Group #92941 — subgroups

Subgroups — #92941

Order 512.

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

normal subgroups only

This group has 186 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	C₂×#1312	C₂	1	1	yes	maximal, characteristic
3	C₈×#108	C₂	1	1	yes	maximal, characteristic
4	#2162	—	2	1	no	—
5	C₂×C₄×C₁₆	—	2	1	no	—
6	C₄×#108	C₄	1	1	yes	characteristic
7	#1564	C₄	1	1	yes	characteristic
8	C₈×M₁₆	—	2	1	no	—
9	C₂×C₈²	C₂²	1	1	yes	characteristic
10	#1370	—	2	1	no	—
11	#1367	C₄	1	2	yes	—
12	#1312	—	2	1	no	—
13	#1312	C₄	1	2	yes	—
14	#1278	—	2	1	no	—
15	C₂²×C₁₆	—	2	2	no	—
16	#431	—	2	1	no	—
17	C₂×#108	C₈	1	2	yes	—
18	C₄×M₁₆	—	2	1	no	—
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	C₄⋊C₁₆	—	2	2	no	—
23	C₂²⋊C₁₆	—	4	1	no	—
24	C₄×C₁₆	—	2	2	no	—
25	C₄×C₁₆	—	2	2	no	—
26	C₄×C₁₆	—	4	1	no	—
27	#341	—	2	1	no	—
28	#335	C₈	1	2	yes	—
29	#334	—	2	1	no	—
30	#322	—	4	1	no	—
31	C₈²	—	4	1	no	—
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	C₂×M₁₆	—	2	2	no	—
37	C₂²×C₈	—	2	1	no	—
38	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₈	M₁₆	1	1	yes	characteristic
40	C₂²×C₈	SD₁₆	1	1	yes	characteristic
41	C₂²×C₈	Q₁₆	1	1	yes	characteristic
42	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₁₆	—	2	4	no	—
44	C₂×C₁₆	—	4	4	no	—
45	C₂×C₁₆	—	4	2	no	—
46	#108	—	2	2	no	—
47	#108	C₁₆	1	4	yes	complemented
48	C₄⋊C₈	—	2	2	no	—
49	C₂²⋊C₈	—	4	1	no	—
50	#97	—	4	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	4	1	no	—
53	C₄×C₈	—	4	1	no	—
54	C₄×C₈	C₄²	1	1	yes	characteristic
55	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₈	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₈	C₂×C₈	1	1	yes	characteristic
61	C₄×C₈	M₁₆	1	1	yes	characteristic
62	C₄×C₈	D₈	1	1	yes	characteristic
63	C₄×C₈	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
65	C₂²×C₄	#103	1	1	yes	characteristic
66	C₂²×C₄	#104	1	1	yes	characteristic
67	M₁₆	—	2	4	no	—
68	M₁₆	—	4	2	no	—
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	4	2	no	—
76	C₂×C₈	—	4	2	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	C₄²	#95	1	1	yes	characteristic
88	C₄²	C₂²⋊C₈	1	1	yes	characteristic
89	C₄²	#102	1	1	yes	characteristic
90	C₄²	#104	1	1	yes	characteristic
91	C₁₆	—	4	4	no	—
92	C₁₆	—	8	2	no	—
93	C₂³	Q₈⋊C₈	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
98	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
99	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
100	C₂×C₄	#336	1	1	yes	characteristic
101	C₂×C₄	#339	1	1	yes	characteristic
102	C₂×C₄	#340	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
104	C₂×C₄	#350	1	2	yes	—
105	C₈	—	2	2	no	—
106	C₈	—	2	2	no	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	4	4	no	—
114	C₈	—	8	2	no	—
115	C₈	C₄×C₁₆	1	1	yes	characteristic
116	C₈	#346	1	1	yes	characteristic
117	C₈	C₄⋊C₁₆	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#1279	1	1	yes	characteristic
121	C₂²	Q₈⋊C₁₆	1	2	yes	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	D₄⋊C₁₆	1	2	yes	—
125	C₄	#1336	1	2	yes	—
126	C₄	#1374	1	1	yes	characteristic
127	C₄	#1379	1	1	yes	characteristic
128	C₄	#1388	1	1	yes	characteristic
129	C₄	#1389	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	#7167	1	1	yes	characteristic
132	C₂	#7177	1	1	yes	characteristic
133	C₂	#7182	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12012

C₂

yes

maximal, characteristic

C₂×#1312

C₂

yes

maximal, characteristic

C₈×#108

C₂

yes

maximal, characteristic

#2162

—

C₂×C₄×C₁₆

—

C₄×#108

C₄

yes

characteristic

#1564

C₄

yes

characteristic

C₈×M₁₆

—

C₂×C₈²

C₂²

yes

characteristic

#1370

—

#1367

C₄

yes

—

#1312

—

#1312

C₄

yes

—

#1278

—

C₂²×C₁₆

—

#431

—

C₂×#108

C₈

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#341

—

#335

C₈

yes

—

#334

—

#322

—

C₈²

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

Q₁₆

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

C₁₆

yes

complemented

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

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

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₂²×C₄

C₂²⋊C₈

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

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

—

C₄²

#95

yes

characteristic

C₄²

C₂²⋊C₈

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₂³

Q₈⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄⋊C₈

yes

characteristic

C₂×C₄

D₄⋊C₈

yes

characteristic

C₂×C₄

Q₈⋊C₈

yes

characteristic

100

C₂×C₄

#336

yes

characteristic

101

C₂×C₄

#339

yes

characteristic

102

C₂×C₄

#340

yes

characteristic

103

C₂×C₄

C₂²⋊C₁₆

yes

—

104

C₂×C₄

#350

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

C₄×C₁₆

yes

characteristic

116

C₈

#346

yes

characteristic

117

C₈

C₄⋊C₁₆

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

#1279

yes

characteristic

121

C₂²

Q₈⋊C₁₆

yes

—

122

C₄

—

123

C₄

—

124

C₄

D₄⋊C₁₆

yes

—

125

C₄

#1336

yes

—

126

C₄

#1374

yes

characteristic

127

C₄

#1379

yes

characteristic

128

C₄

#1388

yes

characteristic

129

C₄

#1389

yes

characteristic

130

C₂

—

131

C₂

#7167

yes

characteristic

132

C₂

#7177

yes

characteristic

133

C₂

#7182

yes

characteristic

	subgroup	G/H
center	C₄²	#102
commutator	C₈	C₄×C₁₆
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1279
prime radical	C₂³	Q₈⋊C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₄²

#102

commutator

C₈

C₄×C₁₆

Frattini, cosocle

C₂×C₈²

C₂²

socle

C₂²

#1279

prime radical

C₂³

Q₈⋊C₈

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
47. #108	C₁₆	semidirect

H ≅ G/N

type

47. #108

C₁₆

semidirect

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

subgroup	index
#12012	2
C₂×#1312	2
C₈×#108	2

subgroup

index

#12012

C₂×#1312

C₈×#108

Subgroups — #92941

Conjugacy classes of subgroups

Complements

Maximal subgroups