Group #146147 — subgroups

Subgroups — #146147

Order 512.

202 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 78 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	#13181	C₂	1	2	yes	maximal
2	#13161	C₂	1	1	yes	maximal, characteristic
3	#12906	C₂	1	1	yes	maximal, characteristic
4	#7269	C₂	1	2	yes	maximal
5	#7160	C₂	1	1	yes	maximal, characteristic
6	#2159	—	2	2	no	—
7	C₂×#367	—	2	1	no	—
8	C₂×#367	—	2	1	no	—
9	C₂×#364	C₂²	1	1	yes	characteristic
10	C₂×#364	C₂²	1	2	yes	—
11	#2151	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1845	C₂²	1	2	yes	—
14	#1772	C₂²	1	1	yes	characteristic
15	#1387	—	2	2	no	—
16	#1379	—	2	1	no	—
17	C₂×M₃₂	—	2	2	no	—
18	C₂×M₃₂	—	2	1	no	—
19	C₂²×C₁₆	D₄	1	1	yes	characteristic
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂×#108	D₄	1	1	yes	characteristic
22	#427	D₄	1	1	yes	characteristic
23	C₂×#107	—	2	2	no	—
24	C₂×#107	—	2	2	no	—
25	#405	—	2	1	no	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	#384	—	2	2	no	—
28	#367	—	4	2	no	—
29	#367	—	4	2	no	—
30	#367	—	4	2	no	—
31	#366	—	4	2	no	—
32	#364	—	2	2	no	—
33	#364	—	2	2	no	—
34	#364	Q₈	1	4	yes	—
35	C₄⋊C₁₆	—	4	2	no	—
36	C₄×C₁₆	—	2	2	no	—
37	#339	—	2	1	no	—
38	C₂×M₁₆	—	2	1	no	—
39	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
41	#117	—	2	1	no	—
42	C₂×C₄⋊C₄	—	4	2	no	—
43	C₂×C₄⋊C₄	—	8	2	no	—
44	C₂×C₄²	#41	1	1	yes	characteristic
45	M₃₂	—	2	4	no	—
46	M₃₂	—	4	2	no	—
47	M₃₂	—	4	1	no	—
48	M₃₂	—	4	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	4	2	no	—
54	C₂×C₁₆	—	4	1	no	—
55	C₂×C₁₆	D₈	1	2	yes	—
56	C₂×C₁₆	Q₁₆	1	2	yes	—
57	#108	—	2	2	no	—
58	#107	—	4	2	no	—
59	#107	—	4	2	no	—
60	#107	—	4	2	no	—
61	#107	—	4	2	no	—
62	#107	D₈	1	2	yes	—
63	#106	—	2	1	no	—
64	#97	—	4	1	no	—
65	C₄×C₈	—	4	1	no	—
66	C₄×C₈	C₂×Q₈	1	2	yes	—
67	C₄×C₈	#41	1	2	yes	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	8	2	no	—
70	C₂²×C₄	#121	1	1	yes	characteristic
71	M₁₆	—	4	1	no	—
72	M₁₆	—	4	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₈	#123	1	1	yes	characteristic
76	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₈	C₂×D₈	1	1	yes	characteristic
78	C₂×C₈	C₂×D₈	1	1	yes	characteristic
79	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₈	#135	1	1	yes	characteristic
81	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₈	#137	1	2	yes	—
83	C₂×C₈	#137	1	1	yes	characteristic
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	8	2	no	—
86	C₄⋊C₄	—	8	2	no	—
87	C₄⋊C₄	—	16	4	no	—
88	C₂²⋊C₄	—	4	1	no	—
89	C₄²	—	2	1	no	—
90	C₄²	—	4	1	no	—
91	C₄²	#125	1	2	yes	—
92	C₁₆	—	2	2	no	—
93	C₁₆	—	2	2	no	—
94	C₁₆	—	4	4	no	—
95	C₁₆	—	8	1	no	—
96	C₂³	#472	1	1	yes	characteristic
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	16	4	no	—
101	C₂×C₄	—	16	2	no	—
102	C₂×C₄	#398	1	1	yes	characteristic
103	C₂×C₄	#459	1	1	yes	characteristic
104	C₂×C₄	#462	1	1	yes	characteristic
105	C₂×C₄	#466	1	1	yes	characteristic
106	C₂×C₄	#470	1	1	yes	characteristic
107	C₂×C₄	#473	1	1	yes	characteristic
108	C₈	—	8	1	no	—
109	C₈	#474	1	2	yes	—
110	C₈	#479	1	2	yes	—
111	C₈	#480	1	1	yes	characteristic
112	C₈	#484	1	1	yes	characteristic
113	C₈	#500	1	1	yes	characteristic
114	C₈	#501	1	1	yes	characteristic
115	C₂²	—	2	2	no	—
116	C₂²	#1667	1	1	yes	characteristic
117	C₂²	#1696	1	1	yes	characteristic
118	C₂²	#2089	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	8	2	no	—
121	C₄	—	32	2	no	—
122	C₄	#1674	1	1	yes	characteristic
123	C₄	#1689	1	1	yes	characteristic
124	C₄	#2083	1	1	yes	characteristic
125	C₄	#2086	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	#10649	1	1	yes	characteristic
129	C₂	#12438	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13181

C₂

yes

maximal

#13161

C₂

yes

maximal, characteristic

#12906

C₂

yes

maximal, characteristic

#7269

C₂

yes

maximal

#7160

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

C₂×#364

C₂²

yes

—

#2151

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1845

C₂²

yes

—

#1772

C₂²

yes

characteristic

#1387

—

#1379

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂×#108

D₄

yes

characteristic

#427

D₄

yes

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

Q₈

yes

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

D₈

yes

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#137

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

#125

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#472

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#398

yes

characteristic

103

C₂×C₄

#459

yes

characteristic

104

C₂×C₄

#462

yes

characteristic

105

C₂×C₄

#466

yes

characteristic

106

C₂×C₄

#470

yes

characteristic

107

C₂×C₄

#473

yes

characteristic

108

C₈

—

109

C₈

#474

yes

—

110

C₈

#479

yes

—

111

C₈

#480

yes

characteristic

112

C₈

#484

yes

characteristic

113

C₈

#500

yes

characteristic

114

C₈

#501

yes

characteristic

115

C₂²

—

116

C₂²

#1667

yes

characteristic

117

C₂²

#1696

yes

characteristic

118

C₂²

#2089

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#1674

yes

characteristic

123

C₄

#1689

yes

characteristic

124

C₄

#2083

yes

characteristic

125

C₄

#2086

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#10649

yes

characteristic

129

C₂

#12438

yes

—

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1667

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#472

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ 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
#13181	2
#13161	2
#12906	2
#7269	2
#7160	2

subgroup

index

Subgroups — #146147

Conjugacy classes of subgroups

Maximal subgroups