Group #13153559 — subgroups

Subgroups — C₂³×3₋¹⁺²⋊C₈ #13153559

Order 1728.

1132 conjugacy classes of subgroups; only the 692 normal subgroups are shown.

normal subgroups only

This group has 1132 conjugacy classes of subgroups — too many to draw. Only its 692 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	C₂³×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂²×3₋¹⁺²⋊C₈	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×C₃⋊C₈	—	3	1	no	maximal
4	C₂³×C₉⋊C₈	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×3₋¹⁺²	C₄	1	1	yes	characteristic
6	C₂²×C₄×3₋¹⁺²	C₄	1	7	yes	—
7	C₂²×C₄×3₋¹⁺²	C₂²	1	7	yes	—
8	C₂×3₋¹⁺²⋊C₈	C₂²	1	28	yes	direct product factor
9	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
10	C₂²×C₃×C₃⋊C₈	—	3	14	no	—
11	C₂³×C₄×C₉	—	2	1	no	—
12	C₂³×C₄×C₉	C₂×C₃	1	1	yes	characteristic
13	C₂²×C₉⋊C₈	C₂×C₃	1	14	yes	complemented
14	C₂³×3₋¹⁺²	C₈	1	8	yes	complemented
15	C₂³×3₋¹⁺²	C₂×C₄	1	7	yes	—
16	C₂×C₄×3₋¹⁺²	C₂×C₄	1	21	yes	—
17	C₂×C₄×3₋¹⁺²	C₂³	1	7	yes	—
18	3₋¹⁺²⋊C₈	C₂³	1	8	yes	direct product factor
19	C₂³×C₃×C₈	—	9	1	no	—
20	C₂³×C₃⋊C₈	—	3	1	no	—
21	C₂⁴×C₃²	Dic₃	1	1	yes	characteristic
22	C₂²×C₃²×C₄	Dic₃	1	7	yes	—
23	C₂²×C₃²×C₄	C₂×D₃	1	7	yes	—
24	C₂⁴×C₉	—	2	1	no	—
25	C₂⁴×C₉	C₃×C₄	1	1	yes	characteristic
26	C₂×C₃×C₃⋊C₈	—	3	28	no	—
27	C₂²×C₄×C₉	—	2	7	no	—
28	C₂²×C₄×C₉	—	2	7	no	—
29	C₂²×C₄×C₉	C₃×C₄	1	7	yes	—
30	C₂²×C₄×C₉	C₂²×C₃	1	7	yes	—
31	C₂×C₉⋊C₈	C₂²×C₃	1	28	yes	complemented
32	C₂²×3₋¹⁺²	C₂×C₈	1	28	yes	complemented
33	C₂²×3₋¹⁺²	C₂²×C₄	1	7	yes	—
34	C₄×3₋¹⁺²	C₂²×C₄	1	7	yes	—
35	C₄×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
36	C₂³×C₃×C₄	—	3	1	no	—
37	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
38	C₂²×C₃×C₈	—	9	14	no	—
39	C₂²×C₃⋊C₈	—	3	14	no	—
40	C₂³×C₃²	C₃⋊C₈	1	8	yes	—
41	C₂³×C₃²	C₂×Dic₃	1	7	yes	—
42	C₂×C₃²×C₄	C₂×Dic₃	1	21	yes	—
43	C₂×C₃²×C₄	C₂²×D₃	1	7	yes	—
44	C₂³×C₉	—	2	8	no	—
45	C₂³×C₉	—	2	7	no	—
46	C₂³×C₉	C₃×C₈	1	8	yes	complemented
47	C₂³×C₉	C₂×C₃×C₄	1	7	yes	—
48	C₃×C₃⋊C₈	—	3	8	no	—
49	C₂×C₄×C₉	—	2	21	no	—
50	C₂×C₄×C₉	—	2	7	no	—
51	C₂×C₄×C₉	C₂×C₃×C₄	1	21	yes	—
52	C₂×C₄×C₉	C₂³×C₃	1	7	yes	—
53	C₉⋊C₈	C₂³×C₃	1	8	yes	complemented
54	C₂³×C₈	—	9	1	no	Sylow
55	C₂×3₋¹⁺²	C₂²×C₈	1	14	yes	complemented
56	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
57	C₂⁴×C₃	—	3	1	no	—
58	C₂⁴×C₃	C₃×Dic₃	1	1	yes	characteristic
59	C₂²×C₃×C₄	—	3	7	no	—
60	C₂²×C₃×C₄	—	3	7	no	—
61	C₂²×C₃×C₄	C₃×Dic₃	1	7	yes	—
62	C₂²×C₃×C₄	C₂×C₃×D₃	1	7	yes	—
63	C₂×C₃×C₈	—	9	28	no	—
64	C₂×C₃⋊C₈	—	3	28	no	—
65	C₂²×C₃²	C₂×C₃⋊C₈	1	28	yes	—
66	C₂²×C₃²	C₂²×Dic₃	1	7	yes	—
67	C₃²×C₄	C₂²×Dic₃	1	7	yes	—
68	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
69	C₂²×C₉	—	2	28	no	—
70	C₂²×C₉	—	2	7	no	—
71	C₂²×C₉	C₂×C₃×C₈	1	28	yes	complemented
72	C₂²×C₉	C₂²×C₃×C₄	1	7	yes	—
73	C₄×C₉	—	2	1	no	—
74	C₄×C₉	—	2	7	no	—
75	C₄×C₉	C₂²×C₃×C₄	1	7	yes	—
76	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
77	C₂³×C₄	AGL(1,ℤ/9)	1	1	yes	characteristic
78	C₂²×C₈	—	9	14	no	—
79	3₋¹⁺²	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₃	—	3	8	no	—
81	C₂³×C₃	—	3	7	no	—
82	C₂³×C₃	C₃×C₃⋊C₈	1	8	yes	—
83	C₂³×C₃	C₂×C₃×Dic₃	1	7	yes	—
84	C₂×C₃×C₄	—	3	21	no	—
85	C₂×C₃×C₄	—	3	7	no	—
86	C₂×C₃×C₄	C₂×C₃×Dic₃	1	21	yes	—
87	C₂×C₃×C₄	C₂²×C₃×D₃	1	7	yes	—
88	C₃×C₈	—	9	8	no	—
89	C₃⋊C₈	—	3	8	no	—
90	C₂×C₃²	C₂²×C₃⋊C₈	1	14	yes	—
91	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
92	C₂×C₉	—	2	14	no	—
93	C₂×C₉	—	2	1	no	—
94	C₂×C₉	C₂²×C₃×C₈	1	14	yes	complemented
95	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
96	C₂⁴	3₋¹⁺²⋊C₄	1	1	yes	characteristic
97	C₂²×C₄	3₋¹⁺²⋊C₄	1	7	yes	—
98	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	7	yes	—
99	C₂×C₈	—	9	28	no	—
100	C₂²×C₃	—	3	7	no	—
101	C₂²×C₃	—	3	28	no	—
102	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	28	yes	—
103	C₂²×C₃	C₂²×C₃×Dic₃	1	7	yes	—
104	C₃×C₄	—	3	1	no	—
105	C₃×C₄	—	3	7	no	—
106	C₃×C₄	C₂²×C₃×Dic₃	1	7	yes	—
107	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
108	C₃²	C₂³×C₃⋊C₈	1	1	yes	characteristic
109	C₉	—	2	1	no	—
110	C₉	C₂³×C₃×C₈	1	1	yes	characteristic, complemented
111	C₂³	3₋¹⁺²⋊C₈	1	8	yes	direct product factor
112	C₂³	C₂×3₋¹⁺²⋊C₄	1	7	yes	—
113	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	21	yes	—
114	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	7	yes	—
115	C₈	—	9	8	no	—
116	C₂×C₃	—	3	14	no	—
117	C₂×C₃	—	3	1	no	—
118	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	14	yes	—
119	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
120	C₂²	C₂×3₋¹⁺²⋊C₈	1	28	yes	direct product factor
121	C₂²	C₂²×3₋¹⁺²⋊C₄	1	7	yes	—
122	C₄	C₂²×3₋¹⁺²⋊C₄	1	7	yes	—
123	C₄	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
124	C₃	—	3	1	no	—
125	C₃	C₂³×C₃×C₃⋊C₈	1	1	yes	characteristic
126	C₂	C₂²×3₋¹⁺²⋊C₈	1	14	yes	direct product factor
127	C₂	C₂³×3₋¹⁺²⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₄×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂²×3₋¹⁺²⋊C₈

C₂

yes

maximal, direct product factor

C₂³×C₃×C₃⋊C₈

—

maximal

C₂³×C₉⋊C₈

C₃

yes

maximal, characteristic, complemented

C₂⁴×3₋¹⁺²

C₄

yes

characteristic

C₂²×C₄×3₋¹⁺²

C₄

yes

—

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂×3₋¹⁺²⋊C₈

C₂²

yes

direct product factor

C₂³×C₃²×C₄

D₃

yes

characteristic

C₂²×C₃×C₃⋊C₈

—

C₂³×C₄×C₉

—

C₂³×C₄×C₉

C₂×C₃

yes

characteristic

C₂²×C₉⋊C₈

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

C₈

yes

complemented

C₂³×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

C₂³

yes

—

3₋¹⁺²⋊C₈

C₂³

yes

direct product factor

C₂³×C₃×C₈

—

C₂³×C₃⋊C₈

—

C₂⁴×C₃²

Dic₃

yes

characteristic

C₂²×C₃²×C₄

Dic₃

yes

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂⁴×C₉

—

C₂⁴×C₉

C₃×C₄

yes

characteristic

C₂×C₃×C₃⋊C₈

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₃×C₄

yes

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂×C₉⋊C₈

C₂²×C₃

yes

complemented

C₂²×3₋¹⁺²

C₂×C₈

yes

complemented

C₂²×3₋¹⁺²

C₂²×C₄

yes

—

C₄×3₋¹⁺²

C₂²×C₄

yes

—

C₄×3₋¹⁺²

C₂⁴

yes

characteristic

C₂³×C₃×C₄

—

C₂³×C₃×C₄

C₃×D₃

yes

characteristic

C₂²×C₃×C₈

—

C₂²×C₃⋊C₈

—

C₂³×C₃²

C₃⋊C₈

yes

—

C₂³×C₃²

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₃×C₈

yes

complemented

C₂³×C₉

C₂×C₃×C₄

yes

—

C₃×C₃⋊C₈

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₂³×C₃

yes

—

C₉⋊C₈

C₂³×C₃

yes

complemented

C₂³×C₈

—

Sylow

C₂×3₋¹⁺²

C₂²×C₈

yes

complemented

C₂×3₋¹⁺²

C₂³×C₄

yes

characteristic

C₂⁴×C₃

—

C₂⁴×C₃

C₃×Dic₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃×Dic₃

yes

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×C₃⋊C₈

yes

—

C₂²×C₃²

C₂²×Dic₃

yes

—

C₃²×C₄

C₂²×Dic₃

yes

—

C₃²×C₄

C₂³×D₃

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×C₈

yes

complemented

C₂²×C₉

C₂²×C₃×C₄

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂²×C₃×C₄

yes

—

C₄×C₉

C₂⁴×C₃

yes

characteristic

C₂³×C₄

AGL(1,ℤ/9)

yes

characteristic

C₂²×C₈

—

3₋¹⁺²

C₂³×C₈

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₃×C₃⋊C₈

yes

—

C₂³×C₃

C₂×C₃×Dic₃

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

C₂×C₃×C₄

C₂²×C₃×D₃

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

C₂²×C₃⋊C₈

yes

—

C₂×C₃²

C₂³×Dic₃

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²×C₃×C₈

yes

complemented

C₂×C₉

C₂³×C₃×C₄

yes

characteristic

C₂⁴

3₋¹⁺²⋊C₄

yes

characteristic

C₂²×C₄

3₋¹⁺²⋊C₄

yes

—

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

C₂×C₈

—

100

C₂²×C₃

—

101

C₂²×C₃

—

102

C₂²×C₃

C₂×C₃×C₃⋊C₈

yes

—

103

C₂²×C₃

C₂²×C₃×Dic₃

yes

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

C₂²×C₃×Dic₃

yes

—

107

C₃×C₄

C₂³×C₃×D₃

yes

characteristic

108

C₃²

C₂³×C₃⋊C₈

yes

characteristic

109

C₉

—

110

C₉

C₂³×C₃×C₈

yes

characteristic, complemented

111

C₂³

3₋¹⁺²⋊C₈

yes

direct product factor

112

C₂³

C₂×3₋¹⁺²⋊C₄

yes

—

113

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

114

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

—

115

C₈

—

116

C₂×C₃

—

117

C₂×C₃

—

118

C₂×C₃

C₂²×C₃×C₃⋊C₈

yes

—

119

C₂×C₃

C₂³×C₃×Dic₃

yes

characteristic

120

C₂²

C₂×3₋¹⁺²⋊C₈

yes

direct product factor

121

C₂²

C₂²×3₋¹⁺²⋊C₄

yes

—

122

C₄

C₂²×3₋¹⁺²⋊C₄

yes

—

123

C₄

C₂³×AGL(1,ℤ/9)

yes

characteristic

124

C₃

—

125

C₃

C₂³×C₃×C₃⋊C₈

yes

characteristic

126

C₂

C₂²×3₋¹⁺²⋊C₈

yes

direct product factor

127

C₂

C₂³×3₋¹⁺²⋊C₄

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₂³×C₃×C₈
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂³×C₄×3₋¹⁺²	C₂
socle	C₂⁴×C₃	C₃×Dic₃
prime radical	C₂⁴×C₃²	Dic₃
cosocle	C₄×C₉	C₂⁴×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂³×C₄

subgroup

G/H

center, hypercenter

C₂³×C₄

AGL(1,ℤ/9)

commutator, hypocenter

C₉

C₂³×C₃×C₈

Frattini

C₃×C₄

C₂³×C₃×D₃

Fitting

C₂³×C₄×3₋¹⁺²

C₂

socle

C₂⁴×C₃

C₃×Dic₃

prime radical

C₂⁴×C₃²

Dic₃

cosocle

C₄×C₉

C₂⁴×C₃

LCS

G ▷ C₉

UCS

C₁ ◁ C₂³×C₄

N	H ≅ G/N	type
2. C₂²×3₋¹⁺²⋊C₈	C₂	direct (N × H)
4. C₂³×C₉⋊C₈	C₃	semidirect
8. C₂×3₋¹⁺²⋊C₈	C₂²	direct (N × H)
13. C₂²×C₉⋊C₈	C₂×C₃	semidirect
14. C₂³×3₋¹⁺²	C₈	semidirect
18. 3₋¹⁺²⋊C₈	C₂³	direct (N × H)
31. C₂×C₉⋊C₈	C₂²×C₃	semidirect
32. C₂²×3₋¹⁺²	C₂×C₈	semidirect
46. C₂³×C₉	C₃×C₈	semidirect
53. C₉⋊C₈	C₂³×C₃	semidirect
55. C₂×3₋¹⁺²	C₂²×C₈	semidirect
71. C₂²×C₉	C₂×C₃×C₈	semidirect
79. 3₋¹⁺²	C₂³×C₈	semidirect
94. C₂×C₉	C₂²×C₃×C₈	semidirect
110. C₉	C₂³×C₃×C₈	semidirect
111. C₂³	3₋¹⁺²⋊C₈	direct (N × H)
120. C₂²	C₂×3₋¹⁺²⋊C₈	direct (N × H)
126. C₂	C₂²×3₋¹⁺²⋊C₈	direct (N × H)

H ≅ G/N

type

2. C₂²×3₋¹⁺²⋊C₈

C₂

direct (N × H)

4. C₂³×C₉⋊C₈

C₃

semidirect

8. C₂×3₋¹⁺²⋊C₈

C₂²

direct (N × H)

13. C₂²×C₉⋊C₈

C₂×C₃

semidirect

14. C₂³×3₋¹⁺²

C₈

semidirect

18. 3₋¹⁺²⋊C₈

C₂³

direct (N × H)

31. C₂×C₉⋊C₈

C₂²×C₃

semidirect

32. C₂²×3₋¹⁺²

C₂×C₈

semidirect

46. C₂³×C₉

C₃×C₈

semidirect

53. C₉⋊C₈

C₂³×C₃

semidirect

55. C₂×3₋¹⁺²

C₂²×C₈

semidirect

71. C₂²×C₉

C₂×C₃×C₈

semidirect

79. 3₋¹⁺²

C₂³×C₈

semidirect

94. C₂×C₉

C₂²×C₃×C₈

semidirect

110. C₉

C₂³×C₃×C₈

semidirect

111. C₂³

3₋¹⁺²⋊C₈

direct (N × H)

120. C₂²

C₂×3₋¹⁺²⋊C₈

direct (N × H)

126. C₂

C₂²×3₋¹⁺²⋊C₈

direct (N × H)

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

subgroup	index
C₂³×C₄×3₋¹⁺²	2
C₂²×3₋¹⁺²⋊C₈	2
C₂³×C₃×C₃⋊C₈	3
C₂³×C₉⋊C₈	3

subgroup

index

C₂³×C₄×3₋¹⁺²

C₂²×3₋¹⁺²⋊C₈

C₂³×C₃×C₃⋊C₈

C₂³×C₉⋊C₈

Subgroups — C₂³×3₋¹⁺²⋊C₈ #13153559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×3−1+2⋊C8 #13153559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×3₋¹⁺²⋊C₈ #13153559