Group #149740 — subgroups

Subgroups — #149740

Order 512.

178 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 72 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	#33475	C₂	1	1	yes	maximal, characteristic
2	#13643	C₂	1	2	yes	maximal
3	#7365	C₂	1	2	yes	maximal
4	#7364	C₂	1	2	yes	maximal
5	#2960	C₄	1	1	yes	characteristic
6	#2183	—	2	2	no	—
7	#2183	—	2	2	no	—
8	#2151	C₄	1	1	yes	characteristic
9	#2151	C₂²	1	1	yes	characteristic
10	#2150	C₂²	1	1	yes	characteristic
11	#2108	C₄	1	1	yes	characteristic
12	#2108	C₂²	1	1	yes	characteristic
13	C₂×#346	C₄	1	1	yes	characteristic
14	#1422	—	2	2	no	—
15	#1400	C₂²	1	4	yes	—
16	#1398	—	2	2	no	—
17	#533	C₈	1	1	yes	characteristic
18	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
19	C₂×M₃₂	C₂³	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
22	#446	—	2	2	no	—
23	C₈×Q₈	—	2	2	no	—
24	#424	C₈	1	1	yes	characteristic
25	#424	C₈	1	1	yes	characteristic
26	#423	C₈	1	1	yes	characteristic
27	#405	C₂×C₄	1	1	yes	characteristic
28	#405	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	M₆₄	—	2	4	no	—
31	C₄⋊C₁₆	—	2	2	no	—
32	C₄⋊C₁₆	—	2	2	no	—
33	C₄⋊C₁₆	—	2	2	no	—
34	#346	—	2	1	no	—
35	#346	—	2	1	no	—
36	#346	—	2	1	no	—
37	#346	D₄	1	2	yes	—
38	C₄×C₁₆	—	2	1	no	—
39	C₄×C₁₆	D₄	1	2	yes	—
40	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
41	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₄⋊Q₈	—	2	1	no	—
45	#125	—	2	1	no	—
46	C₂²⋊Q₈	—	2	2	no	—
47	C₄×Q₈	—	2	2	no	—
48	#117	C₂×C₈	1	1	yes	characteristic
49	#117	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
51	M₃₂	—	4	1	no	—
52	M₃₂	—	4	1	no	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	4	1	no	—
57	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₁₆	C₂×D₄	1	2	yes	—
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	—	2	2	no	—
63	#97	—	2	1	no	—
64	#97	M₁₆	1	2	yes	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	C₂²⋊C₄	1	2	yes	—
69	C₃₂	—	4	4	no	—
70	C₂×Q₈	—	2	2	no	—
71	C₂²×C₄	—	2	1	no	—
72	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
73	M₁₆	—	4	1	no	—
74	M₁₆	—	4	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	C₂²⋊C₈	1	2	yes	—
81	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
83	C₄⋊C₄	—	2	4	no	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	1	no	—
87	C₂²⋊C₄	—	2	1	no	—
88	C₂²⋊C₄	M₃₂	1	2	yes	—
89	C₄²	—	2	1	no	—
90	C₄²	—	2	1	no	—
91	C₄²	C₂²⋊C₈	1	2	yes	—
92	C₄²	M₃₂	1	1	yes	characteristic
93	C₄²	M₃₂	1	1	yes	characteristic
94	C₁₆	—	2	2	no	—
95	C₁₆	—	2	1	no	—
96	C₁₆	—	4	1	no	—
97	C₁₆	—	4	1	no	—
98	C₁₆	#101	1	2	yes	—
99	C₂³	C₂×M₃₂	1	1	yes	characteristic
100	Q₈	—	8	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	2	1	no	—
112	C₈	—	4	1	no	—
113	C₈	—	4	1	no	—
114	C₈	#410	1	1	yes	characteristic
115	C₈	C₂×#101	1	1	yes	characteristic
116	C₂²	—	2	1	no	—
117	C₂²	—	4	1	no	—
118	C₂²	#2112	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	4	1	no	—
121	C₄	—	4	1	no	—
122	C₄	—	4	2	no	—
123	C₄	—	4	2	no	—
124	C₄	#1461	1	1	yes	characteristic
125	C₄	#2115	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	1	no	—
128	C₂	#11494	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33475

C₂

yes

maximal, characteristic

#13643

C₂

yes

maximal

#7365

C₂

yes

maximal

#7364

C₂

yes

maximal

#2960

C₄

yes

—

—

C₄

yes

characteristic

#2151

C₂²

yes

characteristic

#2150

C₂²

yes

characteristic

#2108

C₄

yes

characteristic

#2108

C₂²

yes

characteristic

C₂×#346

C₄

yes

characteristic

#1422

—

#1400

C₂²

yes

—

#1398

—

#533

C₈

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

C₂³

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₈

yes

characteristic

#424

C₈

yes

characteristic

#423

C₈

yes

characteristic

#405

C₂×C₄

yes

characteristic

#405

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

M₆₄

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

D₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₂×M₁₆

C₂×C₈

yes

characteristic

C₂×M₁₆

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×C₈

yes

characteristic

#117

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

M₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

—

C₃₂

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

M₃₂

yes

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₈

yes

—

C₄²

M₃₂

yes

characteristic

C₄²

M₃₂

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

#101

yes

—

C₂³

C₂×M₃₂

yes

characteristic

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

109

C₂×C₄

C₂×M₃₂

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

#410

yes

characteristic

115

C₈

C₂×#101

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

#2112

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#1461

yes

characteristic

125

C₄

#2115

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#11494

yes

characteristic

	subgroup	G/H
center	C₈	#410
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×M₃₂	C₂³
socle	C₂	#11494
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2108 ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#410

commutator

C₂²×C₄

C₂²×C₈

Frattini, cosocle

C₂×M₃₂

C₂³

socle

C₂

#11494

prime radical

C₂³

C₂×M₃₂

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2108 ◁ G

LECS

G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33475	2
#13643	2
#7365	2
#7364	2

subgroup

index

Subgroups — #149740

Conjugacy classes of subgroups

Maximal subgroups