Group #135666 — subgroups

Subgroups — #135666

Order 512.

274 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 138 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	#12898	C₂	1	1	yes	maximal, characteristic
2	#12879	C₂	1	1	yes	maximal, characteristic
3	C₈×#97	C₂	1	1	yes	maximal, characteristic
4	#7151	C₂	1	4	yes	maximal
5	C₂×C₄⋊C₁₆	—	2	2	no	—
6	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	#1744	C₄	1	1	yes	characteristic
9	C₄×#97	C₄	1	1	yes	characteristic
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#322	C₄	1	2	yes	—
12	C₂×C₈²	C₂²	1	2	yes	—
13	#1374	—	2	2	no	—
14	C₂²×C₁₆	—	2	2	no	—
15	C₂²×C₁₆	—	2	2	no	—
16	C₂×#97	C₈	1	2	yes	—
17	C₂×#97	C₈	1	1	yes	characteristic
18	C₂×#97	C₈	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₈	D₄	1	2	yes	—
23	C₂×C₄×C₈	Q₈	1	2	yes	—
24	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
25	C₄³	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₁₆	—	2	4	no	—
27	C₄⋊C₁₆	—	2	4	no	—
28	C₄⋊C₁₆	—	4	4	no	—
29	#336	—	2	2	no	—
30	#322	—	2	4	no	—
31	C₈²	—	2	2	no	—
32	C₈²	—	2	2	no	—
33	C₂²×C₈	—	2	2	no	—
34	C₂²×C₈	C₄⋊C₄	1	2	yes	—
35	C₂²×C₈	C₂×C₈	1	2	yes	—
36	C₂²×C₈	C₂×C₈	1	2	yes	—
37	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄²	C₄⋊C₄	1	2	yes	—
40	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄²	M₁₆	1	2	yes	—
43	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₁₆	—	4	2	no	—
45	C₂×C₁₆	—	4	2	no	—
46	C₂×C₁₆	—	4	4	no	—
47	C₂×C₁₆	—	4	2	no	—
48	C₂×C₁₆	—	4	2	no	—
49	#97	—	2	4	no	—
50	#97	—	2	2	no	—
51	#97	C₁₆	1	4	yes	complemented
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	—	2	2	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	4	2	no	—
59	C₄×C₈	M₁₆	1	4	yes	—
60	C₄×C₈	#41	1	2	yes	—
61	C₄×C₈	#41	1	2	yes	—
62	C₂²×C₄	C₄⋊C₈	1	2	yes	—
63	C₂²×C₄	C₄⋊C₈	1	2	yes	—
64	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
66	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	C₂×C₁₆	1	4	yes	—
78	C₂×C₈	M₃₂	1	4	yes	—
79	C₂×C₈	#124	1	1	yes	characteristic
80	C₂×C₈	#124	1	1	yes	characteristic
81	C₂×C₈	#125	1	2	yes	—
82	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₈	#137	1	2	yes	—
84	C₄²	—	2	2	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	4	2	no	—
88	C₄²	C₂×C₁₆	1	2	yes	—
89	C₄²	M₃₂	1	2	yes	—
90	C₄²	#117	1	2	yes	—
91	C₄²	C₂×M₁₆	1	2	yes	—
92	C₁₆	—	8	2	no	—
93	C₁₆	—	8	2	no	—
94	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
102	C₂×C₄	#382	1	1	yes	characteristic
103	C₂×C₄	#423	1	1	yes	characteristic
104	C₂×C₄	#428	1	2	yes	—
105	C₂×C₄	#430	1	2	yes	—
106	C₂×C₄	#431	1	2	yes	—
107	C₂×C₄	#432	1	2	yes	—
108	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
109	C₂×C₄	C₂×M₃₂	1	2	yes	—
110	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	4	no	—
113	C₈	—	2	4	no	—
114	C₈	—	4	4	no	—
115	C₈	—	4	2	no	—
116	C₈	#489	1	2	yes	—
117	C₈	#490	1	1	yes	characteristic
118	C₈	#491	1	1	yes	characteristic
119	C₂²	#1565	1	2	yes	—
120	C₂²	#1831	1	1	yes	characteristic
121	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
122	C₂²	#2150	1	1	yes	characteristic
123	C₂²	#2153	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	2	4	no	—
126	C₄	—	4	2	no	—
127	C₄	#1569	1	2	yes	—
128	C₄	#1836	1	1	yes	characteristic
129	C₄	#1838	1	1	yes	characteristic
130	C₄	#2162	1	2	yes	—
131	C₄	#2163	1	2	yes	—
132	C₂	#8786	1	1	yes	characteristic
133	C₂	#12015	1	2	yes	—
134	C₂	#12017	1	2	yes	—
135	C₂	#12879	1	1	yes	characteristic
136	C₂	#12884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12898

C₂

yes

maximal, characteristic

#12879

C₂

yes

maximal, characteristic

C₈×#97

C₂

yes

maximal, characteristic

#7151

C₂

yes

maximal

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1744

C₄

yes

characteristic

C₄×#97

C₄

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×#322

C₄

yes

—

C₂×C₈²

C₂²

yes

—

#1374

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#97

C₈

yes

—

C₂×#97

C₈

yes

characteristic

C₂×#97

C₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#336

—

#322

—

C₈²

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

M₁₆

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

#97

C₁₆

yes

complemented

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

C₄⋊C₈

yes

—

C₂²×C₄

C₄⋊C₈

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₁₆

yes

—

C₄²

M₃₂

yes

—

C₄²

#117

yes

—

C₄²

C₂×M₁₆

yes

—

C₁₆

—

C₁₆

—

C₂³

C₂×C₄⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊C₁₆

yes

—

102

C₂×C₄

#382

yes

characteristic

103

C₂×C₄

#423

yes

characteristic

104

C₂×C₄

#428

yes

—

105

C₂×C₄

#430

yes

—

106

C₂×C₄

#431

yes

—

107

C₂×C₄

#432

yes

—

108

C₂×C₄

C₂²×C₁₆

yes

characteristic

109

C₂×C₄

C₂×M₃₂

yes

—

110

C₂×C₄

C₂×M₃₂

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#489

yes

—

117

C₈

#490

yes

characteristic

118

C₈

#491

yes

characteristic

119

C₂²

#1565

yes

—

120

C₂²

#1831

yes

characteristic

121

C₂²

C₂×C₄⋊C₁₆

yes

characteristic

122

C₂²

#2150

yes

characteristic

123

C₂²

#2153

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#1569

yes

—

128

C₄

#1836

yes

characteristic

129

C₄

#1838

yes

characteristic

130

C₄

#2162

yes

—

131

C₄

#2163

yes

—

132

C₂

#8786

yes

characteristic

133

C₂

#12015

yes

—

134

C₂

#12017

yes

—

135

C₂

#12879

yes

characteristic

136

C₂

#12884

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

C₂×D₄

commutator

C₂×C₄

C₂²×C₁₆

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×C₄⋊C₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
51. #97	C₁₆	semidirect

H ≅ G/N

type

51. #97

C₁₆

semidirect

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

subgroup	index
#12898	2
#12879	2
C₈×#97	2
#7151	2

subgroup

index

#12898

#12879

C₈×#97

#7151

Subgroups — #135666

Conjugacy classes of subgroups

Complements

Maximal subgroups