Group #135653 — subgroups

Subgroups — #135653

Order 512.

270 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 146 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×#346	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	1	yes	maximal, characteristic
3	#8806	C₂	1	1	yes	maximal, characteristic
4	#7154	C₂	1	4	yes	maximal
5	C₂×#346	C₄	1	2	yes	—
6	C₂×#346	C₄	1	1	yes	characteristic
7	C₂×#346	C₄	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	2	yes	—
9	#1842	—	2	1	no	—
10	#1831	—	2	1	no	—
11	C₄²×C₈	C₂²	1	1	yes	characteristic
12	C₂×#335	—	2	1	no	—
13	C₂×#335	C₂²	1	2	yes	—
14	C₂×#335	C₂²	1	2	yes	—
15	C₂×#334	—	2	1	no	—
16	C₂²×C₁₆	C₂×C₄	1	2	yes	—
17	C₂²×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₄⋊C₈	—	2	2	no	—
19	C₂×C₄⋊C₈	—	2	2	no	—
20	C₂×C₄⋊C₈	—	4	2	no	—
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
25	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄³	D₄	1	1	yes	characteristic
28	#346	—	2	4	no	—
29	#346	—	2	2	no	—
30	#346	C₈	1	4	yes	complemented
31	C₄×C₁₆	—	2	2	no	—
32	C₄×C₁₆	—	2	2	no	—
33	#336	—	4	2	no	—
34	#335	—	2	4	no	—
35	#335	—	2	4	no	—
36	#335	—	4	2	no	—
37	#334	—	4	2	no	—
38	C₂²×C₈	—	4	2	no	—
39	C₂²×C₈	—	4	2	no	—
40	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄²	D₈	1	2	yes	—
47	C₂×C₄²	Q₁₆	1	2	yes	—
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	C₂×C₈	1	4	yes	—
54	C₂×C₁₆	M₁₆	1	4	yes	—
55	C₄⋊C₈	—	4	2	no	—
56	C₄⋊C₈	—	4	2	no	—
57	C₄⋊C₈	—	4	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	C₄⋊C₈	—	8	4	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	C₂×C₈	1	2	yes	—
64	C₄×C₈	M₁₆	1	2	yes	—
65	C₄×C₈	SD₁₆	1	2	yes	—
66	C₄×C₈	SD₁₆	1	2	yes	—
67	C₄×C₈	#41	1	2	yes	—
68	C₄×C₈	#41	1	2	yes	—
69	C₂²×C₄	#107	1	2	yes	—
70	C₂²×C₄	#107	1	2	yes	—
71	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
73	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	8	2	no	—
79	C₂×C₈	—	8	2	no	—
80	C₂×C₈	—	8	4	no	—
81	C₂×C₈	—	8	2	no	—
82	C₂×C₈	—	8	2	no	—
83	C₂×C₈	C₄⋊C₈	1	2	yes	—
84	C₂×C₈	#106	1	2	yes	—
85	C₂×C₈	#106	1	2	yes	—
86	C₂×C₈	#108	1	2	yes	—
87	C₂×C₈	#117	1	2	yes	—
88	C₂×C₈	#125	1	2	yes	—
89	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₈	C₂×M₁₆	1	2	yes	—
91	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
92	C₄²	—	2	1	no	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	2	no	—
95	C₄²	—	2	2	no	—
96	C₄²	—	4	2	no	—
97	C₄²	C₄⋊C₈	1	2	yes	—
98	C₄²	#108	1	2	yes	—
99	C₄²	#124	1	1	yes	characteristic
100	C₄²	#124	1	1	yes	characteristic
101	C₄²	C₂×SD₁₆	1	1	yes	characteristic
102	C₄²	C₂×SD₁₆	1	1	yes	characteristic
103	C₁₆	—	2	4	no	—
104	C₁₆	—	2	4	no	—
105	C₂³	C₂×#107	1	1	yes	characteristic
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	#335	1	4	yes	—
113	C₂×C₄	#382	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂×C₄	#423	1	1	yes	characteristic
116	C₂×C₄	C₂×#106	1	1	yes	characteristic
117	C₂×C₄	C₂×#108	1	1	yes	characteristic
118	C₂×C₄	C₂×#108	1	1	yes	characteristic
119	C₂×C₄	#486	1	1	yes	characteristic
120	C₂×C₄	#486	1	1	yes	characteristic
121	C₂×C₄	#487	1	1	yes	characteristic
122	C₂×C₄	#487	1	1	yes	characteristic
123	C₂×C₄	#509	1	2	yes	—
124	C₂×C₄	#510	1	2	yes	—
125	C₈	—	2	2	no	—
126	C₈	—	2	2	no	—
127	C₈	—	16	2	no	—
128	C₈	—	16	2	no	—
129	C₈	#334	1	4	yes	—
130	C₈	#431	1	2	yes	—
131	C₈	#432	1	2	yes	—
132	C₂²	C₂×#335	1	1	yes	characteristic
133	C₂²	#1567	1	1	yes	characteristic
134	C₂²	#1834	1	1	yes	characteristic
135	C₂²	#2159	1	2	yes	—
136	C₂²	#2161	1	2	yes	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	4	no	—
140	C₄	—	4	2	no	—
141	C₄	C₂×#334	1	1	yes	characteristic
142	C₄	#1566	1	1	yes	characteristic
143	C₄	#1831	1	1	yes	characteristic
144	C₄	#1837	1	1	yes	characteristic
145	C₄	#2243	1	2	yes	—
146	C₄	#2244	1	1	yes	characteristic
147	C₄	#2245	1	1	yes	characteristic
148	C₂	#8784	1	1	yes	characteristic
149	C₂	#12009	1	2	yes	—
150	C₂	#12025	1	2	yes	—
151	C₂	#12890	1	1	yes	characteristic
152	C₂	#12892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#346

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#8806

C₂

yes

maximal, characteristic

#7154

C₂

yes

maximal

C₂×#346

C₄

yes

—

C₂×#346

C₄

yes

characteristic

C₂×#346

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

#1842

—

#1831

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×#335

C₂²

yes

—

C₂×#335

C₂²

yes

—

C₂×#334

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

—

—

C₈

yes

complemented

C₄×C₁₆

—

C₄×C₁₆

—

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

—

C₂×C₄²

Q₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

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

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#125

yes

—

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₈

yes

—

C₄²

#108

yes

—

C₄²

#124

yes

characteristic

100

C₄²

#124

yes

characteristic

101

C₄²

C₂×SD₁₆

yes

characteristic

102

C₄²

C₂×SD₁₆

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₂³

C₂×#107

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#335

yes

—

113

C₂×C₄

#382

yes

characteristic

114

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

115

C₂×C₄

#423

yes

characteristic

116

C₂×C₄

C₂×#106

yes

characteristic

117

C₂×C₄

C₂×#108

yes

characteristic

118

C₂×C₄

C₂×#108

yes

characteristic

119

C₂×C₄

#486

yes

characteristic

120

C₂×C₄

#486

yes

characteristic

121

C₂×C₄

#487

yes

characteristic

122

C₂×C₄

#487

yes

characteristic

123

C₂×C₄

#509

yes

—

124

C₂×C₄

#510

yes

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#334

yes

—

130

C₈

#431

yes

—

131

C₈

#432

yes

—

132

C₂²

C₂×#335

yes

characteristic

133

C₂²

#1567

yes

characteristic

134

C₂²

#1834

yes

characteristic

135

C₂²

#2159

yes

—

136

C₂²

#2161

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₂×#334

yes

characteristic

142

C₄

#1566

yes

characteristic

143

C₄

#1831

yes

characteristic

144

C₄

#1837

yes

characteristic

145

C₄

#2243

yes

—

146

C₄

#2244

yes

characteristic

147

C₄

#2245

yes

characteristic

148

C₂

#8784

yes

characteristic

149

C₂

#12009

yes

—

150

C₂

#12025

yes

—

151

C₂

#12890

yes

characteristic

152

C₂

#12892

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₈

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×#107

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
30. #346	C₈	semidirect

H ≅ G/N

type

30. #346

C₈

semidirect

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

subgroup	index
C₄×#346	2
#8807	2
#8806	2
#7154	2

subgroup

index

Subgroups — #135653

Conjugacy classes of subgroups

Complements

Maximal subgroups