Subgroups — #11722204

Order 864.

728 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×D₄	C₂	1	1	yes	maximal, characteristic
2	#88864	C₂	1	2	yes	maximal, complemented
3	#63661	—	3	13	no	maximal
4	C₂³×C₃³	C₄	1	1	yes	characteristic, complemented
5	C₂³×C₃³	C₂²	1	1	yes	characteristic
6	C₃³×D₄	—	2	2	no	—
7	C₂×C₃³×C₄	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
9	C₂×C₃²×D₄	D₃	1	13	yes	—
10	#3757	—	3	26	no	—
11	C₂²×C₃³	—	2	1	no	—
12	C₂²×C₃³	—	4	1	no	—
13	C₂²×C₃³	—	4	1	no	—
14	C₂²×C₃³	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃³	D₄	1	2	yes	—
16	C₃³×C₄	—	2	1	no	—
17	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
18	#834	—	9	13	no	—
19	C₂³×C₃²	Dic₃	1	13	yes	complemented
20	C₂³×C₃²	C₂×D₃	1	13	yes	—
21	C₃²×D₄	—	2	26	no	—
22	C₂×C₃²×C₄	Dic₃	1	13	yes	complemented
23	C₂×C₃⋊Dic₃	—	6	26	no	—
24	C₂×C₃³	—	2	2	no	—
25	C₂×C₃³	—	2	1	no	—
26	C₂×C₃³	—	4	1	no	—
27	C₂×C₃³	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₃×D₄	C₃⋊D₃	1	13	yes	—
29	#217	—	9	26	no	—
30	C₂²×C₃²	—	2	13	no	—
31	C₂²×C₃²	—	4	13	no	—
32	C₂²×C₃²	—	4	13	no	—
33	C₂²×C₃²	C₂×Dic₃	1	13	yes	—
34	C₂²×C₃²	#67	1	26	yes	—
35	C₃²×C₄	—	2	13	no	—
36	C₃⋊Dic₃	—	12	26	no	—
37	#99	—	27	1	no	Sylow
38	C₃³	#99	1	1	yes	characteristic, complemented, Sylow
39	C₂³×C₃	C₃⋊Dic₃	1	13	yes	complemented
40	C₂³×C₃	C₂×C₃⋊D₃	1	13	yes	—
41	C₃×D₄	—	2	26	no	—
42	C₂×C₃×C₄	C₃⋊Dic₃	1	13	yes	complemented
43	C₂×Dic₃	—	18	26	no	—
44	C₂×C₃²	—	2	26	no	—
45	C₂×C₃²	—	2	13	no	—
46	C₂×C₃²	—	4	13	no	—
47	C₂×C₃²	#217	1	13	yes	—
48	C₂×D₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
49	C₂²⋊C₄	—	27	2	no	—
50	C₂²×C₃	—	2	13	no	—
51	C₂²×C₃	—	4	13	no	—
52	C₂²×C₃	—	4	13	no	—
53	C₂²×C₃	C₂×C₃⋊Dic₃	1	13	yes	—
54	C₂²×C₃	#638	1	26	yes	—
55	C₃×C₄	—	2	13	no	—
56	Dic₃	—	36	26	no	—
57	C₃²	#834	1	13	yes	complemented
58	C₂³	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, complemented
59	C₂³	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
60	D₄	—	2	2	no	—
61	C₂×C₄	—	54	2	no	—
62	C₂×C₄	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, complemented
63	C₂×C₃	—	2	26	no	—
64	C₂×C₃	—	2	13	no	—
65	C₂×C₃	—	4	13	no	—
66	C₂×C₃	#3757	1	13	yes	—
67	C₂²	—	2	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
71	C₂²	#6312	1	2	yes	—
72	C₄	—	2	1	no	—
73	C₄	—	108	2	no	—
74	C₃	#63661	1	13	yes	complemented
75	C₂	—	2	2	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	#88864	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#88864
commutator	C₂²×C₃³	C₂×C₄
Frattini	C₂³	C₂×C₃⋊(C₃⋊D₃)
Fitting, prime radical	C₂×C₃³×D₄	C₂
socle	C₂×C₃³	C₂²⋊C₄
cosocle	C₂³×C₃³	C₂²
hypercenter	C₂×D₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#99
LCS	G ▷ C₂²×C₃³ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. #88864	C₂	semidirect
4. C₂³×C₃³	C₄	semidirect
7. C₂×C₃³×C₄	C₄	semidirect
19. C₂³×C₃²	Dic₃	semidirect
22. C₂×C₃²×C₄	Dic₃	semidirect
38. C₃³	#99	semidirect
39. C₂³×C₃	C₃⋊Dic₃	semidirect
42. C₂×C₃×C₄	C₃⋊Dic₃	semidirect
57. C₃²	#834	semidirect
58. C₂³	C₃⋊(C₃⋊Dic₃)	semidirect
62. C₂×C₄	C₃⋊(C₃⋊Dic₃)	semidirect
74. C₃	#63661	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×D₄	2
#88864	2
#63661	3