Group #146229 — subgroups

Subgroups — #146229

Order 512.

273 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 88 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	#13212	C₂	1	1	yes	maximal, characteristic
2	#13207	C₂	1	1	yes	maximal, characteristic
3	#12910	C₂	1	1	yes	maximal, characteristic
4	#12904	C₂	1	1	yes	maximal, characteristic
5	#9772	C₂	1	1	yes	maximal, characteristic
6	#8790	C₂	1	1	yes	maximal, characteristic
7	#7140	C₂	1	1	yes	maximal, characteristic
8	C₂×#494	C₂²	1	1	yes	characteristic
9	C₂×#368	—	2	2	no	—
10	C₂×#365	—	2	2	no	—
11	C₂×#362	—	2	2	no	—
12	C₂×#358	—	2	2	no	—
13	C₄×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×#346	C₂²	1	1	yes	characteristic
15	#1848	C₂²	1	1	yes	characteristic
16	#1848	C₂²	1	1	yes	characteristic
17	#1846	C₂²	1	1	yes	characteristic
18	#1546	—	2	2	no	—
19	#1450	C₂²	1	1	yes	characteristic
20	#1286	—	2	2	no	—
21	C₂²×Q₁₆	—	2	1	no	—
22	C₂²×Q₁₆	—	2	1	no	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂×M₃₂	D₄	1	2	yes	—
25	C₂²×C₁₆	D₄	1	2	yes	—
26	#494	—	2	1	no	—
27	#494	—	2	1	no	—
28	#494	—	2	1	no	—
29	#494	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	#368	—	4	4	no	—
42	#365	—	4	4	no	—
43	#362	—	4	4	no	—
44	#358	—	4	4	no	—
45	#346	—	2	1	no	—
46	#346	—	2	1	no	—
47	#346	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	#332	—	4	4	no	—
50	#322	—	2	1	no	—
51	C₈²	—	2	1	no	—
52	C₂²×Q₈	—	4	1	no	—
53	C₂×Q₁₆	—	2	2	no	—
54	C₂×Q₁₆	—	2	2	no	—
55	C₂×Q₁₆	—	4	1	no	—
56	C₂×Q₁₆	—	4	1	no	—
57	C₂×Q₁₆	—	4	2	no	—
58	C₂×Q₁₆	—	4	2	no	—
59	C₂×Q₁₆	—	4	2	no	—
60	C₂×Q₁₆	—	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	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	—	4	1	no	—
67	C₄⋊Q₈	—	4	2	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄⋊C₄	—	4	1	no	—
70	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
71	M₃₂	—	2	2	no	—
72	M₃₂	D₈	1	4	yes	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	SD₁₆	1	4	yes	—
77	#108	—	4	2	no	—
78	#108	—	4	2	no	—
79	#108	—	4	2	no	—
80	#108	—	4	2	no	—
81	#107	—	4	2	no	—
82	#106	—	4	2	no	—
83	C₄⋊C₈	—	2	2	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₂²⋊C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
89	C₄×C₈	#41	1	1	yes	characteristic
90	C₄×C₈	#41	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	#95	—	4	2	no	—
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	8	1	no	—
95	C₂×Q₈	—	8	1	no	—
96	C₂×Q₈	—	8	1	no	—
97	C₂×Q₈	—	8	1	no	—
98	C₂×Q₈	—	8	1	no	—
99	C₂²×C₄	—	8	1	no	—
100	C₂²×C₄	—	8	1	no	—
101	C₂²×C₄	#127	1	1	yes	characteristic
102	C₂²×C₄	#135	1	2	yes	—
103	Q₁₆	—	4	4	no	—
104	Q₁₆	—	4	4	no	—
105	Q₁₆	—	8	2	no	—
106	Q₁₆	—	8	2	no	—
107	M₁₆	—	8	1	no	—
108	M₁₆	—	8	1	no	—
109	M₁₆	—	8	1	no	—
110	M₁₆	—	8	1	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	8	1	no	—
113	C₂×C₈	—	8	1	no	—
114	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₈	#123	1	1	yes	characteristic
117	C₂×C₈	#123	1	1	yes	characteristic
118	C₂×C₈	C₂×D₈	1	2	yes	—
119	C₂×C₈	C₂×SD₁₆	1	2	yes	—
120	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₈	#137	1	2	yes	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	8	2	no	—
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	—	2	4	no	—
131	C₂³	#495	1	1	yes	characteristic
132	Q₈	—	8	2	no	—
133	Q₈	—	8	2	no	—
134	Q₈	—	16	1	no	—
135	Q₈	—	16	1	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	16	1	no	—
139	C₂×C₄	—	16	1	no	—
140	C₂×C₄	—	16	2	no	—
141	C₂×C₄	—	16	1	no	—
142	C₂×C₄	#399	1	1	yes	characteristic
143	C₂×C₄	#478	1	2	yes	—
144	C₂×C₄	#482	1	2	yes	—
145	C₂×C₄	#492	1	1	yes	characteristic
146	C₂×C₄	#493	1	1	yes	characteristic
147	C₂×C₄	#496	1	1	yes	characteristic
148	C₂×C₄	#496	1	1	yes	characteristic
149	C₂×C₄	#497	1	1	yes	characteristic
150	C₈	—	8	2	no	—
151	C₈	—	16	1	no	—
152	C₈	—	16	1	no	—
153	C₈	#474	1	2	yes	—
154	C₈	#476	1	2	yes	—
155	C₈	#480	1	2	yes	—
156	C₈	#481	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	#1706	1	1	yes	characteristic
159	C₂²	#1720	1	1	yes	characteristic
160	C₂²	#2092	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	16	2	no	—
164	C₄	—	32	1	no	—
165	C₄	#1714	1	1	yes	characteristic
166	C₄	#1721	1	1	yes	characteristic
167	C₄	#2093	1	1	yes	characteristic
168	C₄	#2094	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#10672	1	1	yes	characteristic
172	C₂	#13524	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13212

C₂

yes

maximal, characteristic

#13207

C₂

yes

maximal, characteristic

#12910

C₂

yes

maximal, characteristic

#12904

C₂

yes

maximal, characteristic

#9772

C₂

yes

maximal, characteristic

#8790

C₂

yes

maximal, characteristic

#7140

C₂

yes

maximal, characteristic

C₂×#494

C₂²

yes

—

—

—

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1846

C₂²

yes

characteristic

#1546

—

#1450

C₂²

yes

characteristic

#1286

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

—

—

—

Q₈

yes

—

#431

D₄

yes

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

—

C₄×C₁₆

—

#332

—

#322

—

C₈²

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

D₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

#127

yes

characteristic

102

C₂²×C₄

#135

yes

—

103

Q₁₆

—

104

Q₁₆

—

105

Q₁₆

—

106

Q₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂²⋊Q₈

yes

characteristic

115

C₂×C₈

C₂²⋊Q₈

yes

characteristic

116

C₂×C₈

#123

yes

characteristic

117

C₂×C₈

#123

yes

characteristic

118

C₂×C₈

C₂×D₈

yes

—

119

C₂×C₈

C₂×SD₁₆

yes

—

120

C₂×C₈

AGL(1,ℤ/8)

yes

—

121

C₂×C₈

#137

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

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

#495

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#399

yes

characteristic

143

C₂×C₄

#478

yes

—

144

C₂×C₄

#482

yes

—

145

C₂×C₄

#492

yes

characteristic

146

C₂×C₄

#493

yes

characteristic

147

C₂×C₄

#496

yes

characteristic

148

C₂×C₄

#496

yes

characteristic

149

C₂×C₄

#497

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

#474

yes

—

154

C₈

#476

yes

—

155

C₈

#480

yes

—

156

C₈

#481

yes

—

157

C₂²

—

158

C₂²

#1706

yes

characteristic

159

C₂²

#1720

yes

characteristic

160

C₂²

#2092

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#1714

yes

characteristic

166

C₄

#1721

yes

characteristic

167

C₄

#2093

yes

characteristic

168

C₄

#2094

yes

characteristic

169

C₂

—

170

C₂

—

171

C₂

#10672

yes

characteristic

172

C₂

#13524

yes

—

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
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

C₂³

#495

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

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

subgroup	index
#13212	2
#13207	2
#12910	2
#12904	2
#9772	2
#8790	2
#7140	2

subgroup

index

Subgroups — #146229

Conjugacy classes of subgroups

Maximal subgroups