Four sects and the four sons of Imam as-Sadiq

Imam as-Sadiq had four sons, whose followers regarded each of them, as a natural successor to his father. All of them were proved to be false (illegitimate), eventually, except Imam Musa al-Kadhim (may peace be upon him).

Imam Musa al-Kadhim
Imam Musa al-Kadhim was the seventh of the Twelve Imams of Twelver Shi'a Islam. According to the Kitab al-Irshad of Sheikh al-Mufid:

Among the shaykhs of the followers of Abu Abd Allah Ja'far al-Sadiq, peace be on him, his special group (khassa), his inner circle and the trustworthy righteous legal scholars, may God have mercy on them, who report the clear designation of the Imamate by Abu Abd Allah Jafars peace be on him, for his son, Abu al-Hasan Musa, peace be on him, are: al-Mufaddal b. Umar al-Jufi, Mu'adh b. Kathir, Abd al-Rahman b. al-Hajjaj, al-Fayd b. al-Mukhtar, Yaqub al-Sarraj, Sulayman b. Khalid, Safwan al-Jammal... [That designation] is also reported by his two brothers, Ishaq and Ali, sons of Jafar, peace be on him.[12]

Abdullah al-Aftah ibn Ja'far al-Sadiq
Abdullah al-Aftah ibn Ja'far al-Sadiq (d.766 CE / 149 A.H.)[1] was the eldest surviving son of Ja'far al-Sadiq (after al-Sadiq’s death) and the full-brother of Isma'il ibn Jafar.[2] Following Ja'far al-Sadiq’s death, the majority of Ja'far’s followers accepted Abdullah al-Aftah as their new Imam. These followers were known as the Fathites and, according to the Mu'tazili heresiographer Abul-Qasim al-Balkhi al-Ka‘bi (d.319 A.H. / 931 CE), they were the biggest and most important section of the followers of Ja'far al-Sadiq.[5] To support his claims, Abdullah al-Aftah seems to have claimed a 2nd Nass from his father (following Ismā'īl's demise) and his adherents cited a supposed Hadith from Ja'far al-Sadiq to the effect that the Imamate must be transmitted through the eldest son of the Imam. However, when Abdullah al-Aftah died childless[6][7] about 70 days after the death of his father, the bulk of his supporters went over to his brother Musa al-Kadhim

Isma‘il ibn Ja‘far
Isma‘il ibn Ja‘far (Arabic: إسماعيل بن جعفر‎ c. 721 CE/103 AH - 755 CE/138AH) was the eldest son of Imam Ja'far al-Sadiq and the full-brother of Abdullah al-Aftah. Following Ja'far's death, the Shia community split between the element that would become the Twelver Shia, and those who believed the imamate passed through to Ismail's son; the Ismaili branch of Shia Islam is accordingly named for Ismail.[1] According to both the Nizari and Mustaali Shia sects, he is the sixth Imam. Traditions about his death.

Muhammad ibn Ja'far al-Sadiq
Muhammad ibn Ja'far al-Sadiq (Al-Dibaj i.e. the handsome),[1] the younger full brother of Musa al-Kadhim,[2] appeared in Mecca in the year 200 A.H. / 815 C.E. claiming that he was the Awaited Mahdi. He believed in a Zaydl Shia type of Imamate[3] and declared himself as the Caliph of the Muslims and took the oath of allegiance from them and was called the Leader of the faithful. He was recognized as the Imam by a small group of followers. His followers became denominated as the Shumaytiyya (Sumaytiyya) after their leader Yahya ibn Abi’l-Shumayt (al-Sumayt).[4] However, his revolt against the Caliph Al-Ma'mun proved unsuccessful in the very same year it started (i.e. 815 C.E.).[5] He ended his revolt by abdicating and publicly confessing his error and was then banished from the Hejaz.[6]