Group #497104 — subgroups

Subgroups — #497104

Order 512.

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

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 186 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	#33726	C₂	1	1	yes	maximal, characteristic
2	#21733	C₂	1	2	yes	maximal
3	#13286	C₂	1	2	yes	maximal
4	#13113	C₂	1	8	yes	maximal
5	#12891	C₂	1	1	yes	maximal, characteristic
6	#12592	C₂	1	1	yes	maximal, characteristic
7	#3160	C₂²	1	2	yes	—
8	#2961	—	2	2	no	—
9	#2254	C₄	1	2	yes	complemented
10	#2253	C₄	1	4	yes	complemented
11	#2252	C₄	1	2	yes	complemented
12	#2158	C₂²	1	4	yes	—
13	C₂×#346	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#1995	—	2	2	no	—
16	#1946	—	2	2	no	—
17	#1945	C₂²	1	4	yes	—
18	#1848	C₂²	1	2	yes	—
19	#1773	—	2	2	no	—
20	#1772	C₂²	1	4	yes	—
21	#1726	C₂²	1	1	yes	characteristic
22	#1382	C₂²	1	8	yes	—
23	#1381	C₂²	1	4	yes	—
24	#1374	C₂²	1	4	yes	—
25	#533	—	2	2	no	—
26	C₂²×C₁₆	C₂³	1	2	yes	—
27	C₂²×C₁₆	C₂³	1	2	yes	—
28	#500	C₂×C₄	1	4	yes	—
29	#499	—	2	2	no	—
30	#499	C₂×C₄	1	4	yes	—
31	#498	—	2	2	no	—
32	#446	—	4	8	no	—
33	C₂×#108	—	2	2	no	—
34	C₂×#108	C₂³	1	4	yes	—
35	#427	—	2	2	no	—
36	#427	C₂³	1	4	yes	—
37	#424	—	2	4	no	—
38	#423	—	2	2	no	—
39	#405	—	2	4	no	—
40	C₂×#97	D₄	1	2	yes	—
41	C₂×#97	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	2	yes	—
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#376	D₄	1	1	yes	characteristic
45	#367	C₂×C₄	1	8	yes	—
46	#366	C₂×C₄	1	8	yes	—
47	#346	—	2	1	no	—
48	#346	—	2	1	no	—
49	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
50	C₄×C₁₆	C₂×C₄	1	2	yes	—
51	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
52	#339	—	2	4	no	—
53	#339	—	2	4	no	—
54	C₂×M₁₆	—	2	4	no	—
55	C₂²×C₈	C₂×D₄	1	2	yes	—
56	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₈	C₂⁴	1	1	yes	characteristic
58	C₄⋊Q₈	—	2	4	no	—
59	#125	—	2	4	no	—
60	C₂²⋊Q₈	—	4	4	no	—
61	C₄×Q₈	—	4	4	no	—
62	#117	—	2	4	no	—
63	C₂×C₄²	C₂×D₄	1	2	yes	—
64	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
65	C₂×C₁₆	—	2	4	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	2	4	no	—
68	C₂×C₁₆	C₂²×C₄	1	2	yes	—
69	C₂×C₁₆	C₂²×C₄	1	2	yes	—
70	#108	—	2	8	no	—
71	#108	—	4	4	no	—
72	#107	—	2	4	no	—
73	#107	C₂²×C₄	1	8	yes	—
74	#106	—	2	4	no	—
75	#106	—	2	4	no	—
76	C₄⋊C₈	—	4	8	no	—
77	C₄⋊C₈	—	4	4	no	—
78	#97	—	2	4	no	—
79	#97	—	4	4	no	—
80	#97	D₈	1	4	yes	—
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	4	4	no	—
83	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₄×C₈	#41	1	1	yes	characteristic
86	C₄×C₈	#41	1	1	yes	characteristic
87	#95	—	2	2	no	—
88	C₂×Q₈	—	4	4	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	#135	1	2	yes	—
91	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
92	M₁₆	—	4	4	no	—
93	M₁₆	—	4	4	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	4	no	—
97	C₂×C₈	C₄×D₄	1	1	yes	characteristic
98	C₂×C₈	C₄×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₈	1	4	yes	—
100	C₂×C₈	#137	1	4	yes	—
101	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₈	C₂×#41	1	1	yes	characteristic
103	C₂×C₈	2₋¹⁺⁴	1	4	yes	—
104	C₄⋊C₄	—	2	8	no	—
105	C₄⋊C₄	—	4	8	no	—
106	C₄⋊C₄	—	4	4	no	—
107	C₂²⋊C₄	—	4	4	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	4	4	no	—
111	C₄²	C₄×D₄	1	1	yes	characteristic
112	C₄²	C₄×D₄	1	1	yes	characteristic
113	C₄²	C₂×D₈	1	2	yes	—
114	C₁₆	—	2	2	no	—
115	C₁₆	—	2	2	no	—
116	C₁₆	—	4	2	no	—
117	C₂³	C₂×#135	1	1	yes	characteristic
118	Q₈	—	16	4	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	4	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	C₄×D₈	1	2	yes	—
124	C₂×C₄	C₄×D₈	1	2	yes	—
125	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂×C₄	#536	1	2	yes	—
127	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
128	C₂×C₄	C₂×#137	1	2	yes	—
129	C₈	—	2	4	no	—
130	C₈	—	8	4	no	—
131	C₈	#441	1	2	yes	—
132	C₈	#441	1	2	yes	—
133	C₈	#519	1	2	yes	—
134	C₈	#541	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
137	C₂²	#3231	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	8	8	no	—
142	C₄	#2327	1	1	yes	characteristic
143	C₄	C₂×#441	1	1	yes	characteristic
144	C₄	#3226	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	#22247	1	1	yes	characteristic
147	C₂	#33465	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33726

C₂

yes

maximal, characteristic

#21733

C₂

yes

maximal

#13286

C₂

yes

maximal

#13113

C₂

yes

maximal

#12891

C₂

yes

maximal, characteristic

#12592

C₂

yes

maximal, characteristic

#3160

C₂²

yes

—

#2961

—

#2254

C₄

yes

complemented

#2253

C₄

yes

complemented

#2252

C₄

yes

complemented

#2158

C₂²

yes

—

C₂×#346

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

C₂²

yes

—

#1848

C₂²

yes

—

#1773

—

#1772

C₂²

yes

—

#1726

C₂²

yes

characteristic

#1382

C₂²

yes

—

#1381

C₂²

yes

—

#1374

C₂²

yes

—

#533

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

#500

C₂×C₄

yes

—

#499

—

#499

C₂×C₄

yes

—

—

—

—

C₂³

yes

—

#427

—

#427

C₂³

yes

—

#424

—

#423

—

#405

—

C₂×#97

D₄

yes

—

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#376

D₄

yes

characteristic

#367

C₂×C₄

yes

—

#366

C₂×C₄

yes

—

#346

—

#346

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

characteristic

#339

—

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

—

—

—

—

C₂²×C₄

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

D₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#135

yes

—

C₂²×C₄

C₂²×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₂×D₈

yes

—

100

C₂×C₈

#137

yes

—

101

C₂×C₈

C₂³×C₄

yes

characteristic

102

C₂×C₈

C₂×#41

yes

characteristic

103

C₂×C₈

2₋¹⁺⁴

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₄×D₄

yes

characteristic

112

C₄²

C₄×D₄

yes

characteristic

113

C₄²

C₂×D₈

yes

—

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₂³

C₂×#135

yes

characteristic

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₄×D₈

yes

—

124

C₂×C₄

C₄×D₈

yes

—

125

C₂×C₄

C₂×C₄×D₄

yes

characteristic

126

C₂×C₄

#536

yes

—

127

C₂×C₄

C₂²×D₈

yes

characteristic

128

C₂×C₄

C₂×#137

yes

—

129

C₈

—

130

C₈

—

131

C₈

#441

yes

—

132

C₈

#441

yes

—

133

C₈

#519

yes

—

134

C₈

#541

yes

—

135

C₂²

—

136

C₂²

C₂×C₄×D₈

yes

characteristic

137

C₂²

#3231

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#2327

yes

characteristic

143

C₄

C₂×#441

yes

characteristic

144

C₄

#3226

yes

—

145

C₂

—

146

C₂

#22247

yes

characteristic

147

C₂

#33465

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₈

commutator

C₂×C₈

C₂³×C₄

Frattini, cosocle

C₂²×C₈

C₂⁴

socle

C₂²

C₂×C₄×D₈

prime radical

C₂³

C₂×#135

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
9. #2254	C₄	semidirect
10. #2253	C₄	semidirect
11. #2252	C₄	semidirect

H ≅ G/N

type

9. #2254

C₄

semidirect

10. #2253

C₄

semidirect

11. #2252

C₄

semidirect

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

subgroup	index
#33726	2
#21733	2
#13286	2
#13113	2
#12891	2
#12592	2

subgroup

index

Subgroups — #497104

Conjugacy classes of subgroups

Complements

Maximal subgroups