Subgroups — #11723989

Order 864.

868 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#41	C₂	1	1	yes	maximal, characteristic
2	#88855	C₂	1	3	yes	maximal, complemented
3	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	3	yes	maximal, complemented
4	#64157	—	3	13	no	maximal
5	C₃³×Q₈	C₄	1	1	yes	characteristic
6	C₃³×D₄	C₄	1	3	yes	—
7	C₂×C₃³×C₄	C₂²	1	3	yes	—
8	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
9	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	3	yes	—
10	C₃²×#41	D₃	1	13	yes	—
11	#3748	—	3	39	no	—
12	C₂×C₃⋊(C₃⋊C₈)	—	3	39	no	—
13	C₂²×C₃³	C₂×C₄	1	3	yes	—
14	C₃³×C₄	C₂×C₄	1	3	yes	—
15	C₃³×C₄	C₂³	1	1	yes	characteristic
16	#948	—	9	13	no	—
17	C₃²×Q₈	Dic₃	1	13	yes	—
18	C₃²×D₄	Dic₃	1	39	yes	—
19	C₂×C₃²×C₄	C₂×D₃	1	39	yes	—
20	C₃⋊(C₃⋊C₈)	—	3	13	no	—
21	C₃⋊(C₃⋊C₈)	—	3	39	no	—
22	C₂×C₃³	—	2	3	no	—
23	C₂×C₃³	C₂²×C₄	1	1	yes	characteristic
24	C₃×#41	C₃⋊D₃	1	13	yes	—
25	#208	—	9	39	no	—
26	C₂×C₃⋊C₈	—	9	39	no	—
27	C₂²×C₃²	C₂×Dic₃	1	39	yes	—
28	C₃²×C₄	C₂×Dic₃	1	39	yes	—
29	C₃²×C₄	C₂²×D₃	1	13	yes	—
30	#131	—	27	1	no	Sylow
31	C₃³	#131	1	1	yes	characteristic, complemented, Sylow
32	C₃×Q₈	C₃⋊Dic₃	1	13	yes	—
33	C₃×D₄	C₃⋊Dic₃	1	39	yes	—
34	C₂×C₃×C₄	C₂×C₃⋊D₃	1	39	yes	—
35	C₃⋊C₈	—	9	13	no	—
36	C₃⋊C₈	—	9	39	no	—
37	C₂×C₃²	—	2	39	no	—
38	C₂×C₃²	C₂²×Dic₃	1	13	yes	—
39	#41	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
40	M₁₆	—	27	3	no	—
41	C₂×C₈	—	27	3	no	—
42	C₂²×C₃	C₂×C₃⋊Dic₃	1	39	yes	—
43	C₃×C₄	C₂×C₃⋊Dic₃	1	39	yes	—
44	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
45	C₃²	#948	1	13	yes	complemented
46	Q₈	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
47	D₄	C₃⋊(C₃⋊Dic₃)	1	3	yes	—
48	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
49	C₈	—	27	1	no	—
50	C₈	—	27	3	no	—
51	C₂×C₃	—	2	39	no	—
52	C₂×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
53	C₂²	C₂×C₃⋊(C₃⋊Dic₃)	1	3	yes	—
54	C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	3	yes	—
55	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
56	C₃	#64157	1	13	yes	complemented
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle	C₂×C₃³	C₂²×C₄
Fitting, prime radical	C₃³×#41	C₂
cosocle	C₃³×C₄	C₂³
hypercenter	#41	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#131
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #88855	C₂	semidirect
3. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
31. C₃³	#131	semidirect
45. C₃²	#948	semidirect
56. C₃	#64157	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#41	2
#88855	2
C₂×C₃⋊(C₃⋊(C₃⋊C₈))	2
#64157	3