There are quite a lot of them, they roughly correspond to the Eastern Orthodox from who they split to return to the Catholic Church. The Catholic Encyclopedia lists them:

1. Byzantine Catholics

1. The Byzantine Catholics are those who correspond to the Orthodox. They all use the same (Byzantine) Rite; but they are not all organized as one body. They form seven groups:

• the Melkites in Syria and Egypt (about 110,000), under a Patriarch of Antioch who administers, and bears the titles of, Alexandria and Jerusalem too. They have eleven dioceses and use Arabic liturgically with fragments Greek, though any of their priests may (and some do) celebrate entirely in Greek. The old name "Melkite", which meant originally one who accepted the decrees of Chalcedon (and the imperial laws), as against the Jacobites and Copts, is now used only for these Catholics.
• There are a few hundred Catholics of this Rite in Greece and Turkey in Europe. They use Greek liturgically and depend on Latin delegates at Constantinople and Athens.
• One Georgian congregation of Constantinople (last remnant of the old Georgian Church destroyed by Russia), who use their own language and obey the Latin Delegate.
• The Ruthenians, of whom there are nearly four millions in Austria-Hungary and hidden still in corners of Russia. They use Old Slavonic.
• The Bulgarian Catholics (about 13,000), under two vicars Apostolic, who also use old Slavonic.
• Rumanian Catholics (about a million and a half) in Rumania, but chiefly in Transylvania. They have bishops and use their own language in the liturgy.
• The Italo-Greeks (about 50,000), a remnant of the old Church of Greater Greece. They are scattered about Calabria and Sicily, have a famous monastery near Rome (Grotta-ferrata) and colonies at Leghorn, Malta, Algiers, Marseilles, and Corsica, besides a church (St-Julien le Pauvre) at Paris. They use Greek liturgically but, living as they do surrounded by Latins, they have considerably latinized their rites.

This completes the list of Byzantine Catholics, of whom it may be said that the chief want is organization among themselves. There has often been talk of restoring a Catholic (Melkite) Patriarch of Constantinople. It was said that Pope Leo XIII intended to arrange this before he died. If such a revival ever is made, the patriarch would have jurisdiction, or at least a primacy, over all Catholics of his Rite; in this way the scattered unities of Melkites in Syria, Ruthenians in Hungary, Italo-Greeks in Sicily, and so on, would be linked together as are all other Eastern Catholic Churches.

2. Chaldean Catholics

The Chaldees are Eastern Catholics converted from Nestorianism. In the sixteenth and seventeenth centuries a complicated series of quarrels and schisms among the Nestorians led to not very stable unions of first one and then another party with the Holy See. Since that time there has always been a Catholic patriarch of the Chaldees, though several times the person so appointed fell away into schism again and had to be replaced by another. The Chaldees are said now to number about 70,000 souls (Silbernagl, op. cit., 354; but Werner, "Orbis Terr. Cath.", 166, gives the number as 33,000). Their primate lives at Mosul, having the title of Patriarch of Babylon. Under him are two archbishoprics and ten other sees. There are monasteries whose arrangements are very similar to those of the Nestorians. The liturgical books (in Syriac, slightly revised from the Nestorian ones) are printed by the Dominicans at Mosul. Most of their canon law depends on the Bull of Pius IX, "Reversurus" (12 July, 1867), published for the Armenians and extended to the Chaldees by another Bull, "Cum ecclesiastica" (31 Aug., 1869). They have some students at the Propaganda College in Rome.

3. Alexandrian Catholics

The Alexandrian Catholics (Catholic Copts) have had a vicar Apostolic since 1781. Before that (in 1442 and again in 1713) the Coptic patriarch had submitted to Rome, but in neither case was the union of long duration. As the number of Catholics of this Rite has increased very considerably of late years, Leo XIII in 1895 restore the Catholic patriarchate. The patriarch lives at Cairo and rules over about 20,000 Catholic Copts.

4. Abyssinians

The Abyssinians, too, had many relations with Rome in past times, and Latin missionaries built up a considerable Catholic Abyssinian Church. But repeated persecutions and banishment of Catholics prevented this community from becoming a permanent one with a regular hierarchy. Now that the Government is tolerant, some thousands of Abyssinians are Catholics. They have an Apostolic vicar at Keren. If their numbers increase, no doubt they will in time be organized under a Catholic Abuna who should depend on the Catholic Coptic patriarch. Their liturgy, too, is at present in a state of disorganization. It seems that the Monophysite Abyssinian books will need a good deal of revision before they can be used by Catholics. Meanwhile the priests ordained for this rite have a translation of the Roman Mass in their own language, an arrangement that is not meant to be more than a temporary expedient.

5. Syrians

The Catholic Syrian Church dates from 1781. At that time a number of Jacobite bishops, priests, and lay people, who had agreed to reunion with Rome, elected one Ignatius Giarve to succeed the dead Jacobite patriarch, George III. Giarve sent to Rome asking for recognition and a pallium, and submitting in all things to the pope's authority. But he was then deposed by those of his people who clung to Jacobitism, and a Jacobite patriarch was elected. From this time there have been two rival successions. In 1830 the Catholic Syrians were acknowledged by the Turkish Government as a separate millet. The Catholic patriarch lives at Beirut, most of his flock in Mesopotamia. Under him are three archbishops and six other bishops, five monasteries, and about 25,000 families.

6. Uniat Church of Malabar

There is also a Catholic Church of Malabar formed by the Synod of Diamper in 1599. This Church, too, has passed through stormy periods; quite lately, since the Vatican Council, a new schism has been formed form it of about 30,000 people who are in communion with neither the Catholics, nor the Jacobites, nor the Nestorians, nor any one else at all. There are now about 200,000 Malabar Catholics under three vicars Apostolic (at Trichur, Changanacherry, and Ernaculam).

7. Armenians

The Catholic Armenians are an important body numbering altogether about 130,000 souls. Like their Gregorian countrymen they are scattered about the Levant, and they have congregations in Austria and Italy. There have been several more or less temporary reunions of the Armenian Church since the fourteenth century, but in each case a rival Gregorian party set up rival patriarchs and bishops. The head of the Catholic Armenians is the Catholic Armenian Patriarch of Constantinople (since 1830), in whom is joined the patriarchate of Cilicia. He always takes the name Peter, and rules over three titular archbishops and fourteen sees, of which one is Alexandria and one Ispahan in Persia (Werner-- Silbernagl, 346). After much dispute he is now recognized by the Porte as the head of a separate millet, and he also represents before the Government all other Catholic bodies that have as yet no political organization. There are also many Catholic Armenians in Austria-Hungary who are subject in Transylvania to the Latin bishops, but in Galicia to the Armenian Archbishop of Lemberg. In Russia there is an Armenian Catholic See of Artvin immediately subject to the pope. The Mechitarists (Founded by Mechitar of Sebaste in 1711) are an important element of Armenian Catholicism. They are monks who follow the Rule of St. Benedict and have monasteries at San Lazzaro outside Venice, at Vienna, and in many towns in the Balkans, Armenia, and Russia. They have missions all over the Levant, schools, and presses that produce important liturgical, historical, historical, and theological works. Since 1869 all Armenian Catholic priests must be celibate.

8. Maronites

Lastly, the Maronite Church is entirely Catholic. There is much dispute as to its origin and the reason of its separation from the Syrian national Church. It is certain that it was formed around monasteries in the Lebanon founded by a certain John Maro in the fourth century. In spite of the indignant protests of all Maronites there is no doubt that they were separated from the old See of Antioch by the fact that they were Monothelites. They were reunited to the Roman Church in the twelfth century, and then (after a period of wavering) since 1216, when their patriarch, Jeremias II, made his definite submission, they have been unswervingly faithful, alone among all Eastern Churches. As in other cases, the Maronites, too, are allowed to keep their old organization and titles. Their head is the Maronite "Patriarch of Antioch and all the East", successor to Monothelite rivals of the old line, who, therefore, in no way represents the original patriarchate. He is also the civil head of his nation, although he has no berat from the sultan, and lives in a large palace at Bkerki in the Lebanon. He has under him nine sees and several titular bishops. There are many monasteries and convents. The present law of the Maronite Church was drawn up by the great national council held in 1736 at the monastery of Our Lady of the Almond Trees (Deir Saïdat al-Luaize), in the Lebanon. There are about 300,000 Maronites in the Lebanon and scattered along the Syrian coast. They also have colonies in Egypt and Cyprus, and numbers of them have lately begun to emigrate to America. They have a national college at Rome.