What happend after Charlemagne's Death?

When Charlemagne died the Frankish Empire was split in parts by his sons. The Empire was split into 3 parts pursuant to the Treaty of the Verdun East Francia, West Francia, and Central Francia. This all happend by the 9th century. The Tradition of dividing Patrimonies among brothers meant that the Frankish realm was ruled, nominally, as one polity subdivided into several Sub-Kingdoms once he died. Charlemagne died in the year of 814