Group #151981 — subgroups

Subgroups — #151981

Order 512.

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

normal subgroups only

This group has 234 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	C₂×#1422	C₂	1	2	yes	maximal
2	C₂×#1398	C₂	1	2	yes	maximal
3	C₄×#346	C₂	1	1	yes	maximal, characteristic
4	#7468	C₂	1	2	yes	maximal
5	C₂×M₆₄	C₂²	1	4	yes	—
6	C₂×#346	C₄	1	1	yes	characteristic
7	C₂×#346	C₄	1	1	yes	characteristic
8	C₂×#346	C₂²	1	1	yes	characteristic
9	C₂×#346	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄²×C₈	C₄	1	1	yes	characteristic
13	#1422	C₄	1	8	yes	complemented
14	#1398	—	2	4	no	—
15	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₈	C₈	1	1	yes	characteristic
20	C₂×C₄×C₈	C₈	1	1	yes	characteristic
21	C₂×C₄×C₈	C₈	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄³	C₈	1	1	yes	characteristic
26	M₆₄	—	2	4	no	—
27	M₆₄	C₂×C₄	1	8	yes	—
28	C₂×C₃₂	—	2	4	no	—
29	#346	—	2	2	no	—
30	#346	—	2	2	no	—
31	#346	—	2	2	no	—
32	#346	D₄	1	2	yes	—
33	#346	Q₈	1	2	yes	—
34	C₄×C₁₆	C₈	1	4	yes	—
35	C₄×C₁₆	C₂×C₄	1	4	yes	—
36	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄²	—	2	2	no	—
41	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
44	C₂×C₁₆	—	2	1	no	—
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	—	2	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	C₄²	1	2	yes	—
49	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
50	C₂×C₁₆	C₂×C₈	1	2	yes	—
51	C₂×C₁₆	C₂×C₈	1	2	yes	—
52	C₂×C₁₆	M₁₆	1	2	yes	—
53	C₂×C₁₆	C₂²×C₄	1	2	yes	—
54	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
55	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₁₆	#41	1	2	yes	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	C₄²	1	2	yes	—
63	C₄×C₈	C₄⋊C₄	1	2	yes	—
64	C₄×C₈	C₂×C₈	1	2	yes	—
65	C₄×C₈	M₁₆	1	2	yes	—
66	C₃₂	—	2	8	no	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	M₃₂	1	2	yes	—
70	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	C₄×C₈	1	2	yes	—
76	C₂×C₈	C₄⋊C₈	1	2	yes	—
77	C₂×C₈	M₃₂	1	2	yes	—
78	C₂×C₈	M₃₂	1	2	yes	—
79	C₂×C₈	M₃₂	1	2	yes	—
80	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₈	#117	1	1	yes	characteristic
83	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
86	C₄²	—	2	1	no	—
87	C₄²	—	2	1	no	—
88	C₄²	—	2	1	no	—
89	C₄²	—	2	1	no	—
90	C₄²	—	4	4	no	—
91	C₄²	C₄×C₈	1	2	yes	—
92	C₄²	C₄⋊C₈	1	2	yes	—
93	C₁₆	—	2	2	no	—
94	C₁₆	—	2	2	no	—
95	C₁₆	C₄⋊C₈	1	4	yes	—
96	C₁₆	C₄×D₄	1	2	yes	—
97	C₁₆	C₄×Q₈	1	2	yes	—
98	C₂³	C₂×M₃₂	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₂×C₄	#431	1	1	yes	characteristic
108	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
109	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
110	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	#346	1	4	yes	—
114	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₈	C₄×M₁₆	1	1	yes	characteristic
116	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₈	#431	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	C₄×M₃₂	1	1	yes	characteristic
120	C₂²	#2150	1	1	yes	characteristic
121	C₂²	#2163	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	2	no	—
125	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
126	C₄	C₂×#346	1	1	yes	characteristic
127	C₄	#2150	1	1	yes	characteristic
128	C₄	#2163	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	#12653	1	1	yes	characteristic
132	C₂	#13642	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1422

C₂

yes

maximal

C₂×#1398

C₂

yes

maximal

C₄×#346

C₂

yes

maximal, characteristic

#7468

C₂

yes

maximal

C₂×M₆₄

C₂²

yes

—

C₂×#346

C₄

yes

characteristic

C₂×#346

C₄

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄²×C₈

C₄

yes

characteristic

#1422

C₄

yes

complemented

#1398

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

C₂×C₄×C₈

C₈

yes

characteristic

C₂×C₄×C₈

C₈

yes

characteristic

C₂×C₄×C₈

C₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₄³

C₈

yes

characteristic

M₆₄

—

M₆₄

C₂×C₄

yes

—

C₂×C₃₂

—

—

—

—

D₄

yes

—

#346

Q₈

yes

—

C₄×C₁₆

C₈

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

#41

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

M₃₂

yes

—

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×C₈

yes

—

C₄²

C₄⋊C₈

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₄⋊C₈

yes

—

C₁₆

C₄×D₄

yes

—

C₁₆

C₄×Q₈

yes

—

C₂³

C₂×M₃₂

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₄×C₈

yes

characteristic

106

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

107

C₂×C₄

#431

yes

characteristic

108

C₂×C₄

C₂×M₃₂

yes

characteristic

109

C₂×C₄

C₂×M₃₂

yes

characteristic

110

C₂×C₄

C₂×M₃₂

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

#346

yes

—

114

C₈

C₄×C₄⋊C₄

yes

characteristic

115

C₈

C₄×M₁₆

yes

characteristic

116

C₈

C₂×C₄⋊C₈

yes

characteristic

117

C₈

#431

yes

characteristic

118

C₂²

—

119

C₂²

C₄×M₃₂

yes

characteristic

120

C₂²

#2150

yes

characteristic

121

C₂²

#2163

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₄×C₄⋊C₈

yes

characteristic

126

C₄

C₂×#346

yes

characteristic

127

C₄

#2150

yes

characteristic

128

C₄

#2163

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

#12653

yes

characteristic

132

C₂

#13642

yes

—

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	#2163
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#117

commutator

C₂×C₄

C₂×C₄×C₈

Frattini, cosocle

C₂²×C₁₆

C₂³

socle

C₂²

#2163

prime radical

C₂³

C₂×M₃₂

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. #1422	C₄	semidirect

H ≅ G/N

type

13. #1422

C₄

semidirect

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

subgroup	index
C₂×#1422	2
C₂×#1398	2
C₄×#346	2
#7468	2

subgroup

index

Subgroups — #151981

Conjugacy classes of subgroups

Complements

Maximal subgroups