The book
Bayerns Königshaus by Martha Schad says that this gorgeous painting was done in about 1837.
In 1837 Queen Caroline had the following grandchildren:
Ludwig Wilhelm Duke in Bavaria (born 1831)
Helene Duchess in Bavaria (born 1834)
Elisabeth Duchess in Bavaria (born 1837)
Franz Joseph, Archduke of Austria (born 1830)
Ferdinand Maximilian, Archduke of Austria (born 1832)
Carl Ludwig Archduke of Austria (born 1833)
Maria Anna Archduchess of Austria (born 1835)
Marie Princess of Saxony (born 1827)
Albert Prince of Saxony (born 1828)
Elisabeth Princess of Saxony (born 1830)
Ernst Prince of Saxony (born 1831)
Georg Prince of Saxony (born 1832)
Sidonie Princess of Saxony (born 1834)
Anna Princess of Saxony (born 1836)
On this painting we have seven boys and only three girls. So I guess Peter Fendi painted all grandsons of Queen Caroline, who were alive in 1837, but only the eldest daughters of Caroline's daughters Amalie (married to Johann of saxony), Sophie (married to Archduke Franz Carl) and Louise (married to Max in Bavaria). So the children could be: Ludwig Wilhelm in Bavaria, Franz Joseph of Austria and his brothers Ferdinand Maximilian and Carl Ludwig as well as Albert, Georg and Ernst of Saxony. I also think that the eldest sons of Caroline's daughters (Franz Joseph, Albert and Ludwig Wilhelm) wear something blue. The girls could be Helene in Bavaria (eldest daughter of Louise), Maria Anna of Austria (only daughter Sophie) and Marie Princess of Saxony (eldest daughter of Amalie).
My identification (left to right):
Carl Ludwig, Georg, (standing behind them) Ludwig Wilhelm, Franz Joseph, Ernst (looking at his brother Albert), Ferdinand Max, Helene, Maria Anna, Marie, Albert