Group #134852 — subgroups

Subgroups — #134852

Order 512.

262 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#12023	C₂	1	2	yes	maximal
3	#8617	C₂	1	2	yes	maximal
4	#7241	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
6	Q₈×C₁₆	—	2	2	no	—
7	Q₈×C₁₆	—	2	2	no	—
8	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₄	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1801	—	2	2	no	—
13	#1567	C₂²	1	4	yes	—
14	#1372	—	2	2	no	—
15	#1372	—	2	2	no	—
16	#1292	—	2	2	no	—
17	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₈×Q₈	—	2	2	no	—
22	C₈×Q₈	—	2	2	no	—
23	#423	—	2	4	no	—
24	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₈	Q₈	1	2	yes	—
26	C₂×#101	—	4	2	no	—
27	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	C₄⋊C₁₆	—	2	4	no	—
30	C₄⋊C₁₆	—	2	2	no	—
31	C₄×C₁₆	—	2	2	no	—
32	C₄×C₁₆	—	2	2	no	—
33	C₄×C₁₆	D₄	1	2	yes	—
34	C₄×C₁₆	D₄	1	2	yes	—
35	#343	—	4	2	no	—
36	#335	—	2	4	no	—
37	#335	—	2	4	no	—
38	C₂²×Q₈	D₈	1	1	yes	characteristic
39	C₂×M₁₆	—	4	4	no	—
40	C₂²×C₈	C₄⋊C₄	1	2	yes	—
41	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
43	C₄×Q₈	—	2	2	no	—
44	C₄×Q₈	—	2	2	no	—
45	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
46	C₂×C₄⋊C₄	Q₁₆	1	2	yes	—
47	C₂×C₄²	C₄⋊C₄	1	2	yes	—
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	4	1	no	—
52	C₂×C₁₆	—	4	2	no	—
53	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
55	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
56	C₄⋊C₈	—	2	4	no	—
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	4	4	no	—
59	C₄⋊C₈	—	4	4	no	—
60	#101	—	8	4	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	C₂×D₄	1	2	yes	—
64	C₄×C₈	#41	1	2	yes	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	2	2	no	—
67	C₂×Q₈	SD₃₂	1	4	yes	—
68	C₂²×C₄	#107	1	2	yes	—
69	C₂²×C₄	#107	1	2	yes	—
70	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
73	M₁₆	—	8	4	no	—
74	M₁₆	—	8	4	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	4	1	no	—
79	C₂×C₈	—	4	2	no	—
80	C₂×C₈	—	8	4	no	—
81	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈	#117	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	2	2	no	—
88	C₄²	#121	1	2	yes	—
89	C₄²	#135	1	2	yes	—
90	C₁₆	—	2	1	no	—
91	C₁₆	—	2	2	no	—
92	C₁₆	—	4	2	no	—
93	C₁₆	#101	1	2	yes	complemented
94	C₂³	C₂×#107	1	1	yes	characteristic
95	Q₈	—	2	8	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	#367	1	4	yes	—
104	C₂×C₄	#380	1	1	yes	characteristic
105	C₂×C₄	#427	1	1	yes	characteristic
106	C₂×C₄	#466	1	2	yes	—
107	C₂×C₄	#467	1	2	yes	—
108	C₂×C₄	C₂×SD₃₂	1	2	yes	—
109	C₂×C₄	#508	1	2	yes	—
110	C₈	—	2	1	no	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	16	4	no	—
114	C₈	C₂×#101	1	1	yes	characteristic
115	C₈	#413	1	1	yes	characteristic
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	#1808	1	1	yes	characteristic
119	C₂²	C₂×#367	1	1	yes	characteristic
120	C₂²	#2158	1	1	yes	characteristic
121	C₄	—	2	4	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	—	4	2	no	—
125	C₄	#1801	1	1	yes	characteristic
126	C₄	#1813	1	1	yes	characteristic
127	C₄	#2217	1	4	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#8617	1	1	yes	characteristic
131	C₂	#12835	1	1	yes	characteristic
132	C₂	#12840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₁₆

C₂

yes

maximal, characteristic

#12023

C₂

yes

maximal

#8617

C₂

yes

maximal

#7241

C₂

yes

maximal

C₂×C₈×Q₈

C₂²

yes

characteristic

Q₈×C₁₆

—

Q₈×C₁₆

—

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1801

—

#1567

C₂²

yes

—

#1372

—

#1372

—

#1292

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#423

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄⋊C₈

Q₈

yes

—

C₂×#101

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

#343

—

#335

—

#335

—

C₂²×Q₈

D₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

D₈

yes

characteristic

C₂×C₄⋊C₄

Q₁₆

yes

—

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

SD₃₂

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

#121

yes

—

C₄²

#135

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

yes

C₂³

yes

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#367

yes

—

104

C₂×C₄

#380

yes

characteristic

105

C₂×C₄

#427

yes

characteristic

106

C₂×C₄

#466

yes

—

107

C₂×C₄

#467

yes

—

108

C₂×C₄

C₂×SD₃₂

yes

—

109

C₂×C₄

#508

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

C₂×#101

yes

characteristic

115

C₈

#413

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

#1808

yes

characteristic

119

C₂²

C₂×#367

yes

characteristic

120

C₂²

#2158

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#1801

yes

characteristic

126

C₄

#1813

yes

characteristic

127

C₄

#2217

yes

—

128

C₂

—

129

C₂

—

130

C₂

#8617

yes

characteristic

131

C₂

#12835

yes

characteristic

132

C₂

#12840

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#107
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1808

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

C₂×#107

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
93. C₁₆	#101	semidirect

H ≅ G/N

type

93. C₁₆

#101

semidirect

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

subgroup	index
C₂×Q₈×C₁₆	2
#12023	2
#8617	2
#7241	2

subgroup

index

C₂×Q₈×C₁₆

#12023

#8617

#7241

Subgroups — #134852

Conjugacy classes of subgroups

Complements

Maximal subgroups