Group #94017 — subgroups

Subgroups — #94017

Order 512.

150 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 88 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	#13627	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#11399	C₂	1	1	yes	maximal, characteristic
4	C₂×M₆₄	—	2	1	no	—
5	C₂²×C₃₂	—	2	1	no	—
6	C₄×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	#1450	C₄	1	1	yes	characteristic
9	C₄⋊C₃₂	—	2	1	no	—
10	C₄⋊C₃₂	—	2	1	no	—
11	C₄⋊C₃₂	C₄	1	2	yes	—
12	C₄⋊C₃₂	C₄	1	2	yes	—
13	#1311	—	2	1	no	—
14	C₈×C₁₆	—	2	1	no	—
15	C₂×M₃₂	—	2	1	no	—
16	C₂²×C₁₆	D₄	1	1	yes	characteristic
17	C₂²×C₁₆	Q₈	1	1	yes	characteristic
18	#431	C₈	1	1	yes	characteristic
19	#431	C₈	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	M₆₄	—	4	1	no	—
22	M₆₄	—	4	1	no	—
23	C₂×C₃₂	—	2	2	no	—
24	C₂×C₃₂	—	2	2	no	—
25	C₂×C₃₂	—	4	1	no	—
26	C₂×C₃₂	—	4	1	no	—
27	#346	C₈	1	2	yes	—
28	C₄×C₁₆	—	2	1	no	—
29	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₄×C₁₆	D₄	1	1	yes	characteristic
32	C₄×C₁₆	D₄	1	1	yes	characteristic
33	#322	—	2	1	no	—
34	C₈²	—	2	1	no	—
35	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₈	M₁₆	1	1	yes	characteristic
37	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
38	M₃₂	—	2	2	no	—
39	M₃₂	—	2	2	no	—
40	C₂×C₁₆	—	2	2	no	—
41	C₂×C₁₆	—	2	1	no	—
42	C₂×C₁₆	—	2	1	no	—
43	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₁₆	D₈	1	1	yes	characteristic
48	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
49	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
50	C₂×C₁₆	Q₁₆	1	1	yes	characteristic
51	C₄⋊C₈	—	2	1	no	—
52	C₄⋊C₈	C₁₆	1	2	yes	—
53	C₂²⋊C₈	—	2	1	no	—
54	C₂²⋊C₈	C₁₆	1	2	yes	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	C₄²	1	1	yes	characteristic
58	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₈	C₂×C₈	1	1	yes	characteristic
60	C₄×C₈	M₁₆	1	1	yes	characteristic
61	C₃₂	—	4	2	no	—
62	C₃₂	—	4	2	no	—
63	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
64	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
65	C₂²×C₄	M₃₂	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	#95	1	1	yes	characteristic
71	C₂×C₈	#97	1	1	yes	characteristic
72	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
73	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₈	#106	1	1	yes	characteristic
75	C₂×C₈	#107	1	1	yes	characteristic
76	C₂×C₈	#108	1	1	yes	characteristic
77	C₂×C₈	#108	1	1	yes	characteristic
78	C₄²	C₄×C₈	1	1	yes	characteristic
79	C₄²	C₂²⋊C₈	1	1	yes	characteristic
80	C₄²	C₂×C₁₆	1	1	yes	characteristic
81	C₄²	M₃₂	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₁₆	—	2	1	no	—
84	C₁₆	—	2	1	no	—
85	C₁₆	#102	1	1	yes	characteristic
86	C₁₆	#102	1	1	yes	characteristic
87	C₁₆	#103	1	1	yes	characteristic
88	C₁₆	#103	1	1	yes	characteristic
89	C₂³	C₄⋊C₁₆	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	#334	1	1	yes	characteristic
93	C₂×C₄	#335	1	1	yes	characteristic
94	C₂×C₄	#336	1	1	yes	characteristic
95	C₂×C₄	C₄×C₁₆	1	1	yes	characteristic
96	C₂×C₄	#346	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
99	C₂×C₄	C₄⋊C₁₆	1	1	yes	characteristic
100	C₂×C₄	#364	1	1	yes	characteristic
101	C₂×C₄	#364	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	4	2	no	—
105	C₈	—	4	2	no	—
106	C₈	#334	1	1	yes	characteristic
107	C₈	#335	1	1	yes	characteristic
108	C₈	#340	1	1	yes	characteristic
109	C₈	#341	1	1	yes	characteristic
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	#1367	1	1	yes	characteristic
113	C₂²	#1370	1	1	yes	characteristic
114	C₂²	#1374	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	#1277	1	1	yes	characteristic
119	C₄	#1367	1	1	yes	characteristic
120	C₄	#1370	1	1	yes	characteristic
121	C₄	#1376	1	1	yes	characteristic
122	C₄	#1401	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#7151	1	1	yes	characteristic
126	C₂	#7469	1	1	yes	characteristic
127	C₂	#7470	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13627

C₂

yes

maximal, characteristic

C₂×C₄⋊C₃₂

C₂

yes

maximal, characteristic

#11399

C₂

yes

maximal, characteristic

C₂×M₆₄

—

C₂²×C₃₂

—

C₄×M₃₂

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1450

C₄

yes

characteristic

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₄⋊C₃₂

C₄

yes

—

C₄⋊C₃₂

C₄

yes

—

#1311

—

C₈×C₁₆

—

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

Q₈

yes

characteristic

#431

C₈

yes

characteristic

#431

C₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#346

C₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

#322

—

C₈²

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

D₈

yes

characteristic

C₂×C₁₆

SD₁₆

yes

characteristic

C₂×C₁₆

SD₁₆

yes

characteristic

C₂×C₁₆

Q₁₆

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₁₆

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₃₂

—

C₃₂

—

C₂²×C₄

C₄⋊C₈

yes

characteristic

C₂²×C₄

C₂×C₁₆

yes

characteristic

C₂²×C₄

M₃₂

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

C₄×C₈

yes

characteristic

C₄²

C₂²⋊C₈

yes

characteristic

C₄²

C₂×C₁₆

yes

characteristic

C₄²

M₃₂

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

yes

C₁₆

yes

C₁₆

yes

C₁₆

yes

C₂³

C₄⋊C₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×C₁₆

yes

characteristic

C₂×C₄

#346

yes

characteristic

C₂×C₄

C₂²⋊C₁₆

yes

characteristic

C₂×C₄

C₂²⋊C₁₆

yes

characteristic

C₂×C₄

C₄⋊C₁₆

yes

characteristic

100

C₂×C₄

#364

yes

characteristic

101

C₂×C₄

#364

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

#334

yes

characteristic

107

C₈

#335

yes

characteristic

108

C₈

#340

yes

characteristic

109

C₈

#341

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

#1367

yes

characteristic

113

C₂²

#1370

yes

characteristic

114

C₂²

#1374

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#1277

yes

characteristic

119

C₄

#1367

yes

characteristic

120

C₄

#1370

yes

characteristic

121

C₄

#1376

yes

characteristic

122

C₄

#1401

yes

—

123

C₂

—

124

C₂

—

125

C₂

#7151

yes

characteristic

126

C₂

#7469

yes

characteristic

127

C₂

#7470

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	C₄⋊C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₁₆

C₂²⋊C₄

commutator

C₂×C₄

C₄×C₁₆

Frattini, cosocle

C₂×C₄×C₁₆

C₂²

socle

C₂²

#1374

prime radical

C₂³

C₄⋊C₁₆

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13627	2
C₂×C₄⋊C₃₂	2
#11399	2

subgroup

index

#13627

C₂×C₄⋊C₃₂

#11399

Subgroups — #94017

Conjugacy classes of subgroups

Maximal subgroups