Subgroups — C₂×Q₈×C₃⋊(C₃⋊D₃) #11724680

Order 864.

2184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₃⋊(C₃⋊D₃)	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂	1	3	yes	maximal
4	C₂×#6308	C₂	1	3	yes	maximal, complemented
5	C₂×Q₈×C₃⋊D₃	—	3	13	no	maximal
6	C₂²×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
7	C₃³×Q₈	C₂²	1	4	yes	complemented
8	C₂×C₃³×C₄	C₂²	1	3	yes	—
9	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	3	yes	—
10	C₄×C₃⋊(C₃⋊D₃)	C₂²	1	12	yes	—
11	#6308	C₂²	1	12	yes	complemented
12	C₂×C₃²×Q₈	D₃	1	13	yes	complemented
13	Q₈×C₃⋊D₃	—	3	104	no	—
14	C₂×C₄×C₃⋊D₃	—	3	39	no	—
15	C₂×#634	—	3	39	no	—
16	C₂²×C₃³	C₂³	1	1	yes	characteristic
17	C₂×C₃⋊(C₃⋊D₃)	Q₈	1	4	yes	direct product factor
18	C₂×C₃⋊(C₃⋊D₃)	C₂³	1	2	yes	—
19	C₃³×C₄	C₂³	1	6	yes	—
20	C₃⋊(C₃⋊Dic₃)	C₂³	1	6	yes	—
21	C₂×D₃×Q₈	—	9	13	no	—
22	C₂²×C₃⋊D₃	—	3	13	no	—
23	C₃²×Q₈	C₂×D₃	1	52	yes	complemented
24	C₂×C₃²×C₄	C₂×D₃	1	39	yes	—
25	C₂×C₃⋊Dic₃	—	3	39	no	—
26	C₄×C₃⋊D₃	—	3	156	no	—
27	#634	—	3	156	no	—
28	C₂×C₃³	C₂×Q₈	1	2	yes	complemented
29	C₂×C₃³	C₂⁴	1	1	yes	characteristic
30	C₃⋊(C₃⋊D₃)	C₂×Q₈	1	4	yes	direct product factor
31	C₂×C₃×Q₈	C₃⋊D₃	1	13	yes	complemented
32	D₃×Q₈	—	9	104	no	—
33	C₂×C₄×D₃	—	9	39	no	—
34	C₂×Dic₆	—	9	39	no	—
35	C₂²×C₃²	C₂²×D₃	1	13	yes	—
36	C₂×C₃⋊D₃	—	3	52	no	—
37	C₂×C₃⋊D₃	—	3	26	no	—
38	C₃²×C₄	C₂²×D₃	1	78	yes	—
39	C₃⋊Dic₃	—	3	78	no	—
40	C₂²×Q₈	—	27	1	no	Sylow
41	C₃³	C₂²×Q₈	1	1	yes	characteristic, complemented, Sylow
42	C₂²×D₃	—	9	13	no	—
43	C₃×Q₈	C₂×C₃⋊D₃	1	52	yes	complemented
44	C₂×C₃×C₄	C₂×C₃⋊D₃	1	39	yes	—
45	C₂×Dic₃	—	9	39	no	—
46	C₄×D₃	—	9	156	no	—
47	Dic₆	—	9	156	no	—
48	C₂×C₃²	D₃×Q₈	1	26	yes	complemented
49	C₂×C₃²	C₂³×D₃	1	13	yes	—
50	C₃⋊D₃	—	3	52	no	—
51	C₂×Q₈	—	27	8	no	—
52	C₂×Q₈	—	27	3	no	—
53	C₂×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
54	C₂²×C₄	—	27	3	no	—
55	C₂²×C₃	C₂²×C₃⋊D₃	1	13	yes	—
56	C₂×D₃	—	9	26	no	—
57	C₂×D₃	—	9	52	no	—
58	C₃×C₄	C₂²×C₃⋊D₃	1	78	yes	—
59	Dic₃	—	9	78	no	—
60	C₃²	C₂×D₃×Q₈	1	13	yes	complemented
61	C₂³	—	27	1	no	—
62	Q₈	—	27	12	no	—
63	Q₈	C₂×C₃⋊(C₃⋊D₃)	1	4	yes	direct product factor
64	C₂×C₄	—	27	3	no	—
65	C₂×C₄	—	27	12	no	—
66	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
67	C₂×C₃	Q₈×C₃⋊D₃	1	26	yes	complemented
68	C₂×C₃	C₂³×C₃⋊D₃	1	13	yes	—
69	D₃	—	9	52	no	—
70	C₂²	—	27	2	no	—
71	C₂²	—	27	4	no	—
72	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
73	C₄	—	27	6	no	—
74	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	6	yes	—
75	C₃	C₂×Q₈×C₃⋊D₃	1	13	yes	complemented
76	C₂	—	27	4	no	—
77	C₂	Q₈×C₃⋊(C₃⋊D₃)	1	2	yes	direct product factor
78	C₂	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃³	C₂⁴
Frattini	C₂	C₂³×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×Q₈	C₂
socle	C₂²×C₃³	C₂³
prime radical	C₂²×C₃⋊(C₃⋊D₃)	C₂²
hypercenter	C₂×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂²×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×Q₈	C₂	semidirect
2. Q₈×C₃⋊(C₃⋊D₃)	C₂	direct (N × H)
4. C₂×#6308	C₂	semidirect
7. C₃³×Q₈	C₂²	semidirect
11. #6308	C₂²	semidirect
12. C₂×C₃²×Q₈	D₃	semidirect
17. C₂×C₃⋊(C₃⋊D₃)	Q₈	direct (N × H)
23. C₃²×Q₈	C₂×D₃	semidirect
28. C₂×C₃³	C₂×Q₈	semidirect
30. C₃⋊(C₃⋊D₃)	C₂×Q₈	direct (N × H)
31. C₂×C₃×Q₈	C₃⋊D₃	semidirect
41. C₃³	C₂²×Q₈	semidirect
43. C₃×Q₈	C₂×C₃⋊D₃	semidirect
48. C₂×C₃²	D₃×Q₈	semidirect
53. C₂×Q₈	C₃⋊(C₃⋊D₃)	direct (N × H)
60. C₃²	C₂×D₃×Q₈	semidirect
63. Q₈	C₂×C₃⋊(C₃⋊D₃)	direct (N × H)
67. C₂×C₃	Q₈×C₃⋊D₃	semidirect
75. C₃	C₂×Q₈×C₃⋊D₃	semidirect
77. C₂	Q₈×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×C₃³×Q₈	2
Q₈×C₃⋊(C₃⋊D₃)	2
C₂×C₄×C₃⋊(C₃⋊D₃)	2
C₂×#6308	2
C₂×Q₈×C₃⋊D₃	3

Subgroups — C2×Q8×C3⋊(C3⋊D3) #11724680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₃⋊(C₃⋊D₃) #11724680