Subgroups — #14519136

Order 1944.

686 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11758091	C₂	1	2	yes	maximal, complemented
2	C₃²×#1088	C₂	1	1	yes	maximal, characteristic, complemented
3	#1088⋊D₃	—	3	4	no	maximal
4	C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))	—	2	2	no	—
5	C₃²×#272	C₂²	1	1	yes	characteristic
6	D₃×C₃⋊(C₃⋊D₃)	—	6	2	no	—
7	#66432	—	3	8	no	—
8	C₃×#1088	D₃	1	4	yes	complemented
9	C₃²×H(3,𝔽₃)	D₄	1	1	yes	characteristic, complemented, Sylow
10	C₄⋊(C₃⋊(C₃⋊D₃))	—	9	1	no	maximal
11	#1088⋊C₂	—	9	1	no	—
12	C₃⋊(C₃⋊(C₃⋊D₃))	—	2	2	no	—
13	C₃×C₃⋊(C₃⋊D₃)	—	6	2	no	—
14	C₃³×D₃	—	6	2	no	—
15	C₃×#272	C₂×D₃	1	4	yes	—
16	C₃⋊((C₃⋊D₃)⋊C₃)	—	6	8	no	—
17	C₃⋊((C₃⋊D₃)⋊C₃)	—	12	8	no	—
18	C₂×C₃⋊(C₃⋊D₃)	—	9	2	no	—
19	D₃×C₃⋊D₃	—	18	8	no	—
20	D₃×C₃⋊D₃	—	18	18	no	—
21	C₃³×C₄	—	9	1	no	—
22	((C₃⋊D₃)⋊C₃)⋊C₂	—	9	2	no	—
23	#1088	C₃⋊D₃	1	1	yes	characteristic, complemented
24	C₃⁴	—	2	2	no	—
25	C₃×H(3,𝔽₃)	—	4	8	no	—
26	C₃×H(3,𝔽₃)	D₁₂	1	4	yes	complemented
27	C₄⋊(C₃⋊D₃)	—	27	4	no	—
28	C₄⋊(C₃⋊D₃)	—	27	9	no	—
29	C₂×C₃³	—	9	1	no	—
30	C₃⋊(C₃⋊D₃)	—	6	2	no	—
31	C₃⋊(C₃⋊D₃)	—	6	8	no	—
32	C₃⋊(C₃⋊D₃)	—	12	8	no	—
33	C₃⋊(C₃⋊D₃)	—	18	18	no	—
34	C₃×C₃⋊D₃	—	18	8	no	—
35	C₃×C₃⋊D₃	—	18	18	no	—
36	C₃²×D₃	—	6	8	no	—
37	C₃²×D₃	—	6	18	no	—
38	#272	C₂×C₃⋊D₃	1	1	yes	characteristic
39	(C₃⋊D₃)⋊C₃	—	18	2	no	—
40	(C₃⋊D₃)⋊C₃	—	36	8	no	—
41	C₂×C₃⋊D₃	—	27	18	no	—
42	C₂×C₃⋊D₃	—	27	8	no	—
43	D₃²	—	54	2	no	—
44	D₃²	—	54	24	no	—
45	C₃²×C₄	—	9	4	no	—
46	C₃²×C₄	—	9	9	no	—
47	C₃³	—	2	8	no	—
48	C₃³	—	4	8	no	—
49	C₃³	—	6	18	no	—
50	C₃³	GO⁺(4,𝔽₂)	1	1	yes	characteristic
51	H(3,𝔽₃)	—	4	8	no	—
52	H(3,𝔽₃)	—	8	6	no	—
53	H(3,𝔽₃)	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
54	D₁₂	—	81	1	no	—
55	D₁₂	—	81	12	no	—
56	C₂×C₃²	—	9	9	no	—
57	C₂×C₃²	—	9	4	no	—
58	C₃⋊D₃	—	18	18	no	—
59	C₃⋊D₃	—	18	8	no	—
60	C₃⋊D₃	—	18	2	no	—
61	C₃⋊D₃	—	36	8	no	—
62	C₃⋊D₃	—	54	24	no	—
63	C₃⋊D₃	—	108	24	no	—
64	C₃×D₃	—	6	2	no	—
65	C₃×D₃	—	6	24	no	—
66	C₃×D₃	—	54	2	no	—
67	C₃×D₃	—	54	24	no	—
68	C₂×D₃	—	81	2	no	—
69	C₂×D₃	—	81	24	no	—
70	C₂×D₃	—	162	2	no	—
71	C₃×C₄	—	9	1	no	—
72	C₃×C₄	—	9	12	no	—
73	C₃²	—	2	2	no	—
74	C₃²	—	4	8	no	—
75	C₃²	—	6	24	no	—
76	C₃²	—	12	24	no	—
77	C₃²	#1088⋊C₂	1	9	yes	complemented
78	C₃²	#157⋊D₃	1	4	yes	—
79	D₄	—	243	1	no	Sylow
80	C₂×C₃	—	9	1	no	—
81	C₂×C₃	—	9	12	no	—
82	C₂×C₃	—	162	2	no	—
83	D₃	—	6	2	no	—
84	D₃	—	54	2	no	—
85	D₃	—	54	24	no	—
86	D₃	—	324	8	no	—
87	D₃	—	162	2	no	—
88	C₂²	—	243	2	no	—
89	C₄	—	9	1	no	—
90	C₃	—	6	2	no	—
91	C₃	—	12	8	no	—
92	C₃	#1088⋊D₃	1	12	yes	complemented
93	C₃	#157⋊(C₃⋊D₃)	1	1	yes	characteristic
94	C₂	—	9	1	no	—
95	C₂	—	162	2	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle	C₃²×#272	C₂²
Frattini	C₃	#157⋊(C₃⋊D₃)
Fitting, hypocenter	C₃²×H(3,𝔽₃)	D₄
socle	C₃³	GO⁺(4,𝔽₂)
LCS	G ▷ C₃²×#272 ▷ C₃²×H(3,𝔽₃)
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #11758091	C₂	semidirect
2. C₃²×#1088	C₂	semidirect
8. C₃×#1088	D₃	semidirect
9. C₃²×H(3,𝔽₃)	D₄	semidirect
23. #1088	C₃⋊D₃	semidirect
26. C₃×H(3,𝔽₃)	D₁₂	semidirect
53. H(3,𝔽₃)	C₄⋊(C₃⋊D₃)	semidirect
77. C₃²	#1088⋊C₂	semidirect
92. C₃	#1088⋊D₃	semidirect

Maximal subgroups

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

subgroup	index
#11758091	2
C₃²×#1088	2
#1088⋊D₃	3
C₄⋊(C₃⋊(C₃⋊D₃))	9