Group #92939 — subgroups

Subgroups — #92939

Order 512.

193 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 90 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	#12012	C₂	1	1	yes	maximal, characteristic
2	#11410	C₂	1	1	yes	maximal, characteristic
3	C₈×#106	C₂	1	1	yes	maximal, characteristic
4	#2162	—	2	1	no	—
5	#2162	—	2	1	no	—
6	C₄×#106	C₄	1	1	yes	characteristic
7	C₈×C₄⋊C₄	—	2	1	no	—
8	C₂×#334	C₄	1	1	yes	characteristic
9	C₂×C₈²	C₂²	1	1	yes	characteristic
10	#1370	C₄	1	2	yes	complemented
11	#1367	—	2	1	no	—
12	#1312	C₄	1	2	yes	complemented
13	C₈×C₁₆	—	2	1	no	—
14	#1279	—	2	1	no	—
15	C₂×#106	C₈	1	2	yes	—
16	C₂×C₄⋊C₈	—	2	1	no	—
17	C₂×C₄×C₈	—	4	1	no	—
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	C₄×C₄⋊C₄	—	2	1	no	—
22	C₄⋊C₁₆	—	2	2	no	—
23	C₄⋊C₁₆	—	2	2	no	—
24	C₂²⋊C₁₆	—	2	2	no	—
25	C₂²⋊C₁₆	—	4	1	no	—
26	C₄×C₁₆	—	2	2	no	—
27	C₄×C₁₆	—	4	1	no	—
28	#340	—	2	1	no	—
29	#336	—	4	1	no	—
30	#334	—	2	1	no	—
31	#334	C₈	1	2	yes	—
32	C₈²	C₂×C₄	1	1	yes	characteristic
33	C₈²	C₂×C₄	1	1	yes	characteristic
34	C₈²	D₄	1	1	yes	characteristic
35	C₈²	Q₈	1	1	yes	characteristic
36	C₂²×C₈	—	2	1	no	—
37	C₂²×C₈	—	4	1	no	—
38	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₈	M₁₆	1	1	yes	characteristic
40	C₂²×C₈	D₈	1	1	yes	characteristic
41	C₂²×C₈	SD₁₆	1	1	yes	characteristic
42	C₂×C₄⋊C₄	—	2	2	no	—
43	C₂×C₄²	—	4	1	no	—
44	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₁₆	—	2	4	no	—
46	C₂×C₁₆	—	4	2	no	—
47	#106	—	2	2	no	—
48	#106	C₁₆	1	4	yes	complemented
49	C₄⋊C₈	—	2	2	no	—
50	C₄⋊C₈	—	4	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	8	2	no	—
53	C₄×C₈	C₄²	1	1	yes	characteristic
54	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₈	C₂×C₈	1	1	yes	characteristic
60	C₄×C₈	M₁₆	1	1	yes	characteristic
61	C₄×C₈	SD₁₆	1	1	yes	characteristic
62	C₄×C₈	Q₁₆	1	1	yes	characteristic
63	#95	—	4	1	no	—
64	C₂²×C₄	—	4	2	no	—
65	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
66	C₂²×C₄	#102	1	1	yes	characteristic
67	C₂²×C₄	#104	1	1	yes	characteristic
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	—	4	2	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	8	1	no	—
76	C₂×C₈	—	8	1	no	—
77	C₂×C₈	C₄×C₈	1	1	yes	characteristic
78	C₂×C₈	#97	1	1	yes	characteristic
79	C₂×C₈	#102	1	1	yes	characteristic
80	C₂×C₈	#103	1	1	yes	characteristic
81	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
83	C₂×C₈	#106	1	1	yes	characteristic
84	C₂×C₈	#107	1	1	yes	characteristic
85	C₂×C₈	C₂×C₁₆	1	2	yes	—
86	C₂×C₈	M₃₂	1	2	yes	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄²	—	8	1	no	—
90	C₄²	—	8	1	no	—
91	C₄²	#95	1	1	yes	characteristic
92	C₄²	C₂²⋊C₈	1	1	yes	characteristic
93	C₄²	#103	1	1	yes	characteristic
94	C₄²	#104	1	1	yes	characteristic
95	C₁₆	—	4	4	no	—
96	C₁₆	—	8	2	no	—
97	C₂³	D₄⋊C₈	1	1	yes	characteristic
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
105	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
106	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
107	C₂×C₄	#336	1	1	yes	characteristic
108	C₂×C₄	#339	1	1	yes	characteristic
109	C₂×C₄	#340	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
111	C₂×C₄	#350	1	2	yes	—
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	4	1	no	—
119	C₈	—	8	2	no	—
120	C₈	C₄×C₁₆	1	1	yes	characteristic
121	C₈	#346	1	1	yes	characteristic
122	C₈	#360	1	2	yes	—
123	C₈	C₄⋊C₁₆	1	2	yes	—
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	#1279	1	1	yes	characteristic
127	C₂²	D₄⋊C₁₆	1	2	yes	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	8	4	no	—
132	C₄	Q₈⋊C₁₆	1	2	yes	—
133	C₄	#1374	1	1	yes	characteristic
134	C₄	#1379	1	1	yes	characteristic
135	C₄	#1386	1	1	yes	characteristic
136	C₄	#1389	1	1	yes	characteristic
137	C₂	—	2	2	no	—
138	C₂	#7167	1	1	yes	characteristic
139	C₂	#7175	1	1	yes	characteristic
140	C₂	#7181	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12012

C₂

yes

maximal, characteristic

#11410

C₂

yes

maximal, characteristic

C₈×#106

C₂

yes

maximal, characteristic

#2162

—

#2162

—

C₄×#106

C₄

yes

characteristic

C₈×C₄⋊C₄

—

C₂×#334

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1370

C₄

yes

complemented

#1367

—

#1312

C₄

yes

complemented

C₈×C₁₆

—

#1279

—

C₂×#106

C₈

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₄×C₄⋊C₄

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

—

—

—

C₈

yes

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#106

—

#106

C₁₆

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₈

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

C₄×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

#95

yes

characteristic

C₄²

C₂²⋊C₈

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₂³

D₄⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

D₄⋊C₈

yes

characteristic

105

C₂×C₄

Q₈⋊C₈

yes

characteristic

106

C₂×C₄

Q₈⋊C₈

yes

characteristic

107

C₂×C₄

#336

yes

characteristic

108

C₂×C₄

#339

yes

characteristic

109

C₂×C₄

#340

yes

characteristic

110

C₂×C₄

C₂²⋊C₁₆

yes

—

111

C₂×C₄

#350

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

C₄×C₁₆

yes

characteristic

121

C₈

#346

yes

characteristic

122

C₈

#360

yes

—

123

C₈

C₄⋊C₁₆

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

#1279

yes

characteristic

127

C₂²

D₄⋊C₁₆

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

Q₈⋊C₁₆

yes

—

133

C₄

#1374

yes

characteristic

134

C₄

#1379

yes

characteristic

135

C₄

#1386

yes

characteristic

136

C₄

#1389

yes

characteristic

137

C₂

—

138

C₂

#7167

yes

characteristic

139

C₂

#7175

yes

characteristic

140

C₂

#7181

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₄×C₁₆
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1279
prime radical	C₂³	D₄⋊C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₈

#102

commutator

C₈

C₄×C₁₆

Frattini, cosocle

C₂×C₈²

C₂²

socle

C₂²

#1279

prime radical

C₂³

D₄⋊C₈

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
10. #1370	C₄	semidirect
12. #1312	C₄	semidirect
48. #106	C₁₆	semidirect

H ≅ G/N

type

10. #1370

C₄

semidirect

12. #1312

C₄

semidirect

48. #106

C₁₆

semidirect

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

subgroup	index
#12012	2
#11410	2
C₈×#106	2

subgroup

index

#12012

#11410

C₈×#106

Subgroups — #92939

Conjugacy classes of subgroups

Complements

Maximal subgroups