LMFDB - Subgroup search results

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Subgroup label	Subgroup order	Cyclic	Abelian	Solvable

Normal	Characteristic	Perfect	Maximal	Central

Ambient label	Ambient order	Direct product	Semidirect product

Quotient label	Subgroup index	Cyclic quotient	Abelian quotient	Solvable quotient

Maximal quotient	Non-trivial proper	Hall subgroup	Sylow subgroup

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Results (44 matches)

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Label	Subgroup									Ambient			Quotient
Label	Name	Order	Sylow	norm	char	max	cent	cyc		ab	nilp		sup solv	solv	perf	simp	Agp	Zgp	metab	metacyc	Name	Order	Name	Size	max	cyc	ab	nilp	sup solv	solv	simp	Agp	metab
486.213.1.a1	$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		✓	✓					✓	✓	✓					✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_1$	$1$		✓	✓	✓	✓	✓		✓	✓
486.213.2.a1	$C_3^2.C_3^3$	$3^{5}$	$3$	✓	✓	✓				✓	✓	✓					✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_2$	$2$		✓	✓	✓	✓	✓	✓	✓	✓
486.213.3.a1	$C_3^2\times C_{18}$	$2 \cdot 3^{4}$		✓	✓	✓			✓	✓	✓	✓			✓		✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3$	$3$		✓	✓	✓	✓	✓	✓	✓	✓
486.213.3.b1	$C_{18}:C_3^2$	$2 \cdot 3^{4}$		✓		✓				✓	✓	✓					✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3$	$3$		✓	✓	✓	✓	✓	✓	✓	✓
486.213.3.c1	$C_6.\He_3$	$2 \cdot 3^{4}$		✓		✓				✓	✓	✓					✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3$	$3$		✓	✓	✓	✓	✓	✓	✓	✓
486.213.6.a1	$C_3^2\times C_9$	$3^{4}$		✓	✓				✓	✓	✓	✓			✓		✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_6$	$2 \cdot 3$		✓	✓	✓	✓	✓		✓	✓
486.213.6.b1	$C_9:C_3^2$	$3^{4}$		✓						✓	✓	✓					✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_6$	$2 \cdot 3$		✓	✓	✓	✓	✓		✓	✓
486.213.6.c1	$C_3.\He_3$	$3^{4}$		✓						✓	✓	✓					✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_6$	$2 \cdot 3$		✓	✓	✓	✓	✓		✓	✓
486.213.9.a1	$C_3^2\times C_6$	$2 \cdot 3^{3}$		✓	✓				✓	✓	✓	✓			✓		✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3^2$	$3^{2}$			✓	✓	✓	✓		✓	✓
486.213.9.b1	$C_3\times C_{18}$	$2 \cdot 3^{3}$		✓					✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3^2$	$3^{2}$			✓	✓	✓	✓		✓	✓
486.213.9.c1	$C_9:C_6$	$2 \cdot 3^{3}$		✓						✓	✓	✓					✓	?		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3^2$	$3^{2}$			✓	✓	✓	✓		✓	✓
486.213.9.d1	$C_3\times C_{18}$	$2 \cdot 3^{3}$							✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.9.e1	$C_3\times C_{18}$	$2 \cdot 3^{3}$							✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.9.f1	$C_3\times C_{18}$	$2 \cdot 3^{3}$							✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.9.g1	$C_9:C_6$	$2 \cdot 3^{3}$								✓	✓	✓					✓	?		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.18.a1	$C_3^3$	$3^{3}$		✓	✓				✓	✓	✓	✓			✓		✓			$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3\times C_6$	$2 \cdot 3^{2}$			✓	✓	✓	✓		✓	✓
486.213.18.b1	$C_3\times C_9$	$3^{3}$		✓					✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3\times C_6$	$2 \cdot 3^{2}$			✓	✓	✓	✓		✓	✓
486.213.18.c1	$C_9:C_3$	$3^{3}$		✓						✓	✓	✓					✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3\times C_6$	$2 \cdot 3^{2}$			✓	✓	✓	✓		✓	✓
486.213.18.d1	$C_3\times C_9$	$3^{3}$							✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.18.e1	$C_3\times C_9$	$3^{3}$							✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.18.f1	$C_3\times C_9$	$3^{3}$							✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.18.g1	$C_9:C_3$	$3^{3}$								✓	✓	✓					✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.27.a1	$C_3\times C_6$	$2 \cdot 3^{2}$		✓	✓		✓		✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$\He_3$	$3^{3}$				✓	✓	✓			✓
486.213.27.b1	$C_3\times C_6$	$2 \cdot 3^{2}$		✓	✓				✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3^3$	$3^{3}$			✓	✓	✓	✓		✓	✓
486.213.27.c1	$C_3\times C_6$	$2 \cdot 3^{2}$		✓					✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$\He_3$	$3^{3}$				✓	✓	✓			✓
486.213.27.d1	$C_3\times C_6$	$2 \cdot 3^{2}$							✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.27.e1	$C_{18}$	$2 \cdot 3^{2}$						✓	✓	✓	✓	✓			✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.27.f1	$C_{18}$	$2 \cdot 3^{2}$						✓	✓	✓	✓	✓			✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.54.a1	$C_3^2$	$3^{2}$		✓	✓		✓		✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_2\times \He_3$	$2 \cdot 3^{3}$				✓	✓	✓			✓
486.213.54.b1	$C_3^2$	$3^{2}$		✓	✓				✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3^2\times C_6$	$2 \cdot 3^{3}$			✓	✓	✓	✓		✓	✓
486.213.54.c1	$C_3^2$	$3^{2}$		✓					✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_2\times \He_3$	$2 \cdot 3^{3}$				✓	✓	✓			✓
486.213.54.d1	$C_3^2$	$3^{2}$							✓	✓	✓	✓			✓		✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$2 \cdot 3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.54.e1	$C_9$	$3^{2}$						✓	✓	✓	✓	✓			✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$2 \cdot 3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.54.f1	$C_9$	$3^{2}$						✓	✓	✓	✓	✓			✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$2 \cdot 3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.81.a1	$C_6$	$2 \cdot 3$		✓			✓	✓	✓	✓	✓	✓			✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3.\He_3$	$3^{4}$				✓	✓	✓			✓
486.213.81.b1	$C_6$	$2 \cdot 3$		✓	✓		✓	✓	✓	✓	✓	✓			✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3\times \He_3$	$3^{4}$				✓	✓	✓			✓
486.213.81.c1	$C_6$	$2 \cdot 3$						✓	✓	✓	✓	✓			✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$3^{4}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.81.d1	$C_6$	$2 \cdot 3$						✓	✓	✓	✓	✓			✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$3^{4}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.162.a1	$C_3$	$3$		✓			✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_6.\He_3$	$2 \cdot 3^{4}$	✓			✓	✓	✓			✓
486.213.162.b1	$C_3$	$3$		✓	✓		✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_6\times \He_3$	$2 \cdot 3^{4}$	✓			✓	✓	✓			✓
486.213.162.c1	$C_3$	$3$						✓	✓	✓	✓	✓		✓	✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$2 \cdot 3^{4}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.162.d1	$C_3$	$3$						✓	✓	✓	✓	✓		✓	✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$			$2 \cdot 3^{4}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
486.213.243.a1	$C_2$	$2$	$2$	✓	✓		✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3^2.C_3^3$	$3^{5}$	✓			✓	✓	✓			✓
486.213.486.a1	$C_1$	$1$		✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$		$C_3\times C_6.\He_3$	$2 \cdot 3^{5}$				✓	✓	✓			✓

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.