Group #129137 — subgroups

Subgroups — #129137

Order 512.

241 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 136 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	#12668	C₂	1	1	yes	maximal, characteristic
2	#12664	C₂	1	2	yes	maximal
3	#12002	C₂	1	1	yes	maximal, characteristic
4	#11399	C₂	1	1	yes	maximal, characteristic
5	#7180	C₂	1	2	yes	maximal
6	#2155	C₄	1	2	yes	—
7	C₂×#364	C₂²	1	2	yes	—
8	#2108	—	2	2	no	—
9	C₄×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1772	C₂²	1	2	yes	—
12	#1450	C₂²	1	1	yes	characteristic
13	#1385	C₄	1	1	yes	characteristic
14	#1384	C₄	1	1	yes	characteristic
15	#1379	—	2	2	no	—
16	#1369	C₄	1	4	yes	complemented
17	#1311	C₄	1	2	yes	complemented
18	C₈×C₁₆	C₄	1	2	yes	complemented
19	C₂×M₃₂	—	2	2	no	—
20	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
21	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	#431	—	2	1	no	—
25	C₂×#108	C₂×C₄	1	2	yes	—
26	#427	C₂×C₄	1	2	yes	—
27	#405	—	2	2	no	—
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	#365	C₈	1	8	yes	complemented
30	#364	—	2	2	no	—
31	#364	C₂×C₄	1	4	yes	—
32	#346	—	4	2	no	—
33	#346	C₂×C₄	1	2	yes	—
34	C₄×C₁₆	—	4	2	no	—
35	C₄×C₁₆	C₂×C₄	1	2	yes	—
36	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₄×C₁₆	Q₈	1	1	yes	characteristic
39	C₄×C₁₆	Q₈	1	1	yes	characteristic
40	#339	—	2	2	no	—
41	#322	—	2	1	no	—
42	C₈²	C₂×C₄	1	2	yes	—
43	C₂×M₁₆	—	2	2	no	—
44	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	#117	—	2	2	no	—
47	C₂×C₄²	#41	1	1	yes	characteristic
48	M₃₂	—	2	2	no	—
49	M₃₂	—	2	4	no	—
50	M₃₂	—	4	2	no	—
51	M₃₂	C₂×C₈	1	4	yes	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	4	2	no	—
54	C₂×C₁₆	C₄²	1	2	yes	—
55	C₂×C₁₆	C₄²	1	2	yes	—
56	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₁₆	SD₁₆	1	2	yes	—
61	C₂×C₁₆	SD₁₆	1	2	yes	—
62	#108	—	2	2	no	—
63	#108	C₂×C₈	1	4	yes	—
64	#107	—	2	2	no	—
65	#106	C₂×C₈	1	4	yes	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	#97	—	4	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	4	2	no	—
71	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	M₁₆	—	4	2	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₈	C₄×D₄	1	1	yes	characteristic
84	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₈	C₂²×C₈	1	2	yes	—
86	C₂×C₈	#131	1	2	yes	—
87	C₂×C₈	C₂×SD₁₆	1	2	yes	—
88	C₂×C₈	#135	1	2	yes	—
89	C₄⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	4	2	no	—
91	C₄²	—	2	1	no	—
92	C₄²	—	4	2	no	—
93	C₄²	#117	1	1	yes	characteristic
94	C₄²	C₄×Q₈	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₁₆	—	2	1	no	—
97	C₁₆	—	2	1	no	—
98	C₁₆	—	4	4	no	—
99	C₁₆	C₄×C₈	1	4	yes	—
100	C₁₆	#106	1	2	yes	—
101	C₁₆	#106	1	1	yes	characteristic
102	C₁₆	#106	1	1	yes	characteristic
103	C₂³	#443	1	1	yes	characteristic
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄	C₈×D₄	1	2	yes	—
110	C₂×C₄	C₄×SD₁₆	1	2	yes	—
111	C₂×C₄	#443	1	1	yes	characteristic
112	C₈	—	4	4	no	—
113	C₈	—	4	4	no	—
114	C₈	C₂×C₄×C₈	1	1	yes	characteristic
115	C₈	#405	1	1	yes	characteristic
116	C₈	C₂×#106	1	1	yes	characteristic
117	C₈	#427	1	1	yes	characteristic
118	C₈	C₄×SD₁₆	1	2	yes	—
119	C₈	C₈×Q₈	1	2	yes	—
120	C₂²	—	4	1	no	—
121	C₂²	C₈×SD₁₆	1	2	yes	—
122	C₂²	#1772	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	4	1	no	—
125	C₄	—	8	4	no	—
126	C₄	C₈×SD₁₆	1	2	yes	—
127	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
128	C₄	C₄×#106	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	2	2	no	—
131	C₂	C₈×#106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12668

C₂

yes

maximal, characteristic

#12664

C₂

yes

maximal

#12002

C₂

yes

maximal, characteristic

#11399

C₂

yes

maximal, characteristic

#7180

C₂

yes

maximal

#2155

C₄

yes

—

C₂×#364

C₂²

yes

—

#2108

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1772

C₂²

yes

—

#1450

C₂²

yes

characteristic

#1385

C₄

yes

characteristic

#1384

C₄

yes

characteristic

#1379

—

#1369

C₄

yes

complemented

#1311

C₄

yes

complemented

C₈×C₁₆

C₄

yes

complemented

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

#431

—

C₂×#108

C₂×C₄

yes

—

#427

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

C₂³

yes

characteristic

#365

C₈

yes

complemented

#364

—

#364

C₂×C₄

yes

—

#346

—

#346

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

Q₈

yes

characteristic

C₄×C₁₆

Q₈

yes

characteristic

#339

—

#322

—

C₈²

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

C₂×C₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

SD₁₆

yes

—

#108

—

#108

C₂×C₈

yes

—

#107

—

#106

C₂×C₈

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

—

C₂×C₈

#131

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

#135

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

#117

yes

characteristic

C₄²

C₄×Q₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₄×C₈

yes

—

100

C₁₆

#106

yes

—

101

C₁₆

#106

yes

characteristic

102

C₁₆

#106

yes

characteristic

103

C₂³

#443

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

109

C₂×C₄

C₈×D₄

yes

—

110

C₂×C₄

C₄×SD₁₆

yes

—

111

C₂×C₄

#443

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

C₂×C₄×C₈

yes

characteristic

115

C₈

#405

yes

characteristic

116

C₈

C₂×#106

yes

characteristic

117

C₈

#427

yes

characteristic

118

C₈

C₄×SD₁₆

yes

—

119

C₈

C₈×Q₈

yes

—

120

C₂²

—

121

C₂²

C₈×SD₁₆

yes

—

122

C₂²

#1772

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₈×SD₁₆

yes

—

127

C₄

C₈×C₄⋊C₄

yes

characteristic

128

C₄

C₄×#106

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

C₈×#106

yes

characteristic

	subgroup	G/H
center	C₁₆	#106
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	C₈×#106
prime radical	C₂³	#443
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₁₆

#106

commutator

C₈

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₂³

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. #1369	C₄	semidirect
17. #1311	C₄	semidirect
18. C₈×C₁₆	C₄	semidirect
29. #365	C₈	semidirect

H ≅ G/N

type

16. #1369

C₄

semidirect

17. #1311

C₄

semidirect

18. C₈×C₁₆

C₄

semidirect

29. #365

C₈

semidirect

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

subgroup	index
#12668	2
#12664	2
#12002	2
#11399	2
#7180	2

subgroup

index

Subgroups — #129137

Conjugacy classes of subgroups

Complements

Maximal subgroups