian Empire, gathered so much personal prestige and power that the Spartans were seriously afraid he might stage a coup. Their only answer to this was to formally put the fleet under the command of one of their two kings, who could then assign a puppet nauarchos (as king Agesilaos did after Lysander was removed from the office).

The Spartans were fairly unique in their creation of a separate office of admiral. The Athenians appear to have followed Persian practice in not really making a distinction between commanders of land and naval troops. To my knowledge, it is actually the Romans who first create an entire separate command structure for their (permanently established, professional) navies, but someone who knows more about the navies of the Carthaginians or the Hellenistic kingdoms might be able to correct me on that.