Group #129114 — subgroups

Subgroups — #129114

Order 512.

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

normal subgroups only

This group has 280 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	C₂×#1372	C₂	1	1	yes	maximal, characteristic
3	C₂×#1371	C₂	1	1	yes	maximal, characteristic
4	#8274	C₂	1	2	yes	maximal
5	#7154	C₂	1	2	yes	maximal
6	C₂×#346	C₄	1	2	yes	—
7	C₂×#346	C₄	1	2	yes	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	#1767	—	2	2	no	—
11	C₄²×C₈	C₂²	1	1	yes	characteristic
12	C₂×#335	C₂²	1	2	yes	—
13	C₂×#335	C₂²	1	2	yes	—
14	#1372	C₄	1	4	yes	complemented
15	#1371	C₄	1	4	yes	complemented
16	#1277	—	2	2	no	—
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	2	yes	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊C₈	—	2	2	no	—
21	C₂×C₄⋊C₈	—	2	2	no	—
22	C₂×#97	—	4	2	no	—
23	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	Q₈	1	2	yes	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄³	D₄	1	1	yes	characteristic
28	#346	—	2	4	no	—
29	#346	—	2	4	no	—
30	C₄×C₁₆	—	2	1	no	—
31	C₄×C₁₆	—	2	1	no	—
32	C₄×C₁₆	C₂×C₄	1	2	yes	—
33	C₄×C₁₆	C₂×C₄	1	2	yes	—
34	#336	—	4	2	no	—
35	#335	—	2	2	no	—
36	#335	—	2	2	no	—
37	#335	C₂×C₄	1	4	yes	—
38	#335	C₂×C₄	1	4	yes	—
39	C₂²×C₈	—	4	2	no	—
40	C₂²×C₈	—	4	2	no	—
41	C₂²×C₈	C₄⋊C₄	1	2	yes	—
42	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄²	C₄⋊C₄	1	2	yes	—
45	C₂×C₄²	D₈	1	2	yes	—
46	C₂×C₄²	Q₁₆	1	2	yes	—
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	C₄²	1	2	yes	—
52	C₂×C₁₆	C₄²	1	2	yes	—
53	C₂×C₁₆	M₁₆	1	4	yes	—
54	C₂×C₁₆	M₁₆	1	4	yes	—
55	C₄⋊C₈	—	2	4	no	—
56	C₄⋊C₈	—	2	4	no	—
57	C₄⋊C₈	—	4	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	#97	—	8	4	no	—
60	C₄×C₈	—	2	4	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₄×C₈	#41	1	1	yes	characteristic
68	C₄×C₈	#41	1	1	yes	characteristic
69	C₂²×C₄	#107	1	4	yes	—
70	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	—	4	4	no	—
78	C₂×C₈	—	8	2	no	—
79	C₂×C₈	—	8	2	no	—
80	C₂×C₈	—	8	2	no	—
81	C₂×C₈	—	8	2	no	—
82	C₂×C₈	#108	1	4	yes	—
83	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₈	#117	1	1	yes	characteristic
85	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₈	C₂×M₁₆	1	2	yes	—
88	C₂×C₈	C₂×M₁₆	1	2	yes	—
89	C₄²	—	2	2	no	—
90	C₄²	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	4	2	no	—
94	C₄²	C₄×D₄	1	1	yes	characteristic
95	C₄²	C₄×D₄	1	1	yes	characteristic
96	C₄²	#135	1	1	yes	characteristic
97	C₄²	#135	1	1	yes	characteristic
98	C₁₆	—	2	2	no	—
99	C₁₆	—	2	4	no	—
100	C₁₆	#97	1	4	yes	complemented
101	C₂³	C₂×#107	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄	#423	1	2	yes	—
108	C₂×C₄	#427	1	1	yes	characteristic
109	C₂×C₄	C₂×#108	1	2	yes	—
110	C₂×C₄	C₄×D₈	1	1	yes	characteristic
111	C₂×C₄	C₄×D₈	1	1	yes	characteristic
112	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
113	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
114	C₂×C₄	#509	1	2	yes	—
115	C₂×C₄	#510	1	2	yes	—
116	C₈	—	2	4	no	—
117	C₈	—	2	1	no	—
118	C₈	—	2	1	no	—
119	C₈	—	8	4	no	—
120	C₈	—	8	4	no	—
121	C₈	C₂×#97	1	1	yes	characteristic
122	C₈	C₄×M₁₆	1	1	yes	characteristic
123	C₈	#432	1	2	yes	—
124	C₂²	#1567	1	2	yes	—
125	C₂²	C₄×#107	1	1	yes	characteristic
126	C₂²	#2159	1	2	yes	—
127	C₂²	#2161	1	2	yes	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	4	4	no	—
131	C₄	#1568	1	2	yes	—
132	C₄	#1767	1	1	yes	characteristic
133	C₄	C₄×#108	1	1	yes	characteristic
134	C₄	#2175	1	2	yes	—
135	C₄	#2176	1	1	yes	characteristic
136	C₄	#2177	1	1	yes	characteristic
137	C₂	#8274	1	1	yes	characteristic
138	C₂	#12025	1	4	yes	—
139	C₂	#12674	1	1	yes	characteristic
140	C₂	#12676	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#346

C₂

yes

maximal, characteristic

C₂×#1372

C₂

yes

maximal, characteristic

C₂×#1371

C₂

yes

maximal, characteristic

#8274

C₂

yes

maximal

#7154

C₂

yes

maximal

C₂×#346

C₄

yes

—

C₂×#346

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1767

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

C₂×#335

C₂²

yes

—

#1372

C₄

yes

complemented

#1371

C₄

yes

complemented

#1277

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

#335

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

D₈

yes

—

C₂×C₄²

Q₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

M₁₆

yes

—

C₂×C₁₆

M₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₂²×C₄

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

#108

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₄

yes

characteristic

C₄²

C₄×D₄

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

100

C₁₆

#97

yes

complemented

101

C₂³

C₂×#107

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

107

C₂×C₄

#423

yes

—

108

C₂×C₄

#427

yes

characteristic

109

C₂×C₄

C₂×#108

yes

—

110

C₂×C₄

C₄×D₈

yes

characteristic

111

C₂×C₄

C₄×D₈

yes

characteristic

112

C₂×C₄

C₄×Q₁₆

yes

characteristic

113

C₂×C₄

C₄×Q₁₆

yes

characteristic

114

C₂×C₄

#509

yes

—

115

C₂×C₄

#510

yes

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

C₂×#97

yes

characteristic

122

C₈

C₄×M₁₆

yes

characteristic

123

C₈

#432

yes

—

124

C₂²

#1567

yes

—

125

C₂²

C₄×#107

yes

characteristic

126

C₂²

#2159

yes

—

127

C₂²

#2161

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#1568

yes

—

132

C₄

#1767

yes

characteristic

133

C₄

C₄×#108

yes

characteristic

134

C₄

#2175

yes

—

135

C₄

#2176

yes

characteristic

136

C₄

#2177

yes

characteristic

137

C₂

#8274

yes

characteristic

138

C₂

#12025

yes

—

139

C₂

#12674

yes

characteristic

140

C₂

#12676

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
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

C₂×C₄²

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
14. #1372	C₄	semidirect
15. #1371	C₄	semidirect
100. C₁₆	#97	semidirect

H ≅ G/N

type

14. #1372

C₄

semidirect

15. #1371

C₄

semidirect

100. C₁₆

#97

semidirect

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

subgroup	index
C₄×#346	2
C₂×#1372	2
C₂×#1371	2
#8274	2
#7154	2

subgroup

index

Subgroups — #129114

Conjugacy classes of subgroups

Complements

Maximal subgroups