Group #13142436 — subgroups

Subgroups — C₄²×3₋¹⁺²⋊C₄ #13142436

Order 1728.

828 conjugacy classes of subgroups; only the 420 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 420 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×3₋¹⁺²⋊C₄	C₂	1	6	yes	maximal
3	C₃×C₄²×Dic₃	—	3	1	no	maximal
4	C₄²×Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
6	C₂²×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
7	C₄²×3₋¹⁺²	C₄	1	4	yes	complemented
8	C₄×3₋¹⁺²⋊C₄	C₄	1	24	yes	direct product factor
9	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₃	—	3	6	no	—
11	C₂×C₄²×C₉	—	2	1	no	—
12	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
13	C₂×C₄×Dic₉	C₂×C₃	1	6	yes	—
14	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
15	C₂×C₄×3₋¹⁺²	C₂×C₄	1	12	yes	—
16	C₂×C₄×3₋¹⁺²	C₂×C₄	1	6	yes	—
17	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	24	yes	—
18	C₃×C₄³	—	9	1	no	—
19	C₄²×Dic₃	—	3	1	no	—
20	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
21	C₂²×C₃×Dic₃	—	3	4	no	—
22	C₃²×C₄²	Dic₃	1	4	yes	—
23	C₃×C₄×Dic₃	—	3	24	no	—
24	C₂²×C₄×C₉	—	2	3	no	—
25	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
26	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
27	C₄²×C₉	—	2	4	no	—
28	C₄²×C₉	C₃×C₄	1	4	yes	complemented
29	C₄×Dic₉	C₃×C₄	1	24	yes	complemented
30	C₂²×3₋¹⁺²	C₂²×C₄	1	6	yes	—
31	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
32	C₄×3₋¹⁺²	C₄²	1	12	yes	complemented
33	3₋¹⁺²⋊C₄	C₄²	1	16	yes	direct product factor
34	C₂×C₃×C₄²	—	3	1	no	—
35	C₂×C₃×C₄²	—	9	6	no	—
36	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
37	C₂×C₄×Dic₃	—	3	6	no	—
38	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
39	C₂×C₃²×C₄	C₄×D₃	1	12	yes	—
40	C₂×C₃²×C₄	C₂×Dic₃	1	6	yes	—
41	C₂×C₃×Dic₃	—	3	24	no	—
42	C₂³×C₉	—	2	1	no	—
43	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
44	C₂×C₄×C₉	—	2	12	no	—
45	C₂×C₄×C₉	—	2	6	no	—
46	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
47	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
48	C₂×Dic₉	C₂×C₃×C₄	1	24	yes	—
49	C₄³	—	9	1	no	Sylow
50	C₂×3₋¹⁺²	C₂×C₄²	1	4	yes	—
51	C₂×3₋¹⁺²	C₂×C₄²	1	3	yes	—
52	C₂²×C₃×C₄	—	3	3	no	—
53	C₂²×C₃×C₄	—	9	4	no	—
54	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
55	C₂²×Dic₃	—	3	4	no	—
56	C₃×C₄²	—	3	4	no	—
57	C₃×C₄²	—	9	24	no	—
58	C₃×C₄²	C₃×Dic₃	1	4	yes	—
59	C₄×Dic₃	—	3	24	no	—
60	C₂²×C₃²	C₂×C₄×D₃	1	6	yes	—
61	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
62	C₃²×C₄	C₄×Dic₃	1	12	yes	—
63	C₃×Dic₃	—	3	16	no	—
64	C₂²×C₉	—	2	6	no	—
65	C₂²×C₉	—	2	1	no	—
66	C₂²×C₉	C₂²×C₃×C₄	1	6	yes	—
67	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
68	C₄×C₉	—	2	12	no	—
69	C₄×C₉	C₃×C₄²	1	12	yes	complemented
70	Dic₉	C₃×C₄²	1	16	yes	complemented
71	C₂×C₄²	—	9	6	no	—
72	C₂×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
73	3₋¹⁺²	C₄³	1	1	yes	characteristic, complemented, Sylow
74	C₂³×C₃	—	3	1	no	—
75	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
76	C₂×C₃×C₄	—	3	12	no	—
77	C₂×C₃×C₄	—	3	6	no	—
78	C₂×C₃×C₄	—	9	24	no	—
79	C₂×C₃×C₄	C₃×C₄×D₃	1	12	yes	—
80	C₂×C₃×C₄	C₂×C₃×Dic₃	1	6	yes	—
81	C₂×Dic₃	—	3	24	no	—
82	C₂×C₃²	C₄²×D₃	1	4	yes	—
83	C₂×C₃²	C₂×C₄×Dic₃	1	3	yes	—
84	C₂×C₉	—	2	3	no	—
85	C₂×C₉	—	2	4	no	—
86	C₂×C₉	C₂×C₃×C₄²	1	3	yes	—
87	C₂×C₉	C₂×C₃×C₄²	1	4	yes	—
88	C₂²×C₄	—	9	4	no	—
89	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
90	C₄²	—	9	24	no	—
91	C₄²	3₋¹⁺²⋊C₄	1	4	yes	direct product factor
92	C₂²×C₃	—	3	6	no	—
93	C₂²×C₃	—	3	1	no	—
94	C₂²×C₃	C₂×C₃×C₄×D₃	1	6	yes	—
95	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
96	C₃×C₄	—	3	12	no	—
97	C₃×C₄	—	9	16	no	—
98	C₃×C₄	C₃×C₄×Dic₃	1	12	yes	—
99	Dic₃	—	3	16	no	—
100	C₃²	C₄²×Dic₃	1	1	yes	characteristic
101	C₉	—	2	1	no	—
102	C₉	C₃×C₄³	1	1	yes	characteristic, complemented
103	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
104	C₂×C₄	—	9	24	no	—
105	C₂×C₄	C₄×AGL(1,ℤ/9)	1	12	yes	—
106	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	6	yes	—
107	C₂×C₃	—	3	3	no	—
108	C₂×C₃	—	3	4	no	—
109	C₂×C₃	C₃×C₄²×D₃	1	4	yes	—
110	C₂×C₃	C₂×C₃×C₄×Dic₃	1	3	yes	—
111	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	6	yes	—
112	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
113	C₄	—	9	16	no	—
114	C₄	C₄×3₋¹⁺²⋊C₄	1	12	yes	direct product factor
115	C₃	—	3	1	no	—
116	C₃	C₃×C₄²×Dic₃	1	1	yes	characteristic
117	C₂	C₄²×AGL(1,ℤ/9)	1	4	yes	—
118	C₂	C₂×C₄×3₋¹⁺²⋊C₄	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂×C₄×3₋¹⁺²⋊C₄

C₂

yes

maximal

C₃×C₄²×Dic₃

—

maximal

C₄²×Dic₉

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

—

C₄²×3₋¹⁺²

C₄

yes

complemented

C₄×3₋¹⁺²⋊C₄

C₄

yes

direct product factor

C₂×C₃²×C₄²

D₃

yes

characteristic

C₂×C₃×C₄×Dic₃

—

C₂×C₄²×C₉

—

C₂×C₄²×C₉

C₂×C₃

yes

characteristic

C₂×C₄×Dic₉

C₂×C₃

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×3₋¹⁺²⋊C₄

C₂×C₄

yes

—

C₃×C₄³

—

C₄²×Dic₃

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₃²×C₄²

Dic₃

yes

—

C₃×C₄×Dic₃

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×Dic₉

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₃×C₄

yes

complemented

C₄×Dic₉

C₃×C₄

yes

complemented

C₂²×3₋¹⁺²

C₂²×C₄

yes

—

C₂²×3₋¹⁺²

C₂²×C₄

yes

characteristic

C₄×3₋¹⁺²

C₄²

yes

complemented

3₋¹⁺²⋊C₄

C₄²

yes

direct product factor

C₂×C₃×C₄²

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₃×D₃

yes

characteristic

C₂×C₄×Dic₃

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₂³×C₃

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×Dic₉

C₂×C₃×C₄

yes

—

C₄³

—

Sylow

C₂×3₋¹⁺²

C₂×C₄²

yes

—

C₂×3₋¹⁺²

C₂×C₄²

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×Dic₃

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₃×Dic₃

yes

—

C₄×Dic₃

—

C₂²×C₃²

C₂×C₄×D₃

yes

—

C₂²×C₃²

C₂²×Dic₃

yes

characteristic

C₃²×C₄

C₄×Dic₃

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂²×C₃×C₄

yes

—

C₂²×C₉

C₂²×C₃×C₄

yes

characteristic

C₄×C₉

—

C₄×C₉

C₃×C₄²

yes

complemented

Dic₉

C₃×C₄²

yes

complemented

C₂×C₄²

—

C₂×C₄²

AGL(1,ℤ/9)

yes

characteristic

3₋¹⁺²

C₄³

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×C₄×D₃

yes

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

C₂×Dic₃

—

C₂×C₃²

C₄²×D₃

yes

—

C₂×C₃²

C₂×C₄×Dic₃

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂×C₃×C₄²

yes

—

C₂×C₉

C₂×C₃×C₄²

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

C₄²

—

C₄²

3₋¹⁺²⋊C₄

yes

direct product factor

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₃×C₄×D₃

yes

—

C₂²×C₃

C₂²×C₃×Dic₃

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₃×C₄×Dic₃

yes

—

Dic₃

—

100

C₃²

C₄²×Dic₃

yes

characteristic

101

C₉

—

102

C₉

C₃×C₄³

yes

characteristic, complemented

103

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

C₄×AGL(1,ℤ/9)

yes

—

106

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

C₃×C₄²×D₃

yes

—

110

C₂×C₃

C₂×C₃×C₄×Dic₃

yes

—

111

C₂²

C₂×C₄×AGL(1,ℤ/9)

yes

—

112

C₂²

C₂²×3₋¹⁺²⋊C₄

yes

characteristic

113

C₄

—

114

C₄

C₄×3₋¹⁺²⋊C₄

yes

direct product factor

115

C₃

—

116

C₃

C₃×C₄²×Dic₃

yes

characteristic

117

C₂

C₄²×AGL(1,ℤ/9)

yes

—

118

C₂

C₂×C₄×3₋¹⁺²⋊C₄

yes

—

	subgroup	G/H
center, hypercenter	C₂×C₄²	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₃×C₄³
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄²×3₋¹⁺²	C₂
prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₄²

subgroup

G/H

center, hypercenter

C₂×C₄²

AGL(1,ℤ/9)

commutator, hypocenter

C₉

C₃×C₄³

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

C₂×C₄²×3₋¹⁺²

C₂

prime radical

C₂³×C₃²

C₂²×D₃

cosocle

C₂³×C₉

C₂³×C₃

LCS

G ▷ C₉

UCS

C₁ ◁ C₂×C₄²

N	H ≅ G/N	type
4. C₄²×Dic₉	C₃	semidirect
7. C₄²×3₋¹⁺²	C₄	semidirect
8. C₄×3₋¹⁺²⋊C₄	C₄	direct (N × H)
28. C₄²×C₉	C₃×C₄	semidirect
29. C₄×Dic₉	C₃×C₄	semidirect
32. C₄×3₋¹⁺²	C₄²	semidirect
33. 3₋¹⁺²⋊C₄	C₄²	direct (N × H)
69. C₄×C₉	C₃×C₄²	semidirect
70. Dic₉	C₃×C₄²	semidirect
73. 3₋¹⁺²	C₄³	semidirect
91. C₄²	3₋¹⁺²⋊C₄	direct (N × H)
102. C₉	C₃×C₄³	semidirect
114. C₄	C₄×3₋¹⁺²⋊C₄	direct (N × H)

H ≅ G/N

type

4. C₄²×Dic₉

C₃

semidirect

7. C₄²×3₋¹⁺²

C₄

semidirect

8. C₄×3₋¹⁺²⋊C₄

C₄

direct (N × H)

28. C₄²×C₉

C₃×C₄

semidirect

29. C₄×Dic₉

C₃×C₄

semidirect

32. C₄×3₋¹⁺²

C₄²

semidirect

33. 3₋¹⁺²⋊C₄

C₄²

direct (N × H)

69. C₄×C₉

C₃×C₄²

semidirect

70. Dic₉

C₃×C₄²

semidirect

73. 3₋¹⁺²

C₄³

semidirect

91. C₄²

3₋¹⁺²⋊C₄

direct (N × H)

102. C₉

C₃×C₄³

semidirect

114. C₄

C₄×3₋¹⁺²⋊C₄

direct (N × H)

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

subgroup	index
C₂×C₄²×3₋¹⁺²	2
C₂×C₄×3₋¹⁺²⋊C₄	2
C₃×C₄²×Dic₃	3
C₄²×Dic₉	3

subgroup

index

C₂×C₄²×3₋¹⁺²

C₂×C₄×3₋¹⁺²⋊C₄

C₃×C₄²×Dic₃

C₄²×Dic₉

Subgroups — C₄²×3₋¹⁺²⋊C₄ #13142436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C42×3−1+2⋊C4 #13142436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×3₋¹⁺²⋊C₄ #13142436