Subgroups — Q₈×(C₃⋊D₃)⋊C₃ #88721

Order 432.

145 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×(C₃⋊D₃)⋊C₃	C₂	1	3	yes	maximal
3	#634⋊C₃	C₂	1	3	yes	maximal, complemented
4	Q₈×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×D₃×Q₈	—	3	1	no	maximal
6	C₂×(C₃⋊D₃)⋊C₃	C₂²	1	1	yes	characteristic
7	C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
8	(C₃⋊Dic₃)⋊C₃	C₂²	1	3	yes	—
9	C₃²×Q₈	—	2	1	no	—
10	C₃²×Q₈	D₃	1	1	yes	characteristic, complemented
11	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
12	C₄×C₃⋊D₃	C₂×C₃	1	3	yes	—
13	#634	C₂×C₃	1	3	yes	complemented
14	C₃×C₄×D₃	—	3	3	no	—
15	C₃×Dic₆	—	3	3	no	—
16	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
17	(C₃⋊D₃)⋊C₃	Q₈	1	2	yes	direct product factor
18	C₂×C₃×Q₈	—	9	1	no	—
19	D₃×Q₈	—	3	1	no	—
20	D₃×Q₈	—	9	1	no	—
21	C₂×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
22	C₂×C₃×D₃	—	3	1	no	—
23	C₃²×C₄	—	2	3	no	—
24	C₃²×C₄	C₂×D₃	1	3	yes	—
25	C₃²×C₄	C₂²×C₃	1	3	yes	—
26	C₃⋊Dic₃	C₂²×C₃	1	3	yes	—
27	C₃×Dic₃	—	3	3	no	—
28	H(3,𝔽₃)	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
29	C₃×Q₈	—	3	1	no	—
30	C₃×Q₈	—	6	1	no	—
31	C₃×Q₈	—	6	1	no	—
32	C₃×Q₈	—	9	3	no	—
33	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
34	C₂×C₃×C₄	—	9	3	no	—
35	C₄×D₃	—	3	3	no	—
36	C₄×D₃	—	9	3	no	—
37	Dic₆	—	3	3	no	—
38	Dic₆	—	9	3	no	—
39	C₂×C₃²	—	2	1	no	—
40	C₂×C₃²	C₂²×D₃	1	1	yes	characteristic
41	C₂×C₃²	C₂³×C₃	1	1	yes	characteristic
42	C₃⋊D₃	C₃×Q₈	1	2	yes	complemented
43	C₃×D₃	—	3	2	no	—
44	C₂×Q₈	—	9	1	no	Sylow
45	C₂²×C₃	—	9	1	no	—
46	C₂×D₃	—	3	1	no	—
47	C₂×D₃	—	9	1	no	—
48	C₃×C₄	—	3	3	no	—
49	C₃×C₄	—	6	3	no	—
50	C₃×C₄	—	6	3	no	—
51	C₃×C₄	—	9	3	no	—
52	C₃×C₄	C₂×C₃×D₃	1	3	yes	—
53	Dic₃	—	3	3	no	—
54	Dic₃	—	9	3	no	—
55	C₃²	—	2	1	no	—
56	C₃²	D₃×Q₈	1	1	yes	characteristic, complemented
57	C₃²	C₂×C₃×Q₈	1	1	yes	characteristic, complemented
58	Q₈	—	9	3	no	—
59	Q₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, direct product factor
60	C₂×C₄	—	9	3	no	—
61	C₂×C₃	—	3	1	no	—
62	C₂×C₃	—	6	1	no	—
63	C₂×C₃	—	6	1	no	—
64	C₂×C₃	—	9	2	no	—
65	C₂×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
66	D₃	—	3	2	no	—
67	D₃	—	9	2	no	—
68	C₂²	—	9	1	no	—
69	C₄	—	9	3	no	—
70	C₄	C₂×(C₃⋊D₃)⋊C₃	1	3	yes	—
71	C₃	—	3	1	no	—
72	C₃	—	6	1	no	—
73	C₃	—	6	1	no	—
74	C₃	C₃×D₃×Q₈	1	1	yes	characteristic
75	C₂	—	9	2	no	—
76	C₂	C₂²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂×C₃²	C₂³×C₃
Frattini, socle	C₂×C₃	C₂²×C₃×D₃
Fitting	Q₈×H(3,𝔽₃)	C₂
prime radical	C₂×(C₃⋊D₃)⋊C₃	C₂²
hypercenter	Q₈	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×C₃×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×H(3,𝔽₃)	C₂	semidirect
3. #634⋊C₃	C₂	semidirect
4. Q₈×C₃⋊D₃	C₃	semidirect
10. C₃²×Q₈	D₃	semidirect
11. C₃²×Q₈	C₂×C₃	semidirect
13. #634	C₂×C₃	semidirect
17. (C₃⋊D₃)⋊C₃	Q₈	direct (N × H)
28. H(3,𝔽₃)	C₂×Q₈	semidirect
42. C₃⋊D₃	C₃×Q₈	semidirect
56. C₃²	D₃×Q₈	semidirect
57. C₃²	C₂×C₃×Q₈	semidirect
59. Q₈	(C₃⋊D₃)⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×H(3,𝔽₃)	2
C₄×(C₃⋊D₃)⋊C₃	2
#634⋊C₃	2
Q₈×C₃⋊D₃	3
C₃×D₃×Q₈	3

Subgroups — Q8×(C3⋊D3)⋊C3 #88721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×(C₃⋊D₃)⋊C₃ #88721