Group #153382 — subgroups

Subgroups — #153382

Order 512.

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

normal subgroups only

This group has 182 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	#13634	C₂	1	2	yes	maximal
2	#13169	C₂	1	1	yes	maximal, characteristic
3	#12905	C₂	1	1	yes	maximal, characteristic
4	#7478	C₂	1	2	yes	maximal
5	#7467	C₂	1	1	yes	maximal, characteristic
6	C₂×M₆₄	C₂²	1	2	yes	—
7	C₂×#368	C₂²	1	1	yes	characteristic
8	#2158	C₂²	1	1	yes	characteristic
9	C₂×#366	—	2	1	no	—
10	C₂×#366	C₂²	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1945	—	2	1	no	—
13	#1845	—	2	1	no	—
14	#1424	—	2	2	no	—
15	#1423	—	2	2	no	—
16	#1382	—	2	1	no	—
17	C₂²×C₁₆	D₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂³	1	1	yes	characteristic
19	C₂×#108	—	2	1	no	—
20	#427	—	2	1	no	—
21	C₂×#107	—	2	2	no	—
22	C₂×#107	—	4	1	no	—
23	#424	—	4	1	no	—
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	#384	—	4	1	no	—
26	M₆₄	—	2	2	no	—
27	M₆₄	Q₈	1	4	yes	—
28	C₂×C₃₂	—	2	2	no	—
29	#368	—	2	2	no	—
30	#367	—	2	1	no	—
31	#366	—	2	2	no	—
32	#366	—	2	2	no	—
33	#366	—	4	2	no	—
34	#366	D₄	1	1	yes	characteristic
35	#366	D₄	1	1	yes	characteristic
36	C₄×C₁₆	—	2	2	no	—
37	#339	—	4	1	no	—
38	C₂×M₁₆	—	4	1	no	—
39	C₂²×C₈	D₈	1	1	yes	characteristic
40	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
41	#117	—	4	1	no	—
42	C₂×C₄⋊C₄	—	4	2	no	—
43	C₂×C₄⋊C₄	—	8	2	no	—
44	C₂×C₄²	D₈	1	1	yes	characteristic
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	—	2	1	no	—
47	C₂×C₁₆	Q₁₆	1	2	yes	—
48	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
49	C₂×C₁₆	C₂×Q₈	1	2	yes	—
50	C₂×C₁₆	#41	1	2	yes	—
51	C₂×C₁₆	#41	1	1	yes	characteristic
52	#108	—	4	1	no	—
53	#108	—	4	1	no	—
54	#107	—	2	1	no	—
55	#107	—	2	1	no	—
56	#107	—	4	2	no	—
57	#107	—	4	2	no	—
58	#107	—	8	2	no	—
59	#106	—	4	1	no	—
60	C₄⋊C₈	—	8	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₃₂	—	2	4	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	8	2	no	—
66	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
67	M₁₆	—	8	1	no	—
68	M₁₆	—	8	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	8	1	no	—
72	C₂×C₈	D₁₆	1	2	yes	—
73	C₂×C₈	#121	1	1	yes	characteristic
74	C₂×C₈	#123	1	1	yes	characteristic
75	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₈	#137	1	1	yes	characteristic
79	C₄⋊C₄	—	4	1	no	—
80	C₄⋊C₄	—	4	1	no	—
81	C₄⋊C₄	—	8	2	no	—
82	C₄⋊C₄	—	8	2	no	—
83	C₄⋊C₄	—	16	4	no	—
84	C₂²⋊C₄	—	8	1	no	—
85	C₄²	—	2	1	no	—
86	C₄²	—	8	1	no	—
87	C₄²	Q₃₂	1	2	yes	—
88	C₁₆	—	2	2	no	—
89	C₁₆	C₂²⋊Q₈	1	2	yes	—
90	C₁₆	#125	1	2	yes	—
91	C₂³	#509	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	8	1	no	—
96	C₂×C₄	—	8	1	no	—
97	C₂×C₄	—	16	4	no	—
98	C₂×C₄	—	16	2	no	—
99	C₂×C₄	#459	1	1	yes	characteristic
100	C₂×C₄	#480	1	1	yes	characteristic
101	C₂×C₄	#500	1	1	yes	characteristic
102	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
103	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
104	C₂×C₄	#510	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	16	1	no	—
107	C₈	#398	1	1	yes	characteristic
108	C₈	#462	1	1	yes	characteristic
109	C₈	#484	1	1	yes	characteristic
110	C₈	#501	1	1	yes	characteristic
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#2206	1	1	yes	characteristic
114	C₂²	#2236	1	1	yes	characteristic
115	C₂²	#2255	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	16	1	no	—
118	C₄	—	16	1	no	—
119	C₄	—	32	2	no	—
120	C₄	#2083	1	1	yes	characteristic
121	C₄	#2209	1	1	yes	characteristic
122	C₄	#2232	1	1	yes	characteristic
123	C₄	#2252	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#13505	1	1	yes	characteristic
127	C₂	#13692	1	1	yes	characteristic
128	C₂	#13693	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13634

C₂

yes

maximal

#13169

C₂

yes

maximal, characteristic

#12905

C₂

yes

maximal, characteristic

#7478

C₂

yes

maximal

#7467

C₂

yes

maximal, characteristic

C₂×M₆₄

C₂²

yes

—

C₂×#368

C₂²

yes

characteristic

#2158

C₂²

yes

characteristic

C₂×#366

—

C₂×#366

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#384

—

M₆₄

—

M₆₄

Q₈

yes

—

C₂×C₃₂

—

—

—

—

—

—

D₄

yes

characteristic

#366

D₄

yes

characteristic

C₄×C₁₆

—

#339

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

Q₃₂

yes

—

C₁₆

—

C₁₆

C₂²⋊Q₈

yes

—

C₁₆

#125

yes

—

C₂³

#509

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#459

yes

characteristic

100

C₂×C₄

#480

yes

characteristic

101

C₂×C₄

#500

yes

characteristic

102

C₂×C₄

C₂×D₁₆

yes

characteristic

103

C₂×C₄

C₂×Q₃₂

yes

characteristic

104

C₂×C₄

#510

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

#398

yes

characteristic

108

C₈

#462

yes

characteristic

109

C₈

#484

yes

characteristic

110

C₈

#501

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

#2206

yes

characteristic

114

C₂²

#2236

yes

characteristic

115

C₂²

#2255

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#2083

yes

characteristic

121

C₄

#2209

yes

characteristic

122

C₄

#2232

yes

characteristic

123

C₄

#2252

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

#13505

yes

characteristic

127

C₂

#13692

yes

characteristic

128

C₂

#13693

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2206

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

#509

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13634	2
#13169	2
#12905	2
#7478	2
#7467	2

subgroup

index

Subgroups — #153382

Conjugacy classes of subgroups

Maximal subgroups