Subgroups — C₂×C₈×C₃⋊(C₃⋊D₃) #11723940

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	C₂×C₃³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₃⋊(C₃⋊D₃)	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₈×C₃⋊D₃	—	3	13	no	maximal
6	C₂²×C₃⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
7	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
9	C₄×C₃⋊(C₃⋊D₃)	C₄	1	2	yes	—
10	C₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
11	C₃³×C₈	C₂²	1	2	yes	complemented
12	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	complemented
13	C₂×C₄×C₃⋊D₃	—	3	13	no	—
14	C₂×C₃²×C₈	D₃	1	13	yes	complemented
15	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
16	C₈×C₃⋊D₃	—	3	52	no	—
17	C₂²×C₃³	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃⋊(C₃⋊D₃)	C₈	1	4	yes	direct product factor
19	C₂×C₃⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
20	C₃³×C₄	C₂×C₄	1	1	yes	characteristic
21	C₃³×C₄	C₂³	1	1	yes	characteristic
22	C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
23	C₂×D₃×C₈	—	9	13	no	—
24	C₂²×C₃⋊D₃	—	3	13	no	—
25	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
26	C₂×C₃⋊Dic₃	—	3	13	no	—
27	C₄×C₃⋊D₃	—	3	26	no	—
28	C₄×C₃⋊D₃	—	3	26	no	—
29	C₃²×C₈	C₂×D₃	1	26	yes	complemented
30	C₃⋊(C₃⋊C₈)	—	3	26	no	—
31	C₂×C₃³	C₂×C₈	1	2	yes	complemented
32	C₂×C₃³	C₂²×C₄	1	1	yes	characteristic
33	C₃⋊(C₃⋊D₃)	C₂×C₈	1	4	yes	direct product factor
34	C₂×C₄×D₃	—	9	13	no	—
35	C₂×C₃×C₈	C₃⋊D₃	1	13	yes	complemented
36	C₂×C₃⋊C₈	—	9	13	no	—
37	D₃×C₈	—	9	52	no	—
38	C₂²×C₃²	C₄×D₃	1	13	yes	—
39	C₂×C₃⋊D₃	—	3	52	no	—
40	C₂×C₃⋊D₃	—	3	26	no	—
41	C₃²×C₄	C₄×D₃	1	13	yes	—
42	C₃²×C₄	C₂²×D₃	1	13	yes	—
43	C₃⋊Dic₃	—	3	26	no	—
44	C₂²×C₈	—	27	1	no	Sylow
45	C₃³	C₂²×C₈	1	1	yes	characteristic, complemented, Sylow
46	C₂²×D₃	—	9	13	no	—
47	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
48	C₂×Dic₃	—	9	13	no	—
49	C₄×D₃	—	9	26	no	—
50	C₄×D₃	—	9	26	no	—
51	C₃×C₈	C₂×C₃⋊D₃	1	26	yes	complemented
52	C₃⋊C₈	—	9	26	no	—
53	C₂×C₃²	D₃×C₈	1	26	yes	complemented
54	C₂×C₃²	C₂×C₄×D₃	1	13	yes	—
55	C₃⋊D₃	—	3	52	no	—
56	C₂²×C₄	—	27	1	no	—
57	C₂×C₈	—	27	1	no	—
58	C₂×C₈	—	27	4	no	—
59	C₂×C₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
60	C₂²×C₃	C₄×C₃⋊D₃	1	13	yes	—
61	C₂×D₃	—	9	52	no	—
62	C₂×D₃	—	9	26	no	—
63	C₃×C₄	C₄×C₃⋊D₃	1	13	yes	—
64	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
65	Dic₃	—	9	26	no	—
66	C₃²	C₂×D₃×C₈	1	13	yes	complemented
67	C₂³	—	27	1	no	—
68	C₂×C₄	—	27	1	no	—
69	C₂×C₄	—	27	2	no	—
70	C₂×C₄	—	27	2	no	—
71	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
72	C₈	—	27	2	no	—
73	C₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	direct product factor
74	C₂×C₃	C₈×C₃⋊D₃	1	26	yes	complemented
75	C₂×C₃	C₂×C₄×C₃⋊D₃	1	13	yes	—
76	D₃	—	9	52	no	—
77	C₂²	—	27	4	no	—
78	C₂²	—	27	2	no	—
79	C₂²	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
80	C₄	—	27	2	no	—
81	C₄	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
82	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
83	C₃	C₂×C₈×C₃⋊D₃	1	13	yes	complemented
84	C₂	—	27	4	no	—
85	C₂	C₈×C₃⋊(C₃⋊D₃)	1	2	yes	direct product factor
86	C₂	C₂×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₈	C₃⋊(C₃⋊D₃)
commutator, hypocenter	C₃³	C₂²×C₈
Frattini	C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×C₈	C₂
socle	C₂²×C₃³	C₂×C₄
prime radical	C₂²×C₃⋊(C₃⋊D₃)	C₄
cosocle	C₃³×C₄	C₂³
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×C₈	C₂	semidirect
3. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
4. C₈×C₃⋊(C₃⋊D₃)	C₂	direct (N × H)
11. C₃³×C₈	C₂²	semidirect
12. C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	semidirect
14. C₂×C₃²×C₈	D₃	semidirect
18. C₂×C₃⋊(C₃⋊D₃)	C₈	direct (N × H)
29. C₃²×C₈	C₂×D₃	semidirect
31. C₂×C₃³	C₂×C₈	semidirect
33. C₃⋊(C₃⋊D₃)	C₂×C₈	direct (N × H)
35. C₂×C₃×C₈	C₃⋊D₃	semidirect
45. C₃³	C₂²×C₈	semidirect
51. C₃×C₈	C₂×C₃⋊D₃	semidirect
53. C₂×C₃²	D₃×C₈	semidirect
59. C₂×C₈	C₃⋊(C₃⋊D₃)	direct (N × H)
66. C₃²	C₂×D₃×C₈	semidirect
73. C₈	C₂×C₃⋊(C₃⋊D₃)	direct (N × H)
74. C₂×C₃	C₈×C₃⋊D₃	semidirect
83. C₃	C₂×C₈×C₃⋊D₃	semidirect
85. C₂	C₈×C₃⋊(C₃⋊D₃)	direct (N × H)

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
C₂×C₃³×C₈	2
C₂×C₃⋊(C₃⋊(C₃⋊C₈))	2
C₈×C₃⋊(C₃⋊D₃)	2
C₂×C₈×C₃⋊D₃	3

Subgroups — C2×C8×C3⋊(C3⋊D3) #11723940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×C₃⋊(C₃⋊D₃) #11723940