Group #129132 — subgroups

Subgroups — #129132

Order 512.

237 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 112 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	#12594	C₂	1	1	yes	maximal, characteristic
2	#12009	C₂	1	1	yes	maximal, characteristic
3	#12000	C₂	1	1	yes	maximal, characteristic
4	#8275	C₂	1	1	yes	maximal, characteristic
5	#8269	C₂	1	1	yes	maximal, characteristic
6	#7159	C₂	1	1	yes	maximal, characteristic
7	#7158	C₂	1	1	yes	maximal, characteristic
8	C₄×M₃₂	C₂²	1	1	yes	characteristic
9	C₄×M₃₂	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊C₈	—	2	1	no	—
11	C₄×#97	C₂²	1	1	yes	characteristic
12	#1564	C₂²	1	1	yes	characteristic
13	#1564	C₂²	1	1	yes	characteristic
14	C₂×#335	C₂²	1	1	yes	characteristic
15	C₂×#334	C₂²	1	1	yes	characteristic
16	#1452	—	2	1	no	—
17	#1375	C₄	1	4	yes	—
18	#1372	C₄	1	2	yes	complemented
19	#1371	C₄	1	2	yes	complemented
20	#1368	C₄	1	4	yes	complemented
21	#1277	—	2	1	no	—
22	#1276	—	2	1	no	—
23	C₂×M₃₂	C₂×C₄	1	2	yes	—
24	C₂×M₃₂	C₂×C₄	1	2	yes	—
25	#431	—	2	1	no	—
26	#431	—	2	1	no	—
27	C₂×C₄⋊C₈	—	2	1	no	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×#97	C₂×C₄	1	2	yes	—
30	C₂×#97	Q₈	1	2	yes	—
31	C₂×C₄×C₈	—	4	1	no	—
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	C₄³	D₄	1	1	yes	characteristic
35	#346	—	2	1	no	—
36	#346	C₂×C₄	1	2	yes	—
37	C₄×C₁₆	—	2	1	no	—
38	C₄×C₁₆	C₂×C₄	1	2	yes	—
39	#336	—	4	1	no	—
40	#335	—	2	1	no	—
41	#335	—	2	1	no	—
42	#335	C₂×C₄	1	2	yes	—
43	#335	C₂×C₄	1	1	yes	characteristic
44	#335	C₂×C₄	1	1	yes	characteristic
45	#334	—	2	1	no	—
46	#334	—	2	1	no	—
47	#334	C₂×C₄	1	2	yes	—
48	#334	C₂×C₄	1	2	yes	—
49	#322	—	4	2	no	—
50	C₂²×C₈	—	4	1	no	—
51	C₂²×C₈	—	4	1	no	—
52	C₂²×C₈	C₄⋊C₄	1	2	yes	—
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄²	—	2	1	no	—
56	C₂×C₄²	—	2	1	no	—
57	C₂×C₄²	C₄⋊C₄	1	2	yes	—
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	M₃₂	—	2	4	no	—
60	M₃₂	—	2	4	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	C₄²	1	4	yes	—
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	2	1	no	—
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	4	1	no	—
69	C₄⋊C₈	—	4	1	no	—
70	C₂²⋊C₈	—	4	1	no	—
71	C₂²⋊C₈	—	4	1	no	—
72	#97	—	2	4	no	—
73	#97	—	2	4	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	4	1	no	—
77	C₄×C₈	—	4	1	no	—
78	C₄×C₈	—	8	2	no	—
79	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₄×C₈	#41	1	1	yes	characteristic
82	C₄×C₈	#41	1	1	yes	characteristic
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂²×C₄	#137	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	8	1	no	—
97	C₂×C₈	—	8	1	no	—
98	C₂×C₈	—	8	1	no	—
99	C₂×C₈	—	8	1	no	—
100	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₈	#117	1	1	yes	characteristic
102	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₈	#131	1	4	yes	—
105	C₄²	—	2	2	no	—
106	C₄²	—	4	1	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	8	1	no	—
109	C₄²	#108	1	4	yes	—
110	C₄²	C₄×D₄	1	1	yes	characteristic
111	C₄²	C₄×D₄	1	1	yes	characteristic
112	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₄²	#137	1	1	yes	characteristic
114	C₁₆	—	2	4	no	—
115	C₁₆	—	4	2	no	—
116	C₂³	#428	1	1	yes	characteristic
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄	#424	1	2	yes	—
125	C₂×C₄	#428	1	1	yes	characteristic
126	C₂×C₄	C₂×#108	1	2	yes	—
127	C₂×C₄	#440	1	1	yes	characteristic
128	C₂×C₄	#440	1	1	yes	characteristic
129	C₂×C₄	#441	1	1	yes	characteristic
130	C₂×C₄	#442	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	2	1	no	—
133	C₈	—	4	2	no	—
134	C₈	—	8	2	no	—
135	C₈	—	8	2	no	—
136	C₈	—	8	1	no	—
137	C₈	—	8	1	no	—
138	C₈	—	8	2	no	—
139	C₈	#405	1	2	yes	—
140	C₈	#433	1	2	yes	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#1573	1	2	yes	—
144	C₂²	#1776	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	4	no	—
149	C₄	#1574	1	2	yes	—
150	C₄	#1768	1	1	yes	characteristic
151	C₄	C₄×#108	1	1	yes	characteristic
152	C₄	#2180	1	1	yes	characteristic
153	C₄	AGL(1,ℤ/16)	1	1	yes	characteristic
154	C₂	—	2	2	no	—
155	C₂	#8279	1	1	yes	characteristic
156	C₂	#12677	1	1	yes	characteristic
157	C₂	#12678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12594

C₂

yes

maximal, characteristic

#12009

C₂

yes

maximal, characteristic

#12000

C₂

yes

maximal, characteristic

#8275

C₂

yes

maximal, characteristic

#8269

C₂

yes

maximal, characteristic

#7159

C₂

yes

maximal, characteristic

#7158

C₂

yes

maximal, characteristic

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

C₄×C₄⋊C₈

—

C₄×#97

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

C₂×#335

C₂²

yes

characteristic

C₂×#334

C₂²

yes

characteristic

#1452

—

#1375

C₄

yes

—

#1372

C₄

yes

complemented

#1371

C₄

yes

complemented

#1368

C₄

yes

complemented

#1277

—

#1276

—

C₂×M₃₂

C₂×C₄

yes

—

C₂×M₃₂

C₂×C₄

yes

—

#431

—

#431

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

C₂×C₄

yes

—

C₂×#97

Q₈

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

#346

—

#346

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

#335

C₂×C₄

yes

characteristic

#335

C₂×C₄

yes

—

—

C₂×C₄

yes

—

#334

C₂×C₄

yes

—

#322

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂×C₄²

yes

characteristic

101

C₂×C₈

#117

yes

characteristic

102

C₂×C₈

C₄×Q₈

yes

characteristic

103

C₂×C₈

C₄×Q₈

yes

characteristic

104

C₂×C₈

#131

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

#108

yes

—

110

C₄²

C₄×D₄

yes

characteristic

111

C₄²

C₄×D₄

yes

characteristic

112

C₄²

AGL(1,ℤ/8)

yes

characteristic

113

C₄²

#137

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₂³

#428

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

124

C₂×C₄

#424

yes

—

125

C₂×C₄

#428

yes

characteristic

126

C₂×C₄

C₂×#108

yes

—

127

C₂×C₄

#440

yes

characteristic

128

C₂×C₄

#440

yes

characteristic

129

C₂×C₄

#441

yes

characteristic

130

C₂×C₄

#442

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

#405

yes

—

140

C₈

#433

yes

—

141

C₂²

—

142

C₂²

—

143

C₂²

#1573

yes

—

144

C₂²

#1776

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#1574

yes

—

150

C₄

#1768

yes

characteristic

151

C₄

C₄×#108

yes

characteristic

152

C₄

#2180

yes

characteristic

153

C₄

AGL(1,ℤ/16)

yes

characteristic

154

C₂

—

155

C₂

#8279

yes

characteristic

156

C₂

#12677

yes

characteristic

157

C₂

#12678

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1776
prime radical	C₂³	#428
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#442

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#97 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
18. #1372	C₄	semidirect
19. #1371	C₄	semidirect
20. #1368	C₄	semidirect

H ≅ G/N

type

18. #1372

C₄

semidirect

19. #1371

C₄

semidirect

20. #1368

C₄

semidirect

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

subgroup	index
#12594	2
#12009	2
#12000	2
#8275	2
#8269	2
#7159	2
#7158	2

subgroup

index

Subgroups — #129132

Conjugacy classes of subgroups

Complements

Maximal subgroups