Group #141085 — subgroups

Subgroups — #141085

Order 512.

330 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 102 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	#33313	C₂	1	1	yes	maximal, characteristic
2	#12883	C₂	1	1	yes	maximal, characteristic
3	#11941	C₂	1	2	yes	maximal
4	#11940	C₂	1	2	yes	maximal
5	#8267	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	2	yes	—
7	C₂×#498	—	2	1	no	—
8	#2862	—	2	1	no	—
9	#2150	C₂²	1	1	yes	characteristic
10	#2150	C₂²	1	1	yes	characteristic
11	#1846	—	2	2	no	—
12	#1846	C₄	1	4	yes	—
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	2	yes	—
15	#1452	—	2	1	no	—
16	#1376	—	2	2	no	—
17	#1365	—	2	4	no	—
18	#1364	—	2	4	no	—
19	#1276	—	2	1	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×M₃₂	—	2	1	no	—
22	C₂×M₃₂	—	2	1	no	—
23	#500	—	2	2	no	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#498	—	4	2	no	—
28	#498	—	4	2	no	—
29	#431	—	2	2	no	—
30	C₂×#107	—	2	2	no	—
31	C₂×#107	C₂×C₄	1	4	yes	—
32	C₂×#106	—	2	4	no	—
33	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₈	D₄	1	2	yes	—
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	#391	—	4	4	no	—
38	#382	—	2	4	no	—
39	C₄³	D₄	1	1	yes	characteristic
40	C₄⋊C₁₆	—	2	2	no	—
41	C₄⋊C₁₆	—	2	2	no	—
42	#335	D₄	1	4	yes	—
43	#334	—	2	2	no	—
44	#322	—	4	2	no	—
45	C₂²×Q₈	—	4	2	no	—
46	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₄⋊Q₈	—	4	2	no	—
50	C₄⋊Q₈	—	4	2	no	—
51	C₄⋊Q₈	—	4	2	no	—
52	C₄⋊Q₈	—	4	2	no	—
53	C₂×C₄⋊C₄	—	2	4	no	—
54	C₂×C₄⋊C₄	—	4	2	no	—
55	C₂×C₄²	—	2	2	no	—
56	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
58	M₃₂	—	4	2	no	—
59	M₃₂	—	4	2	no	—
60	C₂×C₁₆	—	4	1	no	—
61	C₂×C₁₆	—	4	1	no	—
62	#107	—	2	4	no	—
63	#107	—	2	4	no	—
64	#107	—	4	2	no	—
65	#107	—	4	2	no	—
66	#106	—	4	8	no	—
67	C₄⋊C₈	—	2	4	no	—
68	C₂²⋊C₈	—	4	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	—	2	4	no	—
72	C₄×C₈	—	4	2	no	—
73	C₄×C₈	D₈	1	2	yes	—
74	C₄×C₈	SD₁₆	1	2	yes	—
75	C₄×C₈	C₂×D₄	1	2	yes	—
76	C₄×C₈	#41	1	2	yes	—
77	#95	—	4	8	no	—
78	C₂×Q₈	—	8	8	no	—
79	C₂×Q₈	—	8	2	no	—
80	C₂×Q₈	—	8	2	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	4	4	no	—
83	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	4	4	no	—
89	C₂×C₈	#102	1	4	yes	—
90	C₂×C₈	C₄×D₄	1	2	yes	—
91	C₂×C₈	#124	1	1	yes	characteristic
92	C₂×C₈	#127	1	1	yes	characteristic
93	C₂×C₈	#137	1	2	yes	—
94	C₂×C₈	#137	1	2	yes	—
95	C₄⋊C₄	—	4	4	no	—
96	C₄⋊C₄	—	4	4	no	—
97	C₄⋊C₄	—	8	2	no	—
98	C₄⋊C₄	—	8	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	4	4	no	—
101	C₄²	#101	1	4	yes	—
102	C₄²	#121	1	2	yes	—
103	C₄²	C₂×D₈	1	1	yes	characteristic
104	C₄²	C₂×SD₁₆	1	1	yes	characteristic
105	C₁₆	—	8	1	no	—
106	C₁₆	—	8	1	no	—
107	C₂³	#418	1	1	yes	characteristic
108	Q₈	—	16	4	no	—
109	Q₈	—	16	4	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	8	8	no	—
114	C₂×C₄	—	8	4	no	—
115	C₂×C₄	#390	1	1	yes	characteristic
116	C₂×C₄	C₂×#101	1	2	yes	—
117	C₂×C₄	C₂×#102	1	1	yes	characteristic
118	C₂×C₄	#419	1	2	yes	—
119	C₂×C₄	#459	1	2	yes	—
120	C₂×C₄	#460	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	2	2	no	—
123	C₈	—	8	4	no	—
124	C₈	#491	1	2	yes	—
125	C₈	#497	1	1	yes	characteristic
126	C₈	#497	1	1	yes	characteristic
127	C₂²	—	2	2	no	—
128	C₂²	#1556	1	2	yes	—
129	C₂²	#1982	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	4	4	no	—
132	C₄	—	16	4	no	—
133	C₄	#1546	1	2	yes	—
134	C₄	#1960	1	1	yes	characteristic
135	C₄	#1966	1	1	yes	characteristic
136	C₄	#2212	1	4	yes	—
137	C₂	—	2	2	no	—
138	C₂	#9705	1	1	yes	characteristic
139	C₂	#13260	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33313

C₂

yes

maximal, characteristic

#12883

C₂

yes

maximal, characteristic

#11941

C₂

yes

maximal

#11940

C₂

yes

maximal

#8267

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

—

C₂×#498

—

#2862

—

#2150

C₂²

yes

characteristic

#2150

C₂²

yes

characteristic

#1846

—

#1846

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

—

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

C₂×M₃₂

—

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#391

—

#382

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#335

D₄

yes

—

#334

—

#322

—

C₂²×Q₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

#101

yes

—

102

C₄²

#121

yes

—

103

C₄²

C₂×D₈

yes

characteristic

104

C₄²

C₂×SD₁₆

yes

characteristic

105

C₁₆

—

106

C₁₆

—

107

C₂³

#418

yes

characteristic

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#390

yes

characteristic

116

C₂×C₄

C₂×#101

yes

—

117

C₂×C₄

C₂×#102

yes

characteristic

118

C₂×C₄

#419

yes

—

119

C₂×C₄

#459

yes

—

120

C₂×C₄

#460

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

#491

yes

—

125

C₈

#497

yes

characteristic

126

C₈

#497

yes

characteristic

127

C₂²

—

128

C₂²

#1556

yes

—

129

C₂²

#1982

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#1546

yes

—

134

C₄

#1960

yes

characteristic

135

C₄

#1966

yes

characteristic

136

C₄

#2212

yes

—

137

C₂

—

138

C₂

#9705

yes

characteristic

139

C₂

#13260

yes

—

	subgroup	G/H
center, socle	C₂²	#1982
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1982

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#418

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33313	2
#12883	2
#11941	2
#11940	2
#8267	2

subgroup

index

Subgroups — #141085

Conjugacy classes of subgroups

Maximal subgroups