Group #93267 — subgroups

Subgroups — #93267

Order 512.

256 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 118 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	C₄×#364	C₂	1	1	yes	maximal, characteristic
2	#8266	C₂	1	1	yes	maximal, characteristic
3	#7841	C₂	1	1	yes	maximal, characteristic
4	C₂×#364	C₄	1	2	yes	—
5	C₄×M₃₂	—	2	1	no	—
6	#1744	—	2	1	no	—
7	C₄²×C₈	C₂²	1	1	yes	characteristic
8	#1564	C₄	1	2	yes	—
9	#1450	—	2	1	no	—
10	C₂×#322	C₄	1	2	yes	—
11	C₂×C₈²	—	2	1	no	—
12	#1376	—	2	1	no	—
13	#1276	—	2	1	no	—
14	C₂×M₃₂	—	2	2	no	—
15	#431	—	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	C₂×C₄	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	D₄	1	1	yes	characteristic
24	#364	—	2	2	no	—
25	#364	C₈	1	4	yes	complemented
26	#346	—	4	1	no	—
27	C₄×C₁₆	—	4	1	no	—
28	#336	—	2	4	no	—
29	#335	—	2	2	no	—
30	#334	C₈	1	4	yes	complemented
31	#322	—	2	2	no	—
32	#322	—	4	1	no	—
33	#322	C₈	1	4	yes	complemented
34	C₈²	—	2	4	no	—
35	C₈²	—	4	1	no	—
36	C₂²×C₈	—	2	4	no	—
37	C₂²×C₈	C₄²	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	—	2	2	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	—	2	4	no	—
46	M₃₂	—	4	2	no	—
47	C₂×C₁₆	—	4	2	no	—
48	C₄⋊C₈	—	2	4	no	—
49	C₂²⋊C₈	—	4	2	no	—
50	C₄×C₈	—	2	4	no	—
51	C₄×C₈	—	2	2	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	2	4	no	—
54	C₄×C₈	—	4	2	no	—
55	C₄×C₈	C₂×C₈	1	2	yes	—
56	C₄×C₈	C₂×C₈	1	2	yes	—
57	C₄×C₈	C₂×C₈	1	2	yes	—
58	C₄×C₈	M₁₆	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	—	2	1	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	#95	1	1	yes	characteristic
68	C₂²×C₄	#104	1	2	yes	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	8	no	—
72	C₂×C₈	—	4	8	no	—
73	C₂×C₈	—	4	4	no	—
74	C₂×C₈	C₄×C₈	1	1	yes	characteristic
75	C₂×C₈	C₄×C₈	1	1	yes	characteristic
76	C₂×C₈	C₄×C₈	1	1	yes	characteristic
77	C₂×C₈	#97	1	1	yes	characteristic
78	C₂×C₈	#97	1	1	yes	characteristic
79	C₂×C₈	#97	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊C₈	1	2	yes	—
81	C₂×C₈	#102	1	1	yes	characteristic
82	C₂×C₈	#103	1	1	yes	characteristic
83	C₂×C₈	#104	1	2	yes	—
84	C₂×C₈	C₄⋊C₈	1	2	yes	—
85	C₂×C₈	C₄⋊C₈	1	2	yes	—
86	C₂×C₈	#106	1	1	yes	characteristic
87	C₂×C₈	#107	1	1	yes	characteristic
88	C₂×C₈	#108	1	2	yes	—
89	C₄²	—	2	1	no	—
90	C₄²	—	2	1	no	—
91	C₄²	—	4	4	no	—
92	C₄²	C₂²⋊C₈	1	2	yes	—
93	C₄²	C₂²⋊C₈	1	2	yes	—
94	C₄²	#102	1	1	yes	characteristic
95	C₄²	#103	1	1	yes	characteristic
96	C₄²	C₄⋊C₈	1	2	yes	—
97	C₁₆	—	4	4	no	—
98	C₂³	#339	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	D₄⋊C₈	1	2	yes	—
105	C₂×C₄	Q₈⋊C₈	1	2	yes	—
106	C₂×C₄	#336	1	1	yes	characteristic
107	C₂×C₄	#336	1	1	yes	characteristic
108	C₂×C₄	#336	1	1	yes	characteristic
109	C₂×C₄	#339	1	1	yes	characteristic
110	C₂×C₄	#340	1	1	yes	characteristic
111	C₂×C₄	#341	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	4	8	no	—
114	C₈	—	8	4	no	—
115	C₈	C₈²	1	1	yes	characteristic
116	C₈	#322	1	1	yes	characteristic
117	C₈	#322	1	1	yes	characteristic
118	C₈	#322	1	1	yes	characteristic
119	C₈	D₄⋊C₈	1	2	yes	—
120	C₈	Q₈⋊C₈	1	2	yes	—
121	C₈	#334	1	2	yes	—
122	C₈	#335	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	#1276	1	1	yes	characteristic
125	C₂²	#1278	1	1	yes	characteristic
126	C₂²	#1279	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	2	no	—
131	C₄	#1273	1	1	yes	characteristic
132	C₄	#1277	1	1	yes	characteristic
133	C₄	#1278	1	1	yes	characteristic
134	C₄	#1279	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	#7015	1	1	yes	characteristic
138	C₂	#7181	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#364

C₂

yes

maximal, characteristic

#8266

C₂

yes

maximal, characteristic

#7841

C₂

yes

maximal, characteristic

C₂×#364

C₄

yes

—

C₄×M₃₂

—

#1744

—

C₄²×C₈

C₂²

yes

characteristic

#1564

C₄

yes

—

#1450

—

C₂×#322

C₄

yes

—

C₂×C₈²

—

#1376

—

#1276

—

C₂×M₃₂

—

#431

—

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

#364

—

#364

C₈

yes

complemented

#346

—

C₄×C₁₆

—

#336

—

#335

—

#334

C₈

yes

—

—

C₈

yes

complemented

C₈²

—

C₈²

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

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

—

C₂²×C₄

—

C₂²×C₄

#95

yes

characteristic

C₂²×C₄

#104

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₈

yes

—

C₄²

C₂²⋊C₈

yes

—

C₄²

yes

C₄²

yes

C₄²

C₄⋊C₈

yes

—

C₁₆

—

C₂³

#339

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

D₄⋊C₈

yes

—

105

C₂×C₄

Q₈⋊C₈

yes

—

106

C₂×C₄

#336

yes

characteristic

107

C₂×C₄

#336

yes

characteristic

108

C₂×C₄

#336

yes

characteristic

109

C₂×C₄

#339

yes

characteristic

110

C₂×C₄

#340

yes

characteristic

111

C₂×C₄

#341

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

C₈²

yes

characteristic

116

C₈

#322

yes

characteristic

117

C₈

#322

yes

characteristic

118

C₈

#322

yes

characteristic

119

C₈

D₄⋊C₈

yes

—

120

C₈

Q₈⋊C₈

yes

—

121

C₈

#334

yes

—

122

C₈

#335

yes

—

123

C₂²

—

124

C₂²

#1276

yes

characteristic

125

C₂²

#1278

yes

characteristic

126

C₂²

#1279

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#1273

yes

characteristic

132

C₄

#1277

yes

characteristic

133

C₄

#1278

yes

characteristic

134

C₄

#1279

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#7015

yes

characteristic

138

C₂

#7181

yes

—

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1279
prime radical	C₂³	#339
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#102

commutator

C₈

C₈²

Frattini, cosocle

C₄²×C₈

C₂²

C₂²

C₂³

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
25. #364	C₈	semidirect
30. #334	C₈	semidirect
33. #322	C₈	semidirect

H ≅ G/N

type

25. #364

C₈

semidirect

30. #334

C₈

semidirect

33. #322

C₈

semidirect

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

subgroup	index
C₄×#364	2
#8266	2
#7841	2

subgroup

index

C₄×#364

#8266

#7841

Subgroups — #93267

Conjugacy classes of subgroups

Complements

Maximal subgroups