Group #102290 — subgroups

Subgroups — #102290

Order 512.

448 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 166 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	#33310	C₂	1	1	yes	maximal, characteristic
2	#9898	C₂	1	2	yes	maximal
3	#9694	C₂	1	2	yes	maximal
4	#8806	C₂	1	1	yes	maximal, characteristic
5	#8260	C₂	1	1	yes	maximal, characteristic
6	C₂×#498	—	2	1	no	—
7	C₂×#498	—	2	1	no	—
8	#2868	C₂²	1	2	yes	—
9	#1844	—	2	2	no	—
10	#1844	C₄	1	4	yes	—
11	#1842	C₂²	1	1	yes	characteristic
12	#1842	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₈	C₂²	1	2	yes	—
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×#335	—	2	2	no	—
16	C₂×#332	—	2	4	no	—
17	C₂×#322	—	2	1	no	—
18	#1296	—	2	4	no	—
19	#1273	—	2	1	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂×C₄⋊Q₈	—	2	2	no	—
23	#498	—	4	4	no	—
24	#498	—	4	2	no	—
25	#498	—	4	2	no	—
26	C₂×#106	—	2	4	no	—
27	C₂×#106	—	2	4	no	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄⋊C₈	—	2	2	no	—
30	C₂×C₄⋊C₈	—	2	1	no	—
31	C₂×C₄⋊C₈	—	2	1	no	—
32	C₂×C₄⋊C₈	D₄	1	4	yes	—
33	C₂×C₄×C₈	—	2	2	no	—
34	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	2	yes	—
38	#384	—	2	2	no	—
39	#384	—	2	2	no	—
40	#384	—	2	2	no	—
41	#384	C₂×C₄	1	4	yes	—
42	C₄³	C₂³	1	1	yes	characteristic
43	#336	—	2	2	no	—
44	#335	—	4	4	no	—
45	#332	—	4	8	no	—
46	#322	—	4	2	no	—
47	C₂²×Q₈	—	4	2	no	—
48	C₂²×C₈	—	2	4	no	—
49	C₂²×C₈	—	4	1	no	—
50	C₂²×C₈	—	4	1	no	—
51	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
53	C₄⋊Q₈	—	4	2	no	—
54	C₄⋊Q₈	—	4	2	no	—
55	C₄⋊Q₈	—	4	4	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₄⋊Q₈	—	4	4	no	—
59	C₄⋊Q₈	—	8	4	no	—
60	C₂×C₄⋊C₄	—	2	4	no	—
61	C₂×C₄⋊C₄	—	2	4	no	—
62	C₂×C₄⋊C₄	—	2	4	no	—
63	C₂×C₄⋊C₄	—	4	2	no	—
64	C₂×C₄⋊C₄	—	4	2	no	—
65	C₂×C₄⋊C₄	—	4	2	no	—
66	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₄	1	2	yes	—
70	C₂×C₄²	#41	1	2	yes	—
71	#106	—	4	8	no	—
72	#106	—	4	8	no	—
73	C₄⋊C₈	—	2	4	no	—
74	C₄⋊C₈	—	2	4	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄⋊C₈	—	4	4	no	—
78	C₄⋊C₈	—	4	4	no	—
79	C₄×C₈	—	2	4	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	—	4	2	no	—
82	C₄×C₈	—	4	2	no	—
83	C₄×C₈	D₈	1	2	yes	—
84	C₄×C₈	SD₁₆	1	2	yes	—
85	C₄×C₈	SD₁₆	1	4	yes	—
86	C₄×C₈	Q₁₆	1	4	yes	—
87	C₂×Q₈	—	8	2	no	—
88	C₂×Q₈	—	8	2	no	—
89	C₂×Q₈	—	8	4	no	—
90	C₂×Q₈	—	8	4	no	—
91	C₂²×C₄	—	4	4	no	—
92	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₄	1	2	yes	—
94	C₂²×C₄	#121	1	2	yes	—
95	C₂²×C₄	#124	1	1	yes	characteristic
96	C₂²×C₄	#127	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	4	4	no	—
100	C₂×C₈	—	4	4	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	—	8	1	no	—
104	C₂×C₈	—	8	2	no	—
105	C₂×C₈	—	8	2	no	—
106	C₂×C₈	#101	1	4	yes	—
107	C₂×C₈	#102	1	4	yes	—
108	C₂×C₈	#103	1	4	yes	—
109	C₂×C₈	#103	1	4	yes	—
110	C₄⋊C₄	—	4	4	no	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	4	8	no	—
113	C₄⋊C₄	—	8	4	no	—
114	C₄⋊C₄	—	8	4	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₄⋊C₄	—	8	2	no	—
119	C₄²	—	2	2	no	—
120	C₄²	—	2	4	no	—
121	C₄²	C₂×D₈	1	1	yes	characteristic
122	C₄²	C₂×SD₁₆	1	1	yes	characteristic
123	C₄²	C₂×SD₁₆	1	2	yes	—
124	C₄²	C₂×Q₁₆	1	2	yes	—
125	C₄²	AGL(1,ℤ/8)	1	2	yes	—
126	C₄²	AGL(1,ℤ/8)	1	4	yes	—
127	C₄²	#137	1	4	yes	—
128	C₂³	#390	1	1	yes	characteristic
129	Q₈	—	16	4	no	—
130	Q₈	—	16	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	8	4	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	—	8	4	no	—
137	C₂×C₄	C₂×#101	1	2	yes	—
138	C₂×C₄	C₂×#102	1	1	yes	characteristic
139	C₂×C₄	C₂×#103	1	2	yes	—
140	C₂×C₄	#418	1	1	yes	characteristic
141	C₂×C₄	#440	1	4	yes	—
142	C₂×C₄	#459	1	2	yes	—
143	C₂×C₄	#460	1	2	yes	—
144	C₂×C₄	#468	1	4	yes	—
145	C₂×C₄	#471	1	4	yes	—
146	C₂×C₄	#487	1	2	yes	—
147	C₂×C₄	#492	1	1	yes	characteristic
148	C₂×C₄	#494	1	1	yes	characteristic
149	C₂×C₄	#496	1	2	yes	—
150	C₂×C₄	#497	1	2	yes	—
151	C₈	—	2	4	no	—
152	C₈	—	8	4	no	—
153	C₈	—	16	1	no	—
154	C₈	—	16	1	no	—
155	C₈	#332	1	8	yes	complemented
156	C₂²	#1956	1	1	yes	characteristic
157	C₂²	#1966	1	1	yes	characteristic
158	C₂²	#1973	1	2	yes	—
159	C₂²	#1977	1	2	yes	—
160	C₂²	#1982	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	4	no	—
163	C₄	—	16	4	no	—
164	C₄	C₂×#332	1	2	yes	—
165	C₄	#1549	1	2	yes	—
166	C₄	#1685	1	4	yes	—
167	C₄	#1686	1	4	yes	—
168	C₄	#1715	1	1	yes	characteristic
169	C₄	#1716	1	1	yes	characteristic
170	C₄	#1723	1	2	yes	—
171	C₂	#9694	1	1	yes	characteristic
172	C₂	#9744	1	2	yes	—
173	C₂	#9767	1	1	yes	characteristic
174	C₂	#9806	1	2	yes	—
175	C₂	#9841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33310

C₂

yes

maximal, characteristic

#9898

C₂

yes

maximal

#9694

C₂

yes

maximal

#8806

C₂

yes

maximal, characteristic

#8260

C₂

yes

maximal, characteristic

C₂×#498

—

C₂×#498

—

#2868

C₂²

yes

—

#1844

—

#1844

C₄

yes

—

#1842

C₂²

yes

characteristic

#1842

C₂²

yes

characteristic

C₄×C₄⋊C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₄³

C₂³

yes

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

#41

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄×D₄

yes

—

C₂²×C₄

#121

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

#101

yes

—

107

C₂×C₈

#102

yes

—

108

C₂×C₈

#103

yes

—

109

C₂×C₈

#103

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₂×D₈

yes

characteristic

122

C₄²

C₂×SD₁₆

yes

characteristic

123

C₄²

C₂×SD₁₆

yes

—

124

C₄²

C₂×Q₁₆

yes

—

125

C₄²

AGL(1,ℤ/8)

yes

—

126

C₄²

AGL(1,ℤ/8)

yes

—

127

C₄²

#137

yes

—

128

C₂³

#390

yes

characteristic

129

Q₈

—

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×#101

yes

—

138

C₂×C₄

C₂×#102

yes

characteristic

139

C₂×C₄

C₂×#103

yes

—

140

C₂×C₄

#418

yes

characteristic

141

C₂×C₄

#440

yes

—

142

C₂×C₄

#459

yes

—

143

C₂×C₄

#460

yes

—

144

C₂×C₄

#468

yes

—

145

C₂×C₄

#471

yes

—

146

C₂×C₄

#487

yes

—

147

C₂×C₄

#492

yes

characteristic

148

C₂×C₄

#494

yes

characteristic

149

C₂×C₄

#496

yes

—

150

C₂×C₄

#497

yes

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

#332

yes

complemented

156

C₂²

#1956

yes

characteristic

157

C₂²

#1966

yes

characteristic

158

C₂²

#1973

yes

—

159

C₂²

#1977

yes

—

160

C₂²

#1982

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

C₂×#332

yes

—

165

C₄

#1549

yes

—

166

C₄

#1685

yes

—

167

C₄

#1686

yes

—

168

C₄

#1715

yes

characteristic

169

C₄

#1716

yes

characteristic

170

C₄

#1723

yes

—

171

C₂

#9694

yes

characteristic

172

C₂

#9744

yes

—

173

C₂

#9767

yes

characteristic

174

C₂

#9806

yes

—

175

C₂

#9841

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#390

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₄³

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
155. C₈	#332	semidirect

H ≅ G/N

type

155. C₈

#332

semidirect

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

subgroup	index
#33310	2
#9898	2
#9694	2
#8806	2
#8260	2

subgroup

index

Subgroups — #102290

Conjugacy classes of subgroups

Complements

Maximal subgroups