Group #129129 — subgroups

Subgroups — #129129

Order 512.

264 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 138 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	#12590	C₂	1	1	yes	maximal, characteristic
2	C₂×#1371	C₂	1	2	yes	maximal
3	#8274	C₂	1	1	yes	maximal, characteristic
4	C₄×#335	C₂	1	1	yes	maximal, characteristic
5	#7154	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	#1767	—	2	1	no	—
9	C₄×C₄⋊C₈	—	2	1	no	—
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#335	C₂²	1	2	yes	—
12	C₂×#335	C₂²	1	2	yes	—
13	#1374	C₄	1	4	yes	—
14	#1371	C₄	1	8	yes	complemented
15	#1277	—	2	1	no	—
16	#1277	—	2	1	no	—
17	C₂²×C₁₆	C₂×C₄	1	2	yes	—
18	C₂²×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₈	—	2	2	no	—
20	C₂×C₄⋊C₈	—	2	2	no	—
21	C₂×#97	—	4	1	no	—
22	C₂×C₄×C₈	—	4	1	no	—
23	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	Q₈	1	2	yes	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄³	D₄	1	1	yes	characteristic
28	C₄×C₁₆	—	2	2	no	—
29	C₄×C₁₆	C₂×C₄	1	4	yes	—
30	#336	—	4	1	no	—
31	#336	—	4	1	no	—
32	#335	—	2	2	no	—
33	#335	—	2	2	no	—
34	#335	C₂×C₄	1	4	yes	—
35	#335	C₂×C₄	1	4	yes	—
36	C₂²×C₈	—	4	2	no	—
37	C₂²×C₈	—	4	2	no	—
38	C₂²×C₈	C₄⋊C₄	1	2	yes	—
39	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄²	C₄⋊C₄	1	2	yes	—
42	C₂×C₄²	D₈	1	1	yes	characteristic
43	C₂×C₄²	D₈	1	1	yes	characteristic
44	C₂×C₄²	Q₁₆	1	2	yes	—
45	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₁₆	—	2	4	no	—
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	4	no	—
49	C₂×C₁₆	C₄²	1	4	yes	—
50	C₄⋊C₈	—	2	4	no	—
51	C₄⋊C₈	—	2	4	no	—
52	C₄⋊C₈	—	4	2	no	—
53	C₄⋊C₈	—	4	2	no	—
54	#97	—	8	2	no	—
55	C₄×C₈	—	2	4	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	4	2	no	—
58	C₄×C₈	—	8	2	no	—
59	C₄×C₈	C₂²×C₄	1	2	yes	—
60	C₄×C₈	#41	1	2	yes	—
61	C₂²×C₄	#107	1	2	yes	—
62	C₂²×C₄	#107	1	2	yes	—
63	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
65	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	4	4	no	—
71	C₂×C₈	—	4	4	no	—
72	C₂×C₈	—	8	2	no	—
73	C₂×C₈	—	8	2	no	—
74	C₂×C₈	—	8	2	no	—
75	C₂×C₈	—	8	2	no	—
76	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₈	#117	1	1	yes	characteristic
78	C₂×C₈	C₄×Q₈	1	2	yes	—
79	C₂×C₈	#131	1	4	yes	—
80	C₄²	—	2	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	—	2	2	no	—
83	C₄²	—	4	2	no	—
84	C₄²	D₁₆	1	2	yes	—
85	C₄²	Q₃₂	1	2	yes	—
86	C₄²	C₄×D₄	1	2	yes	—
87	C₄²	#135	1	2	yes	—
88	C₁₆	—	2	4	no	—
89	C₁₆	—	4	2	no	—
90	C₂³	C₂×#107	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	#366	1	4	yes	—
98	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄	#424	1	2	yes	—
100	C₂×C₄	#427	1	1	yes	characteristic
101	C₂×C₄	#430	1	2	yes	—
102	C₂×C₄	C₄×D₈	1	2	yes	—
103	C₂×C₄	C₄×Q₁₆	1	2	yes	—
104	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
105	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
106	C₂×C₄	#508	1	2	yes	—
107	C₈	—	2	2	no	—
108	C₈	—	4	2	no	—
109	C₈	—	8	4	no	—
110	C₈	—	8	4	no	—
111	C₈	#405	1	2	yes	—
112	C₈	#433	1	2	yes	—
113	C₂²	#1571	1	2	yes	—
114	C₂²	C₄×#107	1	1	yes	characteristic
115	C₂²	C₂×#366	1	1	yes	characteristic
116	C₂²	#2158	1	1	yes	characteristic
117	C₂²	#2161	1	2	yes	—
118	C₄	—	2	4	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	2	no	—
121	C₄	#1572	1	2	yes	—
122	C₄	#1768	1	1	yes	characteristic
123	C₄	#1778	1	1	yes	characteristic
124	C₄	C₄×D₁₆	1	2	yes	—
125	C₄	C₄×Q₃₂	1	2	yes	—
126	C₂	#8278	1	1	yes	characteristic
127	C₂	#12032	1	2	yes	—
128	C₂	#12034	1	2	yes	—
129	C₂	C₄×#366	1	1	yes	characteristic
130	C₂	#12676	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12590

C₂

yes

maximal, characteristic

C₂×#1371

C₂

yes

maximal

#8274

C₂

yes

maximal, characteristic

C₄×#335

C₂

yes

maximal, characteristic

#7154

C₂

yes

maximal

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1767

—

C₄×C₄⋊C₈

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

C₂×#335

C₂²

yes

—

#1374

C₄

yes

—

#1371

C₄

yes

complemented

#1277

—

#1277

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

#335

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

Q₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

#131

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

D₁₆

yes

—

C₄²

Q₃₂

yes

—

C₄²

C₄×D₄

yes

—

C₄²

#135

yes

—

C₁₆

—

C₁₆

—

C₂³

C₂×#107

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#366

yes

—

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄

#424

yes

—

100

C₂×C₄

#427

yes

characteristic

101

C₂×C₄

#430

yes

—

102

C₂×C₄

C₄×D₈

yes

—

103

C₂×C₄

C₄×Q₁₆

yes

—

104

C₂×C₄

C₂×D₁₆

yes

characteristic

105

C₂×C₄

C₂×Q₃₂

yes

characteristic

106

C₂×C₄

#508

yes

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

#405

yes

—

112

C₈

#433

yes

—

113

C₂²

#1571

yes

—

114

C₂²

C₄×#107

yes

characteristic

115

C₂²

C₂×#366

yes

characteristic

116

C₂²

#2158

yes

characteristic

117

C₂²

#2161

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#1572

yes

—

122

C₄

#1768

yes

characteristic

123

C₄

#1778

yes

characteristic

124

C₄

C₄×D₁₆

yes

—

125

C₄

C₄×Q₃₂

yes

—

126

C₂

#8278

yes

characteristic

127

C₂

#12032

yes

—

128

C₂

#12034

yes

—

129

C₂

C₄×#366

yes

characteristic

130

C₂

#12676

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₈

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×#107

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
14. #1371	C₄	semidirect

H ≅ G/N

type

14. #1371

C₄

semidirect

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

subgroup	index
#12590	2
C₂×#1371	2
#8274	2
C₄×#335	2
#7154	2

subgroup

index

Subgroups — #129129

Conjugacy classes of subgroups

Complements

Maximal subgroups