Group #93259 — subgroups

Subgroups — #93259

Order 512.

266 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 130 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	#12653	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#7841	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
6	C₂×#346	—	2	1	no	—
7	C₂×C₄×C₁₆	—	2	1	no	—
8	#1744	C₄	1	2	yes	—
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₂×#322	—	2	1	no	—
11	C₂×C₈²	—	2	1	no	—
12	#1374	—	2	1	no	—
13	#1374	—	2	1	no	—
14	C₂²×C₁₆	—	2	2	no	—
15	C₂²×C₁₆	—	2	2	no	—
16	C₂×C₄×C₈	—	2	2	no	—
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
23	C₄³	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₁₆	—	2	2	no	—
25	C₄⋊C₁₆	—	2	2	no	—
26	C₄⋊C₁₆	C₈	1	4	yes	complemented
27	C₄⋊C₁₆	C₈	1	4	yes	complemented
28	#346	—	4	1	no	—
29	#346	—	4	1	no	—
30	C₄×C₁₆	—	4	1	no	—
31	C₄×C₁₆	—	4	1	no	—
32	#336	—	2	2	no	—
33	#336	C₈	1	4	yes	—
34	#322	—	2	4	no	—
35	C₈²	—	2	4	no	—
36	C₂²×C₈	—	2	4	no	—
37	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄²	C₄²	1	1	yes	characteristic
42	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄²	C₂×C₈	1	2	yes	—
45	C₂×C₄²	M₁₆	1	2	yes	—
46	C₂×C₁₆	—	2	4	no	—
47	C₂×C₁₆	—	2	4	no	—
48	C₂×C₁₆	—	4	2	no	—
49	C₂×C₁₆	—	4	2	no	—
50	C₂×C₁₆	—	4	2	no	—
51	C₂×C₁₆	—	4	2	no	—
52	C₄×C₈	—	2	4	no	—
53	C₄×C₈	—	2	4	no	—
54	C₄×C₈	—	2	2	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	C₂×C₈	1	2	yes	—
58	C₄×C₈	C₂×C₈	1	2	yes	—
59	C₄×C₈	M₁₆	1	2	yes	—
60	C₄×C₈	M₁₆	1	2	yes	—
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	C₄×C₈	1	1	yes	characteristic
67	C₂²×C₄	#97	1	1	yes	characteristic
68	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
69	C₂²×C₄	C₄⋊C₈	1	2	yes	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	8	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	C₂²⋊C₈	1	2	yes	—
78	C₂×C₈	C₂²⋊C₈	1	2	yes	—
79	C₂×C₈	#102	1	1	yes	characteristic
80	C₂×C₈	#102	1	1	yes	characteristic
81	C₂×C₈	#103	1	1	yes	characteristic
82	C₂×C₈	#103	1	1	yes	characteristic
83	C₂×C₈	#104	1	2	yes	—
84	C₂×C₈	#104	1	2	yes	—
85	C₂×C₈	C₄⋊C₈	1	2	yes	—
86	C₂×C₈	C₄⋊C₈	1	2	yes	—
87	C₄²	—	2	1	no	—
88	C₄²	—	2	1	no	—
89	C₄²	—	2	4	no	—
90	C₄²	—	4	2	no	—
91	C₄²	C₄×C₈	1	1	yes	characteristic
92	C₄²	C₄×C₈	1	1	yes	characteristic
93	C₄²	#97	1	1	yes	characteristic
94	C₄²	#97	1	1	yes	characteristic
95	C₄²	#106	1	1	yes	characteristic
96	C₄²	#107	1	1	yes	characteristic
97	C₄²	#108	1	2	yes	—
98	C₁₆	—	4	4	no	—
99	C₁₆	—	4	4	no	—
100	C₂³	#336	1	1	yes	characteristic
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	C₈²	1	1	yes	characteristic
108	C₂×C₄	#322	1	1	yes	characteristic
109	C₂×C₄	#322	1	1	yes	characteristic
110	C₂×C₄	#322	1	1	yes	characteristic
111	C₂×C₄	#334	1	2	yes	—
112	C₂×C₄	#335	1	2	yes	—
113	C₂×C₄	#336	1	1	yes	characteristic
114	C₂×C₄	#336	1	1	yes	characteristic
115	C₂×C₄	#339	1	1	yes	characteristic
116	C₂×C₄	#339	1	1	yes	characteristic
117	C₂×C₄	#340	1	1	yes	characteristic
118	C₂×C₄	#341	1	1	yes	characteristic
119	C₂×C₄	#350	1	4	yes	—
120	C₈	—	2	2	no	—
121	C₈	—	2	2	no	—
122	C₈	—	4	8	no	—
123	C₈	D₄⋊C₈	1	2	yes	—
124	C₈	D₄⋊C₈	1	2	yes	—
125	C₈	Q₈⋊C₈	1	2	yes	—
126	C₈	Q₈⋊C₈	1	2	yes	—
127	C₂²	#1273	1	1	yes	characteristic
128	C₂²	#1276	1	1	yes	characteristic
129	C₂²	#1277	1	1	yes	characteristic
130	C₂²	#1378	1	2	yes	—
131	C₂²	#1379	1	2	yes	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	4	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	#1278	1	1	yes	characteristic
140	C₄	#1278	1	1	yes	characteristic
141	C₄	#1279	1	1	yes	characteristic
142	C₄	#1279	1	1	yes	characteristic
143	C₂	#7015	1	1	yes	characteristic
144	C₂	#7140	1	1	yes	characteristic
145	C₂	#7141	1	1	yes	characteristic
146	C₂	#7150	1	2	yes	—
147	C₂	#7153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12653

C₂

yes

maximal, characteristic

C₄×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#7841

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×#346

—

C₂×C₄×C₁₆

—

#1744

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#322

—

C₂×C₈²

—

#1374

—

#1374

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

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

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₈

yes

complemented

C₄⋊C₁₆

C₈

yes

complemented

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#336

—

#336

C₈

yes

—

#322

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

M₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₂²×C₄

#95

yes

characteristic

C₂²×C₄

C₄×C₈

yes

characteristic

C₂²×C₄

#97

yes

characteristic

C₂²×C₄

C₂²⋊C₈

yes

—

C₂²×C₄

C₄⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#104

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×C₈

yes

characteristic

C₄²

C₄×C₈

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

—

C₁₆

—

C₁₆

—

100

C₂³

#336

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₈²

yes

characteristic

108

C₂×C₄

#322

yes

characteristic

109

C₂×C₄

#322

yes

characteristic

110

C₂×C₄

#322

yes

characteristic

111

C₂×C₄

#334

yes

—

112

C₂×C₄

#335

yes

—

113

C₂×C₄

#336

yes

characteristic

114

C₂×C₄

#336

yes

characteristic

115

C₂×C₄

#339

yes

characteristic

116

C₂×C₄

#339

yes

characteristic

117

C₂×C₄

#340

yes

characteristic

118

C₂×C₄

#341

yes

characteristic

119

C₂×C₄

#350

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

D₄⋊C₈

yes

—

124

C₈

D₄⋊C₈

yes

—

125

C₈

Q₈⋊C₈

yes

—

126

C₈

Q₈⋊C₈

yes

—

127

C₂²

#1273

yes

characteristic

128

C₂²

#1276

yes

characteristic

129

C₂²

#1277

yes

characteristic

130

C₂²

#1378

yes

—

131

C₂²

#1379

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#1278

yes

characteristic

140

C₄

#1278

yes

characteristic

141

C₄

#1279

yes

characteristic

142

C₄

#1279

yes

characteristic

143

C₂

#7015

yes

characteristic

144

C₂

#7140

yes

characteristic

145

C₂

#7141

yes

characteristic

146

C₂

#7150

yes

—

147

C₂

#7153

yes

—

	subgroup	G/H
center	C₂²×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

C₂²⋊C₄

commutator

C₂×C₄

C₈²

Frattini, cosocle

C₄²×C₈

C₂²

C₂³

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
26. C₄⋊C₁₆	C₈	semidirect
27. C₄⋊C₁₆	C₈	semidirect

H ≅ G/N

type

26. C₄⋊C₁₆

C₈

semidirect

27. C₄⋊C₁₆

C₈

semidirect

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

subgroup	index
#12653	2
C₄×C₄⋊C₁₆	2
#7841	2

subgroup

index

#12653

C₄×C₄⋊C₁₆

#7841

Subgroups — #93259

Conjugacy classes of subgroups

Complements

Maximal subgroups