Subgroups — #11723913

Order 864.

1064 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
3	#88859	C₂	1	2	yes	maximal
4	#88857	C₂	1	2	yes	maximal, complemented
5	C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
6	#64081	—	3	13	no	maximal
7	C₂²×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
8	C₂×C₃³×C₄	C₂²	1	2	yes	—
9	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊(C₃⋊D₃)	C₄	1	4	yes	complemented
13	C₂×C₄×C₃⋊D₃	—	3	13	no	—
14	C₃²×C₄²	D₃	1	13	yes	complemented
15	#3752	—	3	26	no	—
16	#3750	—	3	26	no	—
17	C₄×C₃⋊Dic₃	—	3	13	no	—
18	C₂²×C₃³	C₂³	1	1	yes	characteristic
19	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
20	C₂×C₃⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
21	C₃³×C₄	—	2	2	no	—
22	C₃³×C₄	C₂×C₄	1	2	yes	—
23	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
24	C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
25	#872	—	9	13	no	—
26	C₂²×C₃⋊D₃	—	3	13	no	—
27	C₂×C₃²×C₄	C₂×D₃	1	26	yes	—
28	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
29	C₂×C₃⋊Dic₃	—	3	13	no	—
30	C₂×C₃⋊Dic₃	—	3	26	no	—
31	C₄×C₃⋊D₃	—	3	52	no	—
32	C₂×C₃³	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₃³	#41	1	2	yes	—
34	C₃⋊(C₃⋊D₃)	—	2	2	no	—
35	C₂×C₄×D₃	—	9	13	no	—
36	C₃×C₄²	C₃⋊D₃	1	13	yes	complemented
37	(C₂×D₃)⋊C₄	—	9	26	no	—
38	Dic₃⋊C₄	—	9	26	no	—
39	C₄×Dic₃	—	9	13	no	—
40	C₂²×C₃²	C₂²×D₃	1	13	yes	—
41	C₂×C₃⋊D₃	—	3	26	no	—
42	C₂×C₃⋊D₃	—	6	26	no	—
43	C₃²×C₄	—	2	26	no	—
44	C₃²×C₄	C₄×D₃	1	26	yes	—
45	C₃⋊Dic₃	—	3	26	no	—
46	C₃⋊Dic₃	—	6	26	no	—
47	#117	—	27	1	no	Sylow
48	C₃³	#117	1	1	yes	characteristic, complemented, Sylow
49	C₂²×D₃	—	9	13	no	—
50	C₂×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
51	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
52	C₂×Dic₃	—	9	13	no	—
53	C₂×Dic₃	—	9	26	no	—
54	C₄×D₃	—	9	52	no	—
55	C₂×C₃²	C₂×C₄×D₃	1	13	yes	—
56	C₂×C₃²	#235	1	26	yes	—
57	C₃⋊D₃	—	6	26	no	—
58	C₂²×C₄	—	27	1	no	—
59	C₄⋊C₄	—	27	2	no	—
60	C₂²⋊C₄	—	27	2	no	—
61	C₄²	—	27	1	no	—
62	C₄²	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
63	C₂²×C₃	C₂²×C₃⋊D₃	1	13	yes	—
64	C₂×D₃	—	9	26	no	—
65	C₂×D₃	—	18	26	no	—
66	C₃×C₄	—	2	26	no	—
67	C₃×C₄	C₄×C₃⋊D₃	1	26	yes	—
68	Dic₃	—	9	26	no	—
69	Dic₃	—	18	26	no	—
70	C₃²	#872	1	13	yes	complemented
71	C₂³	—	27	1	no	—
72	C₂×C₄	—	27	1	no	—
73	C₂×C₄	—	27	4	no	—
74	C₂×C₄	—	27	2	no	—
75	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
76	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
77	C₂×C₃	C₂×C₄×C₃⋊D₃	1	13	yes	—
78	C₂×C₃	#3828	1	26	yes	—
79	D₃	—	18	26	no	—
80	C₂²	—	27	2	no	—
81	C₂²	—	54	2	no	—
82	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	27	2	no	—
85	C₄	—	54	2	no	—
86	C₄	C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
87	C₃	#64081	1	13	yes	complemented
88	C₂	—	54	2	no	—
89	C₂	C₂×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
90	C₂	#89076	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₂²×C₄
Frattini	C₂²	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₄²	C₂
socle, cosocle	C₂²×C₃³	C₂³
prime radical	C₂²×C₃⋊(C₃⋊D₃)	C₂²
hypercenter	C₄²	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#117
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₃³×C₄²	C₂	semidirect
4. #88857	C₂	semidirect
5. C₄×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
12. C₄×C₃⋊(C₃⋊D₃)	C₄	semidirect
14. C₃²×C₄²	D₃	semidirect
36. C₃×C₄²	C₃⋊D₃	semidirect
48. C₃³	#117	semidirect
62. C₄²	C₃⋊(C₃⋊D₃)	semidirect
70. C₃²	#872	semidirect
87. C₃	#64081	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊(C₃⋊D₃)	2
C₃³×C₄²	2
#88859	2
#88857	2
C₄×C₃⋊(C₃⋊Dic₃)	2
#64081	3