Based on the theories of value distribution, normal family and the knowledge of advanced algebra, the normal criteria for the holomorphic curves and their derived curves sharing hyperplanes located in t-subgeneral position was studied. Let $ \mathcal{F} $ be a family of holomorphic curves of a domain $ D \subset \mathbb{C} $ to $ {\mathbb{P}^N}(\mathbb{C}) $, ${H_\ell } =\{ {{\bm{x}} \in {\mathbb{P}^N}(\mathbb{C}):} \left. {\left\langle {{\bm{x}},{{\bm{\alpha}} _\ell }} \right\rangle = 0} \right\}$ be hyperplanes in $ {\mathbb{P}^N}(\mathbb{C}) $ located in t-subgeneral position, where ${{\bm{\alpha}} _\ell } = {\left( {{a_{\ell 0}},{a_{\ell 1}}, \cdots ,{a_{\ell N}}} \right)^{\text{T}}}$, $ \ell = 1,2, \cdots ,3t + 1 $, $ {H_0} = \left\{ {{x_0} = 0} \right\} $, $t\geqslant N$. Assume the following conditions hold for every $ f \in \mathcal{F} $: If $ f(z) \in {H_\ell } $, then $ \nabla f(z) \in {H_\ell } $, $ \ell = 1,2, \cdots ,3t + 1 $; If $f(z) \in\displaystyle \bigcup\limits_{\ell = 1}^{3t + 1} {{H_\ell }}$, then $\dfrac{\left|\langle f(z),{H}_{0}\rangle \right|}{\Vert f(z)\Vert \cdot \Vert {H}_{0}\Vert }\geqslant \delta$, where $ \delta \in \left(0，1\right) $ is a constant. Then $ \mathcal{F} $ is normal on $ D $. For the special case of $ N = 3 $ and $ t = 3,4,5 $, the number of shared hyperplanes can be effectively reduced through this research.