Subgroups — #11721353

Order 864.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#88455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#127	C₃	1	1	yes	maximal, characteristic
3	#127⋊C₉	C₃	1	3	yes	maximal, complemented
4	C₂×#1094	—	4	1	no	maximal
5	C₂×C₃²×D₄	—	3	1	no	—
6	C₂×C₃²×D₄	—	3	1	no	—
7	C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
8	C₄²⋊C₉	C₂×C₃	1	3	yes	complemented
9	#1094	—	4	1	no	—
10	C₃×#127	—	3	1	no	—
11	C₃×#127	C₃²	1	1	yes	characteristic
12	C₂³×C₃²	—	3	1	no	—
13	C₂³×C₃²	A₄	1	1	yes	characteristic
14	C₃²×D₄	—	6	2	no	—
15	C₃²×D₄	—	6	2	no	—
16	C₂×C₃²×C₄	—	3	1	no	—
17	C₂×C₂²⋊C₉	—	4	3	no	—
18	C₂×3₋¹⁺²	—	16	1	no	—
19	C₂×C₃×D₄	—	3	1	no	—
20	C₂×C₃×D₄	—	3	3	no	—
21	C₂×C₃×D₄	—	3	3	no	—
22	C₂×C₃×D₄	—	3	1	no	—
23	C₃×C₄²	—	3	1	no	—
24	C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
25	C₂²×C₃²	—	6	1	no	—
26	C₂²×C₃²	—	6	1	no	—
27	C₂²×C₃²	—	6	2	no	—
28	C₂²×C₃²	C₂×A₄	1	1	yes	characteristic
29	C₃²×C₄	—	3	2	no	—
30	C₂²⋊C₉	—	4	3	no	—
31	#127	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
32	3₋¹⁺²	—	16	1	no	Sylow
33	C₂³×C₃	—	3	3	no	—
34	C₂³×C₃	—	3	1	no	—
35	C₂³×C₃	—	3	1	no	—
36	C₂³×C₃	C₃×A₄	1	1	yes	characteristic
37	C₃×D₄	—	6	2	no	—
38	C₃×D₄	—	6	2	no	—
39	C₃×D₄	—	6	6	no	—
40	C₃×D₄	—	6	6	no	—
41	C₂×C₃×C₄	—	3	3	no	—
42	C₂×C₃×C₄	—	3	1	no	—
43	C₂×C₃²	—	3	1	no	—
44	C₂×C₃²	—	4	1	no	—
45	C₂×C₃²	—	12	1	no	—
46	C₂×C₉	—	16	3	no	—
47	C₂×D₄	—	3	1	no	—
48	C₂×D₄	—	3	1	no	—
49	C₄²	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
50	C₂²×C₃	—	3	1	no	—
51	C₂²×C₃	—	6	1	no	—
52	C₂²×C₃	—	6	3	no	—
53	C₂²×C₃	—	6	3	no	—
54	C₂²×C₃	—	6	6	no	—
55	C₂²×C₃	—	6	1	no	—
56	C₂²×C₃	—	6	2	no	—
57	C₂²×C₃	C₂×C₃×A₄	1	1	yes	characteristic
58	C₃×C₄	—	3	6	no	—
59	C₃×C₄	—	3	2	no	—
60	C₃²	#127⋊C₃	1	1	yes	characteristic
61	C₉	—	16	3	no	—
62	C₂³	—	3	1	no	—
63	C₂³	#1094	1	1	yes	characteristic
64	D₄	—	6	2	no	—
65	D₄	—	6	2	no	—
66	C₂×C₄	—	3	1	no	—
67	C₂×C₃	—	3	3	no	—
68	C₂×C₃	—	3	1	no	—
69	C₂×C₃	—	4	1	no	—
70	C₂×C₃	—	12	1	no	—
71	C₂×C₃	—	12	3	no	—
72	C₂×C₃	—	12	1	no	—
73	C₂²	—	6	1	no	—
74	C₂²	—	6	1	no	—
75	C₂²	—	6	2	no	—
76	C₂²	C₂×#1094	1	1	yes	characteristic
77	C₄	—	3	2	no	—
78	C₃	—	3	1	no	—
79	C₃	C₃×#127⋊C₃	1	1	yes	characteristic
80	C₂	—	3	1	no	—
81	C₂	—	4	1	no	—
82	C₂	—	12	1	no	—

	subgroup	G/H
center	C₃	C₃×#127⋊C₃
commutator, cosocle	C₃×C₄²	C₂×C₃²
Frattini, socle	C₂²×C₃	C₂×C₃×A₄
Fitting, prime radical	C₃²×#127	C₃
hypercenter	C₃²	#127⋊C₃
hypocenter	C₄²	C₂×3₋¹⁺²
LCS	G ▷ C₃×C₄² ▷ C₄²
UCS	C₁ ◁ C₃ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. #88455	C₂	semidirect
3. #127⋊C₉	C₃	semidirect
8. C₄²⋊C₉	C₂×C₃	semidirect
31. #127	3₋¹⁺²	semidirect
49. C₄²	C₂×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
#88455	2
C₃²×#127	3
#127⋊C₉	3
C₂×#1094	4