Group #146257 — subgroups

Subgroups — #146257

Order 512.

291 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 84 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	#13213	C₂	1	1	yes	maximal, characteristic
2	#13206	C₂	1	1	yes	maximal, characteristic
3	#12904	C₂	1	1	yes	maximal, characteristic, complemented
4	#12892	C₂	1	1	yes	maximal, characteristic, complemented
5	#9773	C₂	1	1	yes	maximal, characteristic
6	#8793	C₂	1	1	yes	maximal, characteristic, complemented
7	#7140	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#493	C₂²	1	1	yes	characteristic
9	C₂×#365	—	2	2	no	—
10	C₂×#361	—	2	2	no	—
11	C₂×#357	—	2	2	no	—
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×#346	C₂²	1	1	yes	characteristic
14	#1848	C₂²	1	1	yes	characteristic
15	#1848	C₂²	1	1	yes	characteristic
16	#1846	C₂²	1	1	yes	characteristic
17	#1545	—	2	2	no	—
18	#1450	C₂²	1	1	yes	characteristic
19	#1381	—	2	2	no	—
20	#1286	—	2	2	no	—
21	C₂²×D₈	—	2	1	no	—
22	C₂²×D₈	—	2	1	no	—
23	C₂×#127	—	2	1	no	—
24	C₂×M₃₂	D₄	1	2	yes	—
25	C₂²×C₁₆	D₄	1	2	yes	—
26	#493	—	2	1	no	—
27	#493	—	2	1	no	—
28	#493	—	2	1	no	—
29	#493	Q₈	1	2	yes	—
30	#431	D₄	1	2	yes	—
31	C₂×#108	—	2	1	no	—
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	C₂×#108	—	2	1	no	—
35	C₂×#107	—	2	1	no	—
36	C₂×#106	—	2	1	no	—
37	#423	—	2	1	no	—
38	#423	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#382	—	2	1	no	—
41	#365	—	4	4	no	—
42	#361	—	4	4	no	—
43	#357	—	4	4	no	—
44	#346	—	2	1	no	—
45	#346	—	2	1	no	—
46	#346	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	#331	—	4	4	no	—
49	#322	—	2	1	no	—
50	C₈²	—	2	1	no	—
51	C₂²×D₄	—	4	1	no	—
52	C₂²×D₄	—	4	2	no	—
53	C₂×D₈	—	2	2	no	—
54	C₂×D₈	—	2	2	no	—
55	C₂×D₈	—	4	1	no	—
56	C₂×D₈	—	4	1	no	—
57	C₂×D₈	—	4	2	no	—
58	C₂×D₈	—	4	2	no	—
59	C₂×D₈	—	4	2	no	—
60	C₂×D₈	—	4	2	no	—
61	C₂×M₁₆	—	4	1	no	—
62	C₂×M₁₆	—	4	1	no	—
63	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	#127	—	2	2	no	—
66	#127	—	4	1	no	—
67	#127	—	4	2	no	—
68	C₂×C₄⋊C₄	—	4	1	no	—
69	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
70	M₃₂	—	2	2	no	—
71	M₃₂	SD₁₆	1	4	yes	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	2	2	no	—
76	#108	—	4	2	no	—
77	#108	—	4	2	no	—
78	#108	—	4	2	no	—
79	#108	—	4	2	no	—
80	#107	—	4	2	no	—
81	#106	—	4	2	no	—
82	C₄⋊C₈	—	2	2	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₂²⋊C₈	—	2	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₄×C₈	#41	1	1	yes	characteristic
90	C₄×C₈	#41	1	1	yes	characteristic
91	#95	—	4	2	no	—
92	C₂⁴	—	8	1	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	4	no	—
95	C₂×D₄	—	8	2	no	—
96	C₂×D₄	—	8	1	no	—
97	C₂×D₄	—	8	1	no	—
98	C₂×D₄	—	8	1	no	—
99	C₂×D₄	—	8	1	no	—
100	C₂×D₄	—	8	1	no	—
101	C₂×D₄	—	16	4	no	—
102	C₂²×C₄	—	8	1	no	—
103	C₂²×C₄	#127	1	1	yes	characteristic
104	C₂²×C₄	#135	1	2	yes	—
105	D₈	—	4	4	no	—
106	D₈	—	4	4	no	—
107	D₈	—	8	2	no	—
108	D₈	—	8	2	no	—
109	M₁₆	—	8	1	no	—
110	M₁₆	—	8	1	no	—
111	M₁₆	—	8	1	no	—
112	M₁₆	—	8	1	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	8	1	no	—
115	C₂×C₈	—	8	1	no	—
116	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₈	#123	1	1	yes	characteristic
119	C₂×C₈	#123	1	1	yes	characteristic
120	C₂×C₈	C₂×SD₁₆	1	2	yes	—
121	C₂×C₈	#135	1	2	yes	—
122	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
124	C₄⋊C₄	—	8	1	no	—
125	C₄⋊C₄	—	8	1	no	—
126	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
127	C₄²	#123	1	1	yes	characteristic
128	C₄²	AGL(1,ℤ/8)	1	2	yes	—
129	C₁₆	—	2	4	no	—
130	C₁₆	—	4	2	no	—
131	C₂³	—	8	2	no	—
132	C₂³	—	16	4	no	—
133	C₂³	—	16	1	no	—
134	C₂³	—	16	1	no	—
135	C₂³	#495	1	1	yes	characteristic
136	D₄	—	8	2	no	—
137	D₄	—	8	2	no	—
138	D₄	—	16	4	no	—
139	D₄	—	16	4	no	—
140	D₄	—	16	1	no	—
141	D₄	—	16	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	16	2	no	—
144	C₂×C₄	—	16	1	no	—
145	C₂×C₄	#399	1	1	yes	characteristic
146	C₂×C₄	#478	1	2	yes	—
147	C₂×C₄	#482	1	2	yes	—
148	C₂×C₄	#492	1	1	yes	characteristic
149	C₂×C₄	#495	1	1	yes	characteristic
150	C₂×C₄	#496	1	1	yes	characteristic
151	C₂×C₄	#496	1	1	yes	characteristic
152	C₂×C₄	#496	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	—	16	1	no	—
155	C₈	—	16	1	no	—
156	C₈	#476	1	2	yes	—
157	C₈	#478	1	2	yes	—
158	C₈	#481	1	2	yes	—
159	C₈	#482	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	16	2	no	—
162	C₂²	—	16	4	no	—
163	C₂²	—	32	2	no	—
164	C₂²	—	32	2	no	—
165	C₂²	#1708	1	1	yes	characteristic
166	C₂²	#1717	1	1	yes	characteristic
167	C₂²	#2092	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	32	1	no	—
171	C₄	#1715	1	1	yes	characteristic
172	C₄	#1717	1	1	yes	characteristic
173	C₄	#2092	1	1	yes	characteristic
174	C₄	#2094	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	32	2	no	—
178	C₂	#10677	1	1	yes	characteristic
179	C₂	#13523	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13213

C₂

yes

maximal, characteristic

#13206

C₂

yes

maximal, characteristic

#12904

C₂

yes

maximal, characteristic, complemented

#12892

C₂

yes

maximal, characteristic, complemented

#9773

C₂

yes

maximal, characteristic

#8793

C₂

yes

maximal, characteristic, complemented

#7140

C₂

yes

maximal, characteristic, complemented

C₂×#493

C₂²

yes

—

—

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1846

C₂²

yes

characteristic

#1545

—

#1450

C₂²

yes

characteristic

#1381

—

#1286

—

C₂²×D₈

—

C₂²×D₈

—

C₂×#127

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

—

—

—

Q₈

yes

—

#431

D₄

yes

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

C₄×C₁₆

—

#331

—

#322

—

C₈²

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

SD₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂²×C₄

—

103

C₂²×C₄

#127

yes

characteristic

104

C₂²×C₄

#135

yes

—

105

D₈

—

106

D₈

—

107

D₈

—

108

D₈

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂²⋊Q₈

yes

characteristic

117

C₂×C₈

C₂²⋊Q₈

yes

characteristic

118

C₂×C₈

#123

yes

characteristic

119

C₂×C₈

#123

yes

characteristic

120

C₂×C₈

C₂×SD₁₆

yes

—

121

C₂×C₈

#135

yes

—

122

C₂×C₈

AGL(1,ℤ/8)

yes

—

123

C₂×C₈

AGL(1,ℤ/8)

yes

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄²

C₂²⋊Q₈

yes

characteristic

127

C₄²

#123

yes

characteristic

128

C₄²

AGL(1,ℤ/8)

yes

—

129

C₁₆

—

130

C₁₆

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

#495

yes

characteristic

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#399

yes

characteristic

146

C₂×C₄

#478

yes

—

147

C₂×C₄

#482

yes

—

148

C₂×C₄

#492

yes

characteristic

149

C₂×C₄

#495

yes

characteristic

150

C₂×C₄

#496

yes

characteristic

151

C₂×C₄

#496

yes

characteristic

152

C₂×C₄

#496

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

#476

yes

—

157

C₈

#478

yes

—

158

C₈

#481

yes

—

159

C₈

#482

yes

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#1708

yes

characteristic

166

C₂²

#1717

yes

characteristic

167

C₂²

#2092

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#1715

yes

characteristic

172

C₄

#1717

yes

characteristic

173

C₄

#2092

yes

characteristic

174

C₄

#2094

yes

characteristic

175

C₂

—

176

C₂

—

177

C₂

—

178

C₂

#10677

yes

characteristic

179

C₂

#13523

yes

—

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2092

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂×#493

C₂²

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #12904	C₂	semidirect
4. #12892	C₂	semidirect
6. #8793	C₂	semidirect
7. #7140	C₂	semidirect

H ≅ G/N

type

3. #12904

C₂

semidirect

4. #12892

C₂

semidirect

6. #8793

C₂

semidirect

7. #7140

C₂

semidirect

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

subgroup	index
#13213	2
#13206	2
#12904	2
#12892	2
#9773	2
#8793	2
#7140	2

subgroup

index

Subgroups — #146257

Conjugacy classes of subgroups

Complements

Maximal subgroups