Subgroups — C₄⋊(3₋¹⁺²⋊C₄) #88499

Order 432.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×3₋¹⁺²⋊C₄	C₂	1	2	yes	maximal
3	C₃×C₄⋊Dic₃	—	3	1	no	maximal
4	C₄⋊Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×3₋¹⁺²	C₂²	1	1	yes	characteristic
6	C₄×3₋¹⁺²	C₄	1	2	yes	complemented
7	3₋¹⁺²⋊C₄	—	2	2	no	—
8	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
9	C₂×C₃×Dic₃	—	3	2	no	—
10	C₂×C₄×C₉	—	2	1	no	—
11	C₂×C₄×C₉	C₂×C₃	1	1	yes	characteristic
12	C₂×Dic₉	C₂×C₃	1	2	yes	—
13	C₂×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
14	C₂×3₋¹⁺²	D₄	1	1	yes	characteristic
15	C₂×3₋¹⁺²	Q₈	1	1	yes	characteristic
16	C₃×C₄⋊C₄	—	9	1	no	—
17	C₄⋊Dic₃	—	3	1	no	—
18	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
19	C₃²×C₄	Dic₃	1	2	yes	—
20	C₃×Dic₃	—	6	2	no	—
21	C₂²×C₉	—	2	1	no	—
22	C₂²×C₉	C₂²×C₃	1	1	yes	characteristic
23	C₄×C₉	—	2	2	no	—
24	C₄×C₉	C₃×C₄	1	2	yes	complemented
25	Dic₉	—	2	2	no	—
26	3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃×C₄	—	3	1	no	—
28	C₂×C₃×C₄	—	9	2	no	—
29	C₂×C₃×C₄	C₃×D₃	1	1	yes	characteristic
30	C₂×Dic₃	—	3	2	no	—
31	C₂×C₃²	Dic₆	1	1	yes	characteristic
32	C₂×C₃²	D₁₂	1	1	yes	characteristic
33	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
34	C₂×C₉	—	2	1	no	—
35	C₂×C₉	—	2	1	no	—
36	C₂×C₉	—	2	1	no	—
37	C₂×C₉	C₂×C₃×C₄	1	1	yes	characteristic
38	C₂×C₉	C₃×D₄	1	1	yes	characteristic
39	C₂×C₉	C₃×Q₈	1	1	yes	characteristic
40	C₄⋊C₄	—	9	1	no	Sylow
41	C₂²×C₃	—	3	1	no	—
42	C₂²×C₃	C₂×C₃×D₃	1	1	yes	characteristic
43	C₃×C₄	—	3	2	no	—
44	C₃×C₄	—	18	2	no	—
45	C₃×C₄	C₃×Dic₃	1	2	yes	—
46	Dic₃	—	6	2	no	—
47	C₃²	C₄⋊Dic₃	1	1	yes	characteristic
48	C₉	—	2	1	no	—
49	C₉	C₃×C₄⋊C₄	1	1	yes	characteristic, complemented
50	C₂×C₄	—	9	2	no	—
51	C₂×C₄	AGL(1,ℤ/9)	1	1	yes	characteristic
52	C₂×C₃	—	3	1	no	—
53	C₂×C₃	—	3	1	no	—
54	C₂×C₃	—	3	1	no	—
55	C₂×C₃	C₃×Dic₆	1	1	yes	characteristic
56	C₂×C₃	C₃×D₁₂	1	1	yes	characteristic
57	C₂×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
58	C₂²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
59	C₄	—	18	2	no	—
60	C₄	3₋¹⁺²⋊C₄	1	2	yes	complemented
61	C₃	—	3	1	no	—
62	C₃	C₃×C₄⋊Dic₃	1	1	yes	characteristic
63	C₂	3₋¹⁺²⋊Q₈	1	1	yes	characteristic
64	C₂	C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
65	C₂	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂×C₃×C₄
Frattini, socle	C₂²×C₃	C₂×C₃×D₃
Fitting	C₂×C₄×3₋¹⁺²	C₂
prime radical	C₂²×C₃²	C₂×D₃
cosocle	C₂²×C₉	C₂²×C₃
hypercenter	C₂×C₄	AGL(1,ℤ/9)
hypocenter	C₉	C₃×C₄⋊C₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊Dic₉	C₃	semidirect
6. C₄×3₋¹⁺²	C₄	semidirect
24. C₄×C₉	C₃×C₄	semidirect
26. 3₋¹⁺²	C₄⋊C₄	semidirect
49. C₉	C₃×C₄⋊C₄	semidirect
60. C₄	3₋¹⁺²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×3₋¹⁺²	2
C₂×3₋¹⁺²⋊C₄	2
C₃×C₄⋊Dic₃	3
C₄⋊Dic₉	3

Subgroups — C4⋊(3−1+2⋊C4) #88499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(3₋¹⁺²⋊C₄) #88499