Group #93608 — subgroups

Subgroups — #93608

Order 512.

188 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 58 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	#12882	C₂	1	1	yes	maximal, characteristic
2	#8800	C₂	1	1	yes	maximal, characteristic
3	#8263	C₂	1	1	yes	maximal, characteristic
4	C₂×#364	—	2	1	no	—
5	C₄×C₄⋊C₈	—	2	1	no	—
6	C₄×#97	C₂²	1	1	yes	characteristic
7	C₂×#322	—	2	1	no	—
8	#1377	—	2	1	no	—
9	#1375	C₄	1	2	yes	—
10	#1281	—	2	1	no	—
11	#1276	—	2	1	no	—
12	#1275	C₄	1	1	yes	characteristic
13	#1275	C₄	1	1	yes	characteristic
14	#1273	C₄	1	1	yes	characteristic
15	#1273	C₄	1	1	yes	characteristic
16	C₂×M₃₂	—	2	2	no	—
17	#431	—	2	1	no	—
18	#431	—	2	1	no	—
19	C₂×C₄⋊C₈	—	2	2	no	—
20	C₂×#97	C₂×C₄	1	1	yes	characteristic
21	C₂×#97	C₂×C₄	1	1	yes	characteristic
22	C₂×#97	D₄	1	1	yes	characteristic
23	C₂×#97	Q₈	1	1	yes	characteristic
24	C₂×C₄×C₈	—	2	1	no	—
25	C₂×C₄×C₈	—	2	1	no	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₄³	D₄	1	1	yes	characteristic
29	#364	—	4	1	no	—
30	#364	—	4	1	no	—
31	#336	—	2	2	no	—
32	#322	—	4	1	no	—
33	#322	—	4	1	no	—
34	C₂²×C₈	—	2	2	no	—
35	C₂²×C₈	—	2	2	no	—
36	C₂²×C₈	C₄²	1	1	yes	characteristic
37	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄²	—	2	1	no	—
41	C₂×C₄²	—	2	1	no	—
42	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
45	M₃₂	—	4	2	no	—
46	M₃₂	—	4	2	no	—
47	C₂×C₁₆	—	4	2	no	—
48	C₄⋊C₈	—	4	1	no	—
49	C₄⋊C₈	—	4	1	no	—
50	C₄⋊C₈	—	4	4	no	—
51	C₂²⋊C₈	—	4	1	no	—
52	C₂²⋊C₈	—	4	1	no	—
53	#97	—	2	2	no	—
54	#97	—	4	1	no	—
55	#97	—	4	1	no	—
56	#97	—	4	1	no	—
57	C₄×C₈	—	4	1	no	—
58	C₄×C₈	—	4	1	no	—
59	C₄×C₈	—	4	1	no	—
60	C₄×C₈	—	4	1	no	—
61	C₄×C₈	—	4	1	no	—
62	C₄×C₈	—	4	1	no	—
63	C₄×C₈	—	4	1	no	—
64	C₄×C₈	D₈	1	1	yes	characteristic
65	C₄×C₈	SD₁₆	1	1	yes	characteristic
66	C₄×C₈	SD₁₆	1	1	yes	characteristic
67	C₄×C₈	Q₁₆	1	1	yes	characteristic
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	#95	1	1	yes	characteristic
71	C₂²×C₄	#104	1	1	yes	characteristic
72	C₂²×C₄	#104	1	1	yes	characteristic
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	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	—	4	2	no	—
89	C₂×C₈	—	8	1	no	—
90	C₂×C₈	—	8	1	no	—
91	C₂×C₈	#102	1	1	yes	characteristic
92	C₂×C₈	#103	1	1	yes	characteristic
93	C₂×C₈	#106	1	1	yes	characteristic
94	C₂×C₈	#107	1	1	yes	characteristic
95	C₄²	—	2	1	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	2	1	no	—
99	C₄²	—	4	2	no	—
100	C₄²	—	4	2	no	—
101	C₄²	#102	1	1	yes	characteristic
102	C₄²	#103	1	1	yes	characteristic
103	C₄²	#108	1	2	yes	—
104	C₁₆	—	8	2	no	—
105	C₂³	#339	1	1	yes	characteristic
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	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	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
118	C₂×C₄	#340	1	1	yes	characteristic
119	C₂×C₄	#341	1	1	yes	characteristic
120	C₂×C₄	#344	1	1	yes	characteristic
121	C₂×C₄	#344	1	1	yes	characteristic
122	C₂×C₄	#344	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	4	4	no	—
128	C₈	—	4	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	8	2	no	—
131	C₈	—	16	1	no	—
132	C₈	—	16	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#1288	1	1	yes	characteristic
136	C₂²	#1302	1	1	yes	characteristic
137	C₂²	#1303	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	2	no	—
141	C₄	—	4	1	no	—
142	C₄	—	4	1	no	—
143	C₄	#1286	1	1	yes	characteristic
144	C₄	#1287	1	1	yes	characteristic
145	C₄	#1301	1	1	yes	characteristic
146	C₄	#1308	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#7041	1	1	yes	characteristic
150	C₂	#7225	1	1	yes	characteristic
151	C₂	#7226	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12882

C₂

yes

maximal, characteristic

#8800

C₂

yes

maximal, characteristic

#8263

C₂

yes

maximal, characteristic

C₂×#364

—

C₄×C₄⋊C₈

—

C₄×#97

C₂²

yes

—

—

C₄

yes

—

#1281

—

#1276

—

#1275

C₄

yes

characteristic

#1275

C₄

yes

characteristic

#1273

C₄

yes

characteristic

#1273

C₄

yes

characteristic

C₂×M₃₂

—

#431

—

#431

—

C₂×C₄⋊C₈

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

Q₈

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₄³

D₄

yes

—

—

—

—

—

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

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊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

characteristic

C₂²×C₄

—

C₂²×C₄

—

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

#102

yes

characteristic

102

C₄²

#103

yes

characteristic

103

C₄²

#108

yes

—

104

C₁₆

—

105

C₂³

#339

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

H(3,ℤ/4)

yes

characteristic

118

C₂×C₄

#340

yes

characteristic

119

C₂×C₄

#341

yes

characteristic

120

C₂×C₄

#344

yes

characteristic

121

C₂×C₄

#344

yes

characteristic

122

C₂×C₄

#344

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₂²

—

134

C₂²

—

135

C₂²

#1288

yes

characteristic

136

C₂²

#1302

yes

characteristic

137

C₂²

#1303

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#1286

yes

characteristic

144

C₄

#1287

yes

characteristic

145

C₄

#1301

yes

characteristic

146

C₄

#1308

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#7041

yes

characteristic

150

C₂

#7225

yes

characteristic

151

C₂

#7226

yes

characteristic

	subgroup	G/H
center	C₂×C₄	H(3,ℤ/4)
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₄×#97	C₂²
socle	C₂²	#1288
prime radical	C₂³	#339
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

H(3,ℤ/4)

commutator

C₂²×C₈

C₄²

Frattini, cosocle

C₄×#97

C₂²

C₂²

C₂³

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G

LECS

G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12882	2
#8800	2
#8263	2

subgroup

index

#12882

#8800

#8263

Subgroups — #93608

Conjugacy classes of subgroups

Maximal subgroups