Subgroups — #72234692

Order 6144.

At least 6512 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has at least 6512 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225756	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#15947	D₃	1	1	yes	characteristic, complemented
3	C₂⁸	S₄	1	2	yes	complemented
4	C₂³×#95	S₄	1	1	yes	characteristic, complemented
5	#15947	S₄	1	4	yes	complemented
6	C₂⁶	#1020	1	1	yes	characteristic
7	C₂⁶	#1020	1	4	yes	complemented
8	C₂×#95	#1020	1	2	yes	complemented
9	C₂⁵	#5934	1	1	yes	characteristic
10	C₂⁴	#85557	1	1	yes	characteristic
11	C₂⁴	#85559	1	2	yes	—
12	C₂⁴	#87588	1	1	yes	characteristic
13	C₂³	#11714279	1	2	yes	—
14	C₂³	#11715850	1	1	yes	characteristic
15	C₂²	#72209771	1	1	yes	characteristic
16	C₂²	#72209884	1	2	yes	complemented
17	C₂²	#72234218	1	1	yes	characteristic
18	C₂	#72234229	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#72234229
commutator, cosocle, hypocenter	#72225756	C₂
Frattini	C₂⁴	#87588
Fitting	C₂²×#15947	D₃
socle	C₂⁵	#5934
hypercenter	C₂²	#72234218
LCS	G ▷ #72225756
UCS	C₁ ◁ C₂ ◁ C₂²

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.