Subgroups — (3₋¹⁺²⋊C₄)⋊C₄ #88498

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	1	yes	maximal, characteristic
3	C₂×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₃⋊C₄	—	3	1	no	maximal
5	Dic₉⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₄×3₋¹⁺²	—	2	1	no	—
8	3₋¹⁺²⋊C₄	—	2	1	no	—
9	3₋¹⁺²⋊C₄	C₄	1	2	yes	complemented
10	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
11	C₂×C₃×Dic₃	—	3	1	no	—
12	C₂×C₃×Dic₃	—	3	1	no	—
13	C₂×C₄×C₉	—	2	1	no	—
14	C₂×C₄×C₉	C₂×C₃	1	1	yes	characteristic
15	C₂×Dic₉	C₂×C₃	1	1	yes	characteristic
16	C₂×Dic₉	C₂×C₃	1	1	yes	characteristic
17	C₂×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
18	C₂×3₋¹⁺²	D₄	1	1	yes	characteristic
19	C₂×3₋¹⁺²	Q₈	1	1	yes	characteristic
20	C₃×C₄⋊C₄	—	9	1	no	—
21	Dic₃⋊C₄	—	3	1	no	—
22	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
23	C₃²×C₄	—	2	1	no	—
24	C₃×Dic₃	—	3	2	no	—
25	C₃×Dic₃	—	6	1	no	—
26	C₂²×C₉	—	2	1	no	—
27	C₂²×C₉	C₂²×C₃	1	1	yes	characteristic
28	C₄×C₉	—	2	2	no	—
29	C₄×C₉	—	2	1	no	—
30	Dic₉	—	2	1	no	—
31	Dic₉	C₃×C₄	1	2	yes	complemented
32	3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃×C₄	—	3	1	no	—
34	C₂×C₃×C₄	—	9	1	no	—
35	C₂×C₃×C₄	—	9	1	no	—
36	C₂×C₃×C₄	C₃×D₃	1	1	yes	characteristic
37	C₂×Dic₃	—	3	1	no	—
38	C₂×Dic₃	—	3	1	no	—
39	C₂×C₃²	Dic₆	1	1	yes	characteristic
40	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
41	C₂×C₃²	#67	1	1	yes	characteristic
42	C₂×C₉	—	2	1	no	—
43	C₂×C₉	—	2	1	no	—
44	C₂×C₉	—	2	1	no	—
45	C₂×C₉	C₂×C₃×C₄	1	1	yes	characteristic
46	C₂×C₉	C₃×D₄	1	1	yes	characteristic
47	C₂×C₉	C₃×Q₈	1	1	yes	characteristic
48	C₄⋊C₄	—	9	1	no	Sylow
49	C₂²×C₃	—	3	1	no	—
50	C₂²×C₃	C₂×C₃×D₃	1	1	yes	characteristic
51	C₃×C₄	—	2	1	no	—
52	C₃×C₄	—	6	1	no	—
53	C₃×C₄	—	9	2	no	—
54	C₃×C₄	—	18	1	no	—
55	Dic₃	—	3	2	no	—
56	Dic₃	—	6	1	no	—
57	C₃²	Dic₃⋊C₄	1	1	yes	characteristic
58	C₉	—	2	1	no	—
59	C₉	C₃×C₄⋊C₄	1	1	yes	characteristic, complemented
60	C₂×C₄	—	9	1	no	—
61	C₂×C₄	—	9	1	no	—
62	C₂×C₄	AGL(1,ℤ/9)	1	1	yes	characteristic
63	C₂×C₃	—	3	1	no	—
64	C₂×C₃	—	3	1	no	—
65	C₂×C₃	—	3	1	no	—
66	C₂×C₃	C₃×Dic₆	1	1	yes	characteristic
67	C₂×C₃	C₃×C₄×D₃	1	1	yes	characteristic
68	C₂×C₃	C₃×#67	1	1	yes	characteristic
69	C₂²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	9	2	no	—
72	C₄	—	18	1	no	—
73	C₃	—	3	1	no	—
74	C₃	C₃×Dic₃⋊C₄	1	1	yes	characteristic
75	C₂	3₋¹⁺²⋊Q₈	1	1	yes	characteristic
76	C₂	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
77	C₂	#611⋊C₃	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
5. Dic₉⋊C₄	C₃	semidirect
9. 3₋¹⁺²⋊C₄	C₄	semidirect
31. Dic₉	C₃×C₄	semidirect
32. 3₋¹⁺²	C₄⋊C₄	semidirect
59. C₉	C₃×C₄⋊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×Dic₃⋊C₄	3
Dic₉⋊C₄	3

Subgroups — (3−1+2⋊C4)⋊C4 #88498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (3₋¹⁺²⋊C₄)⋊C₄ #88498