Three Key Ancestors

According to Carlos Parada's Greek Mythology Link, the key ancestors of the major Greek families were three: Deucalion (& Pyrrha), Atlas (through his daughters the Pleiades), and Io.

Atlas 

The Pelopides 

Pelops 1 was son of Tantalus 1, son of Zeus and the Pleiad Dione 3, daughter of Atlas. Pelops 1's wife Hippodamia 3 was daughter of the Pleiad Sterope 3. The PLEIADES are daughters of Atlas. The Pelopides ruled Mycenae until the return of the HERACLIDES (descendants of Heracles 1, a descendant of Io). Pelops 1 came from Phrygia to Hellas, whereas Dardanus 1 emigrated from Hellas to Phrygia.

The Trojans 

The Trojans were also descendants of Atlas. Dardanus 1, son of Electra 3, daughter of Atlas, is at the beginning of the house of Troy, for Dardanus 1 is father of Erichthonius 1, father of Tros 1 (after whom the Trojans are called), father of Ilus 2 (founder of Ilium, that is, Troy), father of Laomedon 1, father of Priam 1, who was king when the city was destroyed. 

Some Thebans 

Among the Thebans, the usurpers Nycteus 2 and Lycus 5 were said to have come from Euboea, but they too might be descendants of Atlas, and so could beAmphion 1, grandson of Nycteus 2. Amphion 1 married Pelops 1's sister, the mother of the NIOBIDS.

Deucalion

The descendants of Deucalion 1 (and Aeolus 1) founded and ruled Thessalian cities such as Pherae, Phthia and Iolcus, but were periodically influential inThebes, Argos, Athens, and Messenia. 

Deucalion 1, the first mortal of this line and son of the Titan Prometheus 1, is father of Hellen 1, eponym of the Hellenes. From Hellen 1 sprang Dorus 1 (eponym of the Dorians), Xuthus 1, and Aeolus 1. Xuthus 1 is father of Achaeus 1 (eponym of the Achaeans), and of Ion 1 (eponym of the Ionians). The Thessalian king Aeolus 1 (different from the keeper of the winds) had many important descendants.

Io 

Ancestors and founders of important cities auch as Mycenae, Thebes, and Argos were descendants of Io. These also controlled Crete, Laconia, and perhaps Arcadia. The HERACLIDES were descendants of Io. Their house evolved first in north-east Africa, and in the mideast (Phoenicia). 

Io is the first mortal of this line. She is usually regarded as daughter of the river god Inachus, her other genealogies being more uncertain. After Io comes Epaphus 1, king of Egypt and father of Libya. Her descendants are Agenor 1, Belus 1, and Lelex 2. From Agenor 1 descended Europa and Cadmus, which is to say the houses of Crete and Thebes respectively. From Belus 1 descended Aegyptus 1 and Danaus 1, that is, the houses of Argos, and Mycenae. 

Perseus 1 (descendant of Danaus 1 and Aegyptus 1), and his own descendants reigned in Mycenae. But during Eurystheus' time or after him Mycenae came under the rule of the Pelopides, who are descendants of Atlas. The Pelopides were expelled by the HERACLIDES after the Trojan War.

Interestingly, tradition says the Athenians were a distinct group:

The Athenians do not belong originally to any of the mentioned primary families. They were the children of Gaia, or else "sons of the soil." Later Aegeus 1 married the daughter of a Pelopid, and Theseus married a descendant of Io. Even later the throne was seized by Melanthus 1, a descendant of Deucalion 1. 

The Colchians (for example Medea) are descendants of Helius, and so are the first Corinthians [see Corinth]. The Troezenians had their own origin, but afterwards Troezen was ruled by the descendants of Atlas.

Parada's page of ancestors has detailed lists of members from each family.