These will exist only if their holy orders do.

The Knights Templar (catholic)
The Knights Hospitaller
The Teutonic Order
The Order of Calatrava
The Order of Santiago
The Livonian Brothers of the Sword (HIP only)

The Brotherhood of the Holy Sepulchre (orthodox)

The Order of Saint Anthony (miaphysite)
The Knights of Saint Mark (HIP / CK2plus only)

The Order of Saint Addai (nestorian)

The Army of Lucifer (messalian, HIP / CK2plus only)

The Order of St. Blaise (apostolic, HIP / CK2plus only)

The Bektashi Order (sunni)
The Arsiyah (HIP only)
Al-Ansar (CK2plus only)
The Ghazis (CK2plus only)

The Hashshashin (shia, already have a society)
The Saqaliba (HIP only)
The Ghazis (CK2plus only)

The Haruriyyah (ibadi)

The Zealots (jewish)
The Guardians of the Temple (CK2plus only)

The Immortals (zoroastrian)

Army of Light (mahichean)

The Jomsvikings (germanic/norse)
The Braedralag (CK2plus only)

The Harii (germanic, CK2plus only)

The Lords of the Sky (tengri)
The Great Host (CK2plus only)

The Spirit Guardians (west african)

The Warriors of Perun (slavic)

The Chosen of Perkunas (romuva)
The Guardians of the Sacred Flame (CK2plus only)

The Sons of Kaleva (suomenusko)
The Disciples of the Old Gods(CK2plus only)

The Cult of Huitzilopochtli (aztec)

Chosen of Ashoka (buddhist)

Followers of Arjuna (hindu)

Bulls of Rishabha (jain)

Knights of the Sun (zunist)

The Fianna (celtic, CK2plus only)
The Sons of Gwydion (CK2plus only)
The Red Dragon Knights (CK2plus only)

The Anubis Guard (kemetic, CK2plus only)
The Horusian Guard (CK2plus only)

The Herculeans (hellenic, CK2plus only)
The Sacred Band (CK2plus only)

These will always exist, if the rules allow it.

The Round Table (North/western cultures like Spanish and German)
The Sandpit (South/western cultures like African and Bedouin)
The Ring of Honor (North/eastern cultures like Norse and Magyar)
The Wheel of Life and Death (South/eastern cultures like Hindustani and Han)

The Warriors Guild (Extra, nothing special)
The Berserkers (Extra, higher death risk and rewards)
The Amazons (Extra, female only)