The court in the southern German state of Bavaria, where one of the defendants lived, handed down sentences of up to four years to the group, a spokesman told AFP.
The judges found that the 27-year-old, who lived in Bavaria and whose name was not released, smuggled mainly Syrians from the Turkish city of Izmir to Greece five separate times.
He and a 34-year-old accomplice, also from Aleppo, recruited the third man, 24, to steer the boat on one of these occasions, promising him free passage on the vessel.
In September 2015, the third man was steering an inflatable dinghy with at least 46 people on board towards the Greek island of Lesbos when it collided with a freighter and capsized.
Thirteen of the passengers died, including an unknown number of children.
The presiding judge called the tragedy "foreseeable and avoidable" as he handed down verdicts of people trafficking leading to loss of life, the court spokesman told AFP.
The court sentenced the 27-year-old to four years in prison while the 24-year-old who drove the boat received a two-and-a-half-year sentence.
The last defendant, who mainly managed the group's money, was given a two-year suspended sentence.
Prosecutors had called for jail terms of up to 12 and a half years.
Migration to Europe soared in 2015, as people desperate to flee the worsening conflict in Syria risked their lives attempting the dangerous crossing.
The International Organization of Migration registered a huge jump in the number of arrivals by sea that year, including 8,53,650 on Greek shores.
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