Subgroups — #11722189

Order 864.

420 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	#88855	C₂	1	2	yes	maximal
3	#63646	—	3	13	no	maximal
4	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
5	C₃³×C₈	C₄	1	2	yes	—
6	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
7	C₂×C₃²×C₈	D₃	1	13	yes	—
8	#3748	—	3	26	no	—
9	C₂²×C₃³	Q₈	1	1	yes	characteristic
10	C₃³×C₄	C₂×C₄	1	1	yes	characteristic
11	C₃³×C₄	D₄	1	1	yes	characteristic
12	#819	—	9	13	no	—
13	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
14	C₃²×C₈	Dic₃	1	26	yes	—
15	C₃⋊(C₃⋊C₈)	—	6	26	no	—
16	C₂×C₃³	—	2	1	no	—
17	C₂×C₃³	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₃×C₈	C₃⋊D₃	1	13	yes	—
19	#208	—	9	26	no	—
20	C₂²×C₃²	Dic₆	1	13	yes	—
21	C₃²×C₄	D₁₂	1	13	yes	—
22	C₃²×C₄	C₂×Dic₃	1	13	yes	—
23	#108	—	27	1	no	Sylow
24	C₃³	#108	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
26	C₃×C₈	C₃⋊Dic₃	1	26	yes	—
27	C₃⋊C₈	—	18	26	no	—
28	C₂×C₃²	—	2	13	no	—
29	C₂×C₃²	C₄⋊Dic₃	1	13	yes	—
30	M₁₆	—	27	2	no	—
31	C₂×C₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
32	C₂²×C₃	#634	1	13	yes	—
33	C₃×C₄	C₄⋊(C₃⋊D₃)	1	13	yes	—
34	C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
35	C₃²	#819	1	13	yes	complemented
36	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
37	C₈	—	54	2	no	—
38	C₈	C₃⋊(C₃⋊Dic₃)	1	2	yes	—
39	C₂×C₃	—	2	13	no	—
40	C₂×C₃	C₄⋊(C₃⋊Dic₃)	1	13	yes	—
41	C₂²	#6308	1	1	yes	characteristic
42	C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
43	C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
44	C₃	#63646	1	13	yes	complemented
45	C₂	—	2	1	no	—
46	C₂	C₄⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊(C₃⋊(C₃⋊D₃))
commutator	C₃³×C₄	C₂×C₄
Frattini	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×C₈	C₂
socle	C₂×C₃³	C₄⋊C₄
prime radical	C₂²×C₃³	Q₈
cosocle	C₂×C₃³×C₄	C₂²
hypercenter	C₂×C₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#108
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
24. C₃³	#108	semidirect
35. C₃²	#819	semidirect
44. C₃	#63646	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×C₈	2
#88855	2
#63646	3