Could you give me some hints how we could find a normal series and all the composition series of $D_4$ ?
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A normal series of $G$ is $$G\geq G\geq G^{(1)} \geq G^{(2)} \geq G^{(3)} \geq \dots \geq G^{(n)} \geq \dots $$ where all these are normal subgroups of $G$, right?
But how many normal subgroups do we have to find?