The classical and quantum correlations sharing between modes of the Dirac fields in the noninertial frame are investigated. It is shown that: (i) The classical correlation for the Dirac fields decreases as the acceleration increases, which is different from the result of the scalar field that the classical correlation is independent of the acceleration; (ii) There is no simple dominating relation between the quantum correlation and entanglement for the Dirac fields, which is unlike the scalar case where the quantum correlation is always over and above the entanglement; (iii) As the acceleration increases, the correlations between modes $I$ and $II$ and between modes $A$ and $II$ increase, but the correlations between modes $A$ and $I$ decrease.
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