Group #135665 — subgroups

Subgroups — #135665

Order 512.

210 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 106 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	#12881	C₂	1	2	yes	maximal
2	#12539	C₂	1	1	yes	maximal, characteristic
3	#7161	C₂	1	2	yes	maximal
4	#7160	C₂	1	2	yes	maximal
5	C₂×#364	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×C₄×C₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1726	C₂²	1	1	yes	characteristic
13	#1377	—	2	2	no	—
14	#1375	—	2	2	no	—
15	C₂×M₃₂	—	2	4	no	—
16	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×#97	D₄	1	1	yes	characteristic
21	C₂×#97	D₄	1	1	yes	characteristic
22	C₂×#97	Q₈	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	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	#376	Q₈	1	1	yes	characteristic
27	#364	—	2	4	no	—
28	#364	—	2	4	no	—
29	#346	—	2	2	no	—
30	#346	—	2	2	no	—
31	#346	C₈	1	4	yes	—
32	C₄×C₁₆	—	2	2	no	—
33	C₄×C₁₆	—	2	2	no	—
34	C₄×C₁₆	—	2	2	no	—
35	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
42	M₃₂	—	4	4	no	—
43	M₃₂	—	4	4	no	—
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	—	2	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	4	4	no	—
53	C₂×C₁₆	C₂×C₈	1	2	yes	—
54	C₂×C₁₆	C₂×C₈	1	2	yes	—
55	C₂×C₁₆	M₁₆	1	2	yes	—
56	C₂×C₁₆	M₁₆	1	2	yes	—
57	#97	—	2	2	no	—
58	#97	—	2	2	no	—
59	#97	—	2	2	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	C₂×C₈	1	2	yes	—
63	C₄×C₈	M₁₆	1	2	yes	—
64	C₄×C₈	#41	1	2	yes	—
65	C₄×C₈	#41	1	2	yes	—
66	#95	—	2	2	no	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	#108	1	2	yes	—
70	C₂²×C₄	C₂×C₄⋊C₄	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	#108	1	2	yes	—
77	C₂×C₈	#117	1	1	yes	characteristic
78	C₂×C₈	#117	1	1	yes	characteristic
79	C₂×C₈	#124	1	2	yes	—
80	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₈	#135	1	2	yes	—
85	C₂×C₈	#135	1	2	yes	—
86	C₄²	—	2	1	no	—
87	C₄²	—	2	1	no	—
88	C₄²	—	2	1	no	—
89	C₄²	—	2	1	no	—
90	C₄²	C₄⋊C₈	1	2	yes	—
91	C₄²	#125	1	2	yes	—
92	C₁₆	—	2	2	no	—
93	C₁₆	—	2	2	no	—
94	C₁₆	—	2	2	no	—
95	C₁₆	—	2	2	no	—
96	C₁₆	—	8	4	no	—
97	C₂³	C₂×#108	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	#382	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₂×C₄	#423	1	1	yes	characteristic
107	C₂×C₄	#427	1	1	yes	characteristic
108	C₂×C₄	#427	1	1	yes	characteristic
109	C₂×C₄	C₂×#108	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	#431	1	1	yes	characteristic
113	C₈	#431	1	1	yes	characteristic
114	C₈	#432	1	1	yes	characteristic
115	C₈	#432	1	1	yes	characteristic
116	C₈	#488	1	4	yes	—
117	C₂²	—	2	2	no	—
118	C₂²	#1564	1	1	yes	characteristic
119	C₂²	#1568	1	1	yes	characteristic
120	C₂²	#1837	1	1	yes	characteristic
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	—	4	2	no	—
124	C₄	#1564	1	1	yes	characteristic
125	C₄	#1568	1	1	yes	characteristic
126	C₄	#1831	1	1	yes	characteristic
127	C₄	#1835	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#8785	1	1	yes	characteristic
131	C₂	#12027	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12881

C₂

yes

maximal

#12539

C₂

yes

maximal, characteristic

#7161

C₂

yes

maximal

#7160

C₂

yes

maximal

C₂×#364

C₂²

yes

—

C₂×#346

C₄

yes

characteristic

C₂×#346

C₄

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1726

C₂²

yes

characteristic

#1377

—

#1375

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

Q₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#376

Q₈

yes

—

—

—

—

C₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

C₂×C₁₆

M₁₆

yes

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#108

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₂×C₈

#135

yes

—

C₂×C₈

#135

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₈

yes

—

C₄²

#125

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₂×#108

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#382

yes

characteristic

105

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

106

C₂×C₄

#423

yes

characteristic

107

C₂×C₄

#427

yes

characteristic

108

C₂×C₄

#427

yes

characteristic

109

C₂×C₄

C₂×#108

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

#431

yes

characteristic

113

C₈

#431

yes

characteristic

114

C₈

#432

yes

characteristic

115

C₈

#432

yes

characteristic

116

C₈

#488

yes

—

117

C₂²

—

118

C₂²

#1564

yes

characteristic

119

C₂²

#1568

yes

characteristic

120

C₂²

#1837

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#1564

yes

characteristic

125

C₄

#1568

yes

characteristic

126

C₄

#1831

yes

characteristic

127

C₄

#1835

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#8785

yes

characteristic

131

C₂

#12027

yes

—

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1568
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#427

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ 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
#12881	2
#12539	2
#7161	2
#7160	2

subgroup

index

Subgroups — #135665

Conjugacy classes of subgroups

Maximal subgroups