Group #93604 — subgroups

Subgroups — #93604

Order 512.

206 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 86 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	#12879	C₂	1	1	yes	maximal, characteristic
2	#8805	C₂	1	1	yes	maximal, characteristic
3	#8785	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₁₆	—	2	1	no	—
5	#1842	—	2	1	no	—
6	#1831	—	2	1	no	—
7	C₄²×C₈	C₂²	1	1	yes	characteristic
8	C₂×#334	—	2	1	no	—
9	C₂×#334	C₄	1	2	yes	—
10	#1374	—	2	1	no	—
11	#1374	C₄	1	2	yes	—
12	#1277	—	2	1	no	—
13	#1277	C₄	1	2	yes	—
14	C₂²×C₁₆	—	2	2	no	—
15	C₂×C₄⋊C₈	—	2	2	no	—
16	C₂×C₄⋊C₈	—	2	2	no	—
17	C₂×C₄⋊C₈	—	4	1	no	—
18	C₂×C₄⋊C₈	—	4	1	no	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
25	C₄³	D₄	1	1	yes	characteristic
26	C₄⋊C₁₆	—	4	1	no	—
27	C₄⋊C₁₆	—	4	1	no	—
28	#336	—	4	1	no	—
29	#336	—	4	1	no	—
30	#334	—	4	2	no	—
31	#334	—	4	1	no	—
32	#334	—	4	1	no	—
33	C₂²×C₈	—	4	2	no	—
34	C₂²×C₈	—	4	2	no	—
35	C₂²×C₈	C₄²	1	1	yes	characteristic
36	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄²	D₈	1	1	yes	characteristic
43	C₂×C₄²	SD₁₆	1	1	yes	characteristic
44	C₂×C₄²	SD₁₆	1	1	yes	characteristic
45	C₂×C₄²	Q₁₆	1	1	yes	characteristic
46	C₂×C₁₆	—	4	2	no	—
47	C₂×C₁₆	—	4	2	no	—
48	C₂×C₁₆	—	4	2	no	—
49	C₄⋊C₈	—	4	2	no	—
50	C₄⋊C₈	—	4	2	no	—
51	C₄⋊C₈	—	4	2	no	—
52	C₄⋊C₈	—	4	2	no	—
53	C₄⋊C₈	—	8	2	no	—
54	C₄⋊C₈	—	8	2	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	—	4	1	no	—
59	C₄×C₈	—	4	1	no	—
60	C₄×C₈	—	4	1	no	—
61	C₄×C₈	D₈	1	1	yes	characteristic
62	C₄×C₈	SD₁₆	1	1	yes	characteristic
63	C₄×C₈	SD₁₆	1	1	yes	characteristic
64	C₄×C₈	Q₁₆	1	1	yes	characteristic
65	C₂²×C₄	#95	1	1	yes	characteristic
66	C₂²×C₄	#102	1	1	yes	characteristic
67	C₂²×C₄	#102	1	1	yes	characteristic
68	C₂²×C₄	#103	1	1	yes	characteristic
69	C₂²×C₄	#103	1	1	yes	characteristic
70	C₂²×C₄	#106	1	1	yes	characteristic
71	C₂²×C₄	#107	1	1	yes	characteristic
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	8	2	no	—
82	C₂×C₈	—	8	2	no	—
83	C₂×C₈	—	8	2	no	—
84	C₂×C₈	—	8	2	no	—
85	C₂×C₈	—	8	2	no	—
86	C₂×C₈	—	8	2	no	—
87	C₂×C₈	#102	1	1	yes	characteristic
88	C₂×C₈	#103	1	1	yes	characteristic
89	C₂×C₈	#106	1	1	yes	characteristic
90	C₂×C₈	#107	1	1	yes	characteristic
91	C₂×C₈	#108	1	2	yes	—
92	C₄²	—	2	1	no	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	1	no	—
95	C₄²	—	2	1	no	—
96	C₄²	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	4	1	no	—
100	C₄²	#100	1	1	yes	characteristic
101	C₄²	#101	1	1	yes	characteristic
102	C₄²	#102	1	1	yes	characteristic
103	C₄²	#103	1	1	yes	characteristic
104	C₁₆	—	8	2	no	—
105	C₂³	#340	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	#331	1	1	yes	characteristic
120	C₂×C₄	#332	1	1	yes	characteristic
121	C₂×C₄	#332	1	1	yes	characteristic
122	C₂×C₄	#333	1	1	yes	characteristic
123	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
124	C₂×C₄	#340	1	1	yes	characteristic
125	C₂×C₄	#341	1	1	yes	characteristic
126	C₂×C₄	#343	1	1	yes	characteristic
127	C₂×C₄	#344	1	1	yes	characteristic
128	C₂×C₄	#344	1	1	yes	characteristic
129	C₂×C₄	#360	1	2	yes	—
130	C₂×C₄	#365	1	2	yes	—
131	C₂×C₄	#368	1	2	yes	—
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	2	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	16	2	no	—
139	C₈	—	16	2	no	—
140	C₂²	#1286	1	1	yes	characteristic
141	C₂²	#1296	1	1	yes	characteristic
142	C₂²	#1301	1	1	yes	characteristic
143	C₂²	#1381	1	1	yes	characteristic
144	C₂²	#1383	1	1	yes	characteristic
145	C₂²	#1386	1	1	yes	characteristic
146	C₂²	#1389	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	4	1	no	—
155	C₄	#1286	1	1	yes	characteristic
156	C₄	#1290	1	1	yes	characteristic
157	C₄	#1301	1	1	yes	characteristic
158	C₄	#1307	1	1	yes	characteristic
159	C₂	#7040	1	1	yes	characteristic
160	C₂	#7219	1	1	yes	characteristic
161	C₂	#7221	1	1	yes	characteristic
162	C₂	#7276	1	1	yes	characteristic
163	C₂	#7278	1	1	yes	characteristic
164	C₂	#7290	1	1	yes	characteristic
165	C₂	#7294	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12879

C₂

yes

maximal, characteristic

#8805

C₂

yes

maximal, characteristic

#8785

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

—

#1842

—

#1831

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#334

—

C₂×#334

C₄

yes

—

#1374

—

#1374

C₄

yes

—

#1277

—

#1277

C₄

yes

—

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

Q₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

#100

yes

characteristic

101

C₄²

#101

yes

characteristic

102

C₄²

#102

yes

characteristic

103

C₄²

#103

yes

characteristic

104

C₁₆

—

105

C₂³

#340

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#331

yes

characteristic

120

C₂×C₄

#332

yes

characteristic

121

C₂×C₄

#332

yes

characteristic

122

C₂×C₄

#333

yes

characteristic

123

C₂×C₄

H(3,ℤ/4)

yes

characteristic

124

C₂×C₄

#340

yes

characteristic

125

C₂×C₄

#341

yes

characteristic

126

C₂×C₄

#343

yes

characteristic

127

C₂×C₄

#344

yes

characteristic

128

C₂×C₄

#344

yes

characteristic

129

C₂×C₄

#360

yes

—

130

C₂×C₄

#365

yes

—

131

C₂×C₄

#368

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₂²

#1286

yes

characteristic

141

C₂²

#1296

yes

characteristic

142

C₂²

#1301

yes

characteristic

143

C₂²

#1381

yes

characteristic

144

C₂²

#1383

yes

characteristic

145

C₂²

#1386

yes

characteristic

146

C₂²

#1389

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#1286

yes

characteristic

156

C₄

#1290

yes

characteristic

157

C₄

#1301

yes

characteristic

158

C₄

#1307

yes

characteristic

159

C₂

#7040

yes

characteristic

160

C₂

#7219

yes

characteristic

161

C₂

#7221

yes

characteristic

162

C₂

#7276

yes

characteristic

163

C₂

#7278

yes

characteristic

164

C₂

#7290

yes

characteristic

165

C₂

#7294

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#340
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₄²×C₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#340

commutator

C₂²×C₈

C₄²

Frattini, cosocle

C₄²×C₈

C₂²

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12879	2
#8805	2
#8785	2

subgroup

index

#12879

#8805

#8785

Subgroups — #93604

Conjugacy classes of subgroups

Maximal subgroups