Group #129156 — subgroups

Subgroups — #129156

Order 512.

246 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 130 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	C₄×#368	C₂	1	1	yes	maximal, characteristic
2	#12008	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#8293	C₂	1	2	yes	maximal
5	#7172	C₂	1	2	yes	maximal
6	C₂×#368	C₄	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₄×#108	C₂²	1	2	yes	—
10	#1564	C₂²	1	2	yes	—
11	#1451	—	2	2	no	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1381	C₄	1	2	yes	—
14	#1372	C₄	1	2	yes	—
15	#1371	C₄	1	2	yes	—
16	C₈×C₁₆	C₄	1	2	yes	—
17	C₈×C₁₆	C₄	1	2	yes	—
18	#1278	—	2	2	no	—
19	C₂²×C₁₆	—	2	1	no	—
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
22	#431	—	2	2	no	—
23	C₂×#108	C₂×C₄	1	4	yes	—
24	C₄×M₁₆	—	2	2	no	—
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#368	—	2	4	no	—
29	C₄×C₁₆	—	2	1	no	—
30	C₄×C₁₆	C₂×C₄	1	2	yes	—
31	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
35	#341	—	2	2	no	—
36	#335	C₂×C₄	1	4	yes	—
37	#334	—	2	2	no	—
38	#322	—	4	2	no	—
39	#322	—	4	2	no	—
40	C₈²	C₂×C₄	1	2	yes	—
41	C₈²	Q₈	1	2	yes	—
42	C₂×M₁₆	—	2	4	no	—
43	C₂²×C₈	—	2	1	no	—
44	C₂²×C₈	Q₁₆	1	2	yes	—
45	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₈	#41	1	1	yes	characteristic
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	C₂×C₁₆	—	2	4	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	C₄²	1	1	yes	characteristic
52	C₂×C₁₆	C₄²	1	1	yes	characteristic
53	C₂×C₁₆	C₄²	1	1	yes	characteristic
54	C₂×C₁₆	C₄²	1	1	yes	characteristic
55	C₂×C₁₆	M₁₆	1	4	yes	—
56	#108	—	2	4	no	—
57	#108	—	2	4	no	—
58	C₄⋊C₈	—	2	4	no	—
59	C₂²⋊C₈	—	4	2	no	—
60	#97	—	4	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	4	2	no	—
63	C₄×C₈	—	4	2	no	—
64	C₄×C₈	C₄⋊C₄	1	2	yes	—
65	C₄×C₈	C₄⋊C₄	1	2	yes	—
66	C₄×C₈	D₈	1	1	yes	characteristic
67	C₄×C₈	D₈	1	1	yes	characteristic
68	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₄×C₈	#41	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
74	C₂²×C₄	#135	1	1	yes	characteristic
75	M₁₆	—	4	4	no	—
76	M₁₆	—	4	4	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	4	4	no	—
82	C₂×C₈	—	4	4	no	—
83	C₂×C₈	#107	1	2	yes	—
84	C₂×C₈	#107	1	2	yes	—
85	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₈	#117	1	1	yes	characteristic
87	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₈	C₂×M₁₆	1	2	yes	—
90	C₂×C₈	#131	1	2	yes	—
91	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄²	C₄×D₄	1	1	yes	characteristic
93	C₄²	C₂×D₈	1	1	yes	characteristic
94	C₄²	#135	1	1	yes	characteristic
95	C₁₆	—	2	4	no	—
96	C₁₆	—	2	1	no	—
97	C₁₆	—	2	1	no	—
98	C₁₆	#97	1	2	yes	—
99	C₁₆	#97	1	2	yes	—
100	C₂³	C₄×Q₁₆	1	1	yes	characteristic
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₂×#107	1	1	yes	characteristic
105	C₂×C₄	#427	1	1	yes	characteristic
106	C₂×C₄	#436	1	2	yes	—
107	C₂×C₄	C₄×D₈	1	1	yes	characteristic
108	C₂×C₄	C₄×D₈	1	1	yes	characteristic
109	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
110	C₂×C₄	#443	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	8	4	no	—
116	C₈	—	8	4	no	—
117	C₈	#368	1	4	yes	complemented
118	C₈	C₂×#97	1	1	yes	characteristic
119	C₈	#405	1	1	yes	characteristic
120	C₈	#446	1	2	yes	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#1591	1	2	yes	—
124	C₂²	C₄×#107	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	#1589	1	2	yes	—
128	C₄	#1770	1	1	yes	characteristic
129	C₄	#1772	1	1	yes	characteristic
130	C₄	C₂×#368	1	2	yes	—
131	C₄	#2178	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	#8282	1	1	yes	characteristic
134	C₂	#12670	1	1	yes	characteristic
135	C₂	C₄×#368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#368

C₂

yes

maximal, characteristic

#12008

C₂

yes

maximal, characteristic

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

#8293

C₂

yes

maximal

#7172

C₂

yes

maximal

C₂×#368

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#108

C₂²

yes

—

#1564

C₂²

yes

—

#1451

—

C₂×C₈²

C₂²

yes

characteristic

#1381

C₄

yes

—

#1372

C₄

yes

—

#1371

C₄

yes

—

C₈×C₁₆

C₄

yes

—

C₈×C₁₆

C₄

yes

—

#1278

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

—

C₂×#108

C₂×C₄

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#368

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

#341

—

#335

C₂×C₄

yes

—

#334

—

#322

—

#322

—

C₈²

C₂×C₄

yes

—

C₈²

Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

Q₁₆

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

M₁₆

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

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

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂²×C₄

#135

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₄×D₄

yes

characteristic

C₄²

C₂×D₈

yes

characteristic

C₄²

#135

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

#97

yes

—

C₁₆

#97

yes

—

100

C₂³

C₄×Q₁₆

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

104

C₂×C₄

C₂×#107

yes

characteristic

105

C₂×C₄

#427

yes

characteristic

106

C₂×C₄

#436

yes

—

107

C₂×C₄

C₄×D₈

yes

characteristic

108

C₂×C₄

C₄×D₈

yes

characteristic

109

C₂×C₄

C₄×Q₁₆

yes

characteristic

110

C₂×C₄

#443

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

#368

yes

complemented

118

C₈

C₂×#97

yes

characteristic

119

C₈

#405

yes

characteristic

120

C₈

#446

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#1591

yes

—

124

C₂²

C₄×#107

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

#1589

yes

—

128

C₄

#1770

yes

characteristic

129

C₄

#1772

yes

characteristic

130

C₄

C₂×#368

yes

—

131

C₄

#2178

yes

—

132

C₂

—

133

C₂

#8282

yes

characteristic

134

C₂

#12670

yes

characteristic

135

C₂

C₄×#368

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#107
prime radical	C₂³	C₄×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄²

C₂×D₈

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle

C₂²

C₄×#107

prime radical

C₂³

C₄×Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
117. C₈	#368	semidirect

H ≅ G/N

type

117. C₈

#368

semidirect

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

subgroup	index
C₄×#368	2
#12008	2
C₂×C₈×C₁₆	2
#8293	2
#7172	2

subgroup

index

C₂×C₈×C₁₆

Subgroups — #129156

Conjugacy classes of subgroups

Complements

Maximal subgroups