Group #93273 — subgroups

Subgroups — #93273

Order 512.

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

normal subgroups only

This group has 236 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	C₄×#364	C₂	1	1	yes	maximal, characteristic
3	C₄×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#364	C₄	1	2	yes	—
5	C₂×#364	C₄	1	2	yes	—
6	C₄×M₃₂	—	2	1	no	—
7	C₄×M₃₂	—	2	1	no	—
8	C₂×#346	—	2	1	no	—
9	C₂×#346	C₄	1	2	yes	—
10	C₂×C₄×C₁₆	—	2	1	no	—
11	C₄²×C₈	C₂²	1	1	yes	characteristic
12	#1376	—	2	1	no	—
13	#1376	—	2	1	no	—
14	C₂×M₃₂	—	2	2	no	—
15	C₂×M₃₂	—	2	2	no	—
16	C₂²×C₁₆	—	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	D₄	1	1	yes	characteristic
23	C₄³	Q₈	1	1	yes	characteristic
24	#364	—	2	2	no	—
25	#364	—	2	2	no	—
26	#364	C₈	1	4	yes	—
27	#364	C₈	1	4	yes	—
28	#346	—	2	4	no	—
29	#346	—	2	2	no	—
30	#346	—	4	1	no	—
31	#346	—	4	1	no	—
32	#346	C₈	1	4	yes	—
33	C₄×C₁₆	—	2	4	no	—
34	C₄×C₁₆	—	4	1	no	—
35	C₄×C₁₆	—	4	1	no	—
36	C₂²×C₈	C₄²	1	1	yes	characteristic
37	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄²	—	2	2	no	—
41	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
44	M₃₂	—	2	4	no	—
45	M₃₂	—	2	4	no	—
46	M₃₂	—	4	2	no	—
47	M₃₂	—	4	2	no	—
48	C₂×C₁₆	—	2	4	no	—
49	C₂×C₁₆	—	2	4	no	—
50	C₂×C₁₆	—	2	4	no	—
51	C₂×C₁₆	—	4	2	no	—
52	C₂×C₁₆	—	4	2	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	—	2	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	#108	1	2	yes	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	C₄×C₈	1	1	yes	characteristic
72	C₂×C₈	C₄×C₈	1	1	yes	characteristic
73	C₂×C₈	C₄×C₈	1	1	yes	characteristic
74	C₂×C₈	#97	1	1	yes	characteristic
75	C₂×C₈	#97	1	1	yes	characteristic
76	C₂×C₈	#97	1	1	yes	characteristic
77	C₂×C₈	C₂²⋊C₈	1	2	yes	—
78	C₂×C₈	C₂²⋊C₈	1	2	yes	—
79	C₂×C₈	C₂²⋊C₈	1	2	yes	—
80	C₂×C₈	#102	1	1	yes	characteristic
81	C₂×C₈	#102	1	1	yes	characteristic
82	C₂×C₈	#103	1	1	yes	characteristic
83	C₂×C₈	#103	1	1	yes	characteristic
84	C₂×C₈	#104	1	2	yes	—
85	C₂×C₈	#104	1	2	yes	—
86	C₄²	—	2	1	no	—
87	C₄²	—	2	1	no	—
88	C₄²	—	4	4	no	—
89	C₄²	C₄⋊C₈	1	2	yes	—
90	C₄²	C₄⋊C₈	1	2	yes	—
91	C₄²	C₄⋊C₈	1	2	yes	—
92	C₄²	#106	1	1	yes	characteristic
93	C₄²	#107	1	1	yes	characteristic
94	C₁₆	—	2	8	no	—
95	C₁₆	—	4	4	no	—
96	C₁₆	—	4	4	no	—
97	C₂³	#341	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	#334	1	2	yes	—
104	C₂×C₄	#335	1	2	yes	—
105	C₂×C₄	#336	1	1	yes	characteristic
106	C₂×C₄	#336	1	1	yes	characteristic
107	C₂×C₄	#336	1	1	yes	characteristic
108	C₂×C₄	#339	1	1	yes	characteristic
109	C₂×C₄	#339	1	1	yes	characteristic
110	C₂×C₄	#340	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	C₈²	1	1	yes	characteristic
113	C₈	#322	1	1	yes	characteristic
114	C₈	#322	1	1	yes	characteristic
115	C₈	#322	1	1	yes	characteristic
116	C₈	D₄⋊C₈	1	2	yes	—
117	C₈	D₄⋊C₈	1	2	yes	—
118	C₈	Q₈⋊C₈	1	2	yes	—
119	C₈	Q₈⋊C₈	1	2	yes	—
120	C₂²	—	2	2	no	—
121	C₂²	#1277	1	1	yes	characteristic
122	C₂²	#1278	1	1	yes	characteristic
123	C₂²	#1278	1	1	yes	characteristic
124	C₄	—	2	4	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	—	4	2	no	—
128	C₄	#1273	1	1	yes	characteristic
129	C₄	#1276	1	1	yes	characteristic
130	C₄	#1279	1	1	yes	characteristic
131	C₄	#1279	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#7015	1	1	yes	characteristic
135	C₂	#7162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#364

C₂

yes

maximal, characteristic

C₄×#364

C₂

yes

maximal, characteristic

C₄×#346

C₂

yes

maximal, characteristic

C₂×#364

C₄

yes

—

C₂×#364

C₄

yes

—

C₄×M₃₂

—

C₄×M₃₂

—

C₂×#346

—

C₂×#346

C₄

yes

—

C₂×C₄×C₁₆

—

C₄²×C₈

C₂²

yes

characteristic

#1376

—

#1376

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

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

D₄

yes

characteristic

C₄³

Q₈

yes

—

—

C₈

yes

—

#364

C₈

yes

—

—

—

—

—

C₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

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

#108

yes

—

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

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

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₈

yes

—

C₄²

C₄⋊C₈

yes

—

C₄²

C₄⋊C₈

yes

—

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#341

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#334

yes

—

104

C₂×C₄

#335

yes

—

105

C₂×C₄

#336

yes

characteristic

106

C₂×C₄

#336

yes

characteristic

107

C₂×C₄

#336

yes

characteristic

108

C₂×C₄

#339

yes

characteristic

109

C₂×C₄

#339

yes

characteristic

110

C₂×C₄

#340

yes

characteristic

111

C₈

—

112

C₈

C₈²

yes

characteristic

113

C₈

#322

yes

characteristic

114

C₈

#322

yes

characteristic

115

C₈

#322

yes

characteristic

116

C₈

D₄⋊C₈

yes

—

117

C₈

D₄⋊C₈

yes

—

118

C₈

Q₈⋊C₈

yes

—

119

C₈

Q₈⋊C₈

yes

—

120

C₂²

—

121

C₂²

#1277

yes

characteristic

122

C₂²

#1278

yes

characteristic

123

C₂²

#1278

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#1273

yes

characteristic

129

C₄

#1276

yes

characteristic

130

C₄

#1279

yes

characteristic

131

C₄

#1279

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#7015

yes

characteristic

135

C₂

#7162

yes

—

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1277
prime radical	C₂³	#341
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

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

subgroup	index
C₄×#364	2
C₄×#346	2

subgroup

index

C₄×#364

C₄×#346

Subgroups — #93273

Conjugacy classes of subgroups

Maximal subgroups