Group #146035 — subgroups

Subgroups — #146035

Order 512.

208 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 74 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	#13266	C₂	1	1	yes	maximal, characteristic
2	#12907	C₂	1	1	yes	maximal, characteristic
3	#8677	C₂	1	2	yes	maximal
4	#7279	C₂	1	2	yes	maximal
5	#7155	C₂	1	1	yes	maximal, characteristic
6	#2960	C₂²	1	1	yes	characteristic
7	C₂×#367	—	2	1	no	—
8	C₂×#366	—	2	1	no	—
9	#2150	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1846	C₂²	1	2	yes	—
12	#1564	C₂²	1	2	yes	—
13	#1518	—	2	2	no	—
14	#1379	—	2	1	no	—
15	#1379	—	2	1	no	—
16	#1286	—	2	2	no	—
17	#533	D₄	1	1	yes	characteristic
18	C₂×M₃₂	—	2	1	no	—
19	C₂²×C₁₆	D₄	1	1	yes	characteristic
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	#446	—	2	2	no	—
22	C₈×Q₈	—	2	2	no	—
23	#431	—	2	2	no	—
24	C₂×#107	—	2	2	no	—
25	C₂×#106	—	2	2	no	—
26	#424	—	2	1	no	—
27	#423	D₄	1	1	yes	characteristic
28	#405	—	2	1	no	—
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	#382	—	2	2	no	—
31	#367	—	4	2	no	—
32	#366	—	4	2	no	—
33	C₄⋊C₁₆	—	2	2	no	—
34	C₄×C₁₆	—	2	2	no	—
35	#335	—	2	2	no	—
36	#334	—	2	2	no	—
37	#328	—	4	4	no	—
38	C₂×M₁₆	—	2	1	no	—
39	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₈	#41	1	1	yes	characteristic
41	C₄⋊Q₈	Q₁₆	1	2	yes	—
42	#125	—	2	1	no	—
43	C₂²⋊Q₈	—	2	2	no	—
44	C₄×Q₈	—	2	2	no	—
45	#117	—	2	1	no	—
46	C₂×C₄⋊C₄	—	4	2	no	—
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	M₃₂	—	4	1	no	—
49	M₃₂	—	4	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	—	4	1	no	—
55	C₂×C₁₆	SD₁₆	1	2	yes	—
56	C₂×C₁₆	Q₁₆	1	2	yes	—
57	#107	—	4	2	no	—
58	#107	—	4	2	no	—
59	#106	—	8	2	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	—	4	2	no	—
62	C₄⋊C₈	—	4	2	no	—
63	C₂²⋊C₈	—	4	2	no	—
64	#97	—	4	1	no	—
65	C₄×C₈	—	4	1	no	—
66	C₄×C₈	—	4	2	no	—
67	C₄×C₈	#41	1	2	yes	—
68	C₄×C₈	#41	1	2	yes	—
69	#95	—	4	2	no	—
70	#95	—	4	2	no	—
71	C₂×Q₈	—	2	2	no	—
72	C₂²×C₄	—	8	2	no	—
73	C₂²×C₄	#121	1	1	yes	characteristic
74	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
75	M₁₆	—	4	1	no	—
76	M₁₆	—	4	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	8	2	no	—
81	C₂×C₈	#123	1	1	yes	characteristic
82	C₂×C₈	#124	1	1	yes	characteristic
83	C₂×C₈	#126	1	2	yes	—
84	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂×C₈	#137	1	1	yes	characteristic
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	8	2	no	—
92	C₄⋊C₄	—	8	2	no	—
93	C₂²⋊C₄	—	4	1	no	—
94	C₄²	—	4	1	no	—
95	C₄²	#123	1	2	yes	—
96	C₄²	C₂×Q₁₆	1	1	yes	characteristic
97	C₄²	#135	1	1	yes	characteristic
98	C₁₆	—	2	2	no	—
99	C₁₆	—	2	2	no	—
100	C₁₆	—	8	1	no	—
101	C₂³	#469	1	1	yes	characteristic
102	Q₈	—	8	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	16	4	no	—
106	C₂×C₄	—	16	2	no	—
107	C₂×C₄	#397	1	1	yes	characteristic
108	C₂×C₄	#460	1	1	yes	characteristic
109	C₂×C₄	#462	1	1	yes	characteristic
110	C₂×C₄	#467	1	1	yes	characteristic
111	C₂×C₄	#472	1	1	yes	characteristic
112	C₂×C₄	#473	1	1	yes	characteristic
113	C₂×C₄	#482	1	2	yes	—
114	C₂×C₄	#484	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	—	8	1	no	—
117	C₈	—	16	2	no	—
118	C₈	#483	1	1	yes	characteristic
119	C₈	#484	1	1	yes	characteristic
120	C₈	#487	1	1	yes	characteristic
121	C₈	#489	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	#1688	1	1	yes	characteristic
124	C₂²	#1694	1	1	yes	characteristic
125	C₂²	#2076	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	2	no	—
128	C₄	—	8	2	no	—
129	C₄	—	32	2	no	—
130	C₄	#1670	1	1	yes	characteristic
131	C₄	#1676	1	1	yes	characteristic
132	C₄	#2072	1	1	yes	characteristic
133	C₄	#2078	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#10618	1	1	yes	characteristic
137	C₂	#13500	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13266

C₂

yes

maximal, characteristic

#12907

C₂

yes

maximal, characteristic

#8677

C₂

yes

maximal

#7279

C₂

yes

maximal

#7155

C₂

yes

maximal, characteristic

#2960

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1846

C₂²

yes

—

#1564

C₂²

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

#446

—

C₈×Q₈

—

—

—

—

—

D₄

yes

characteristic

#405

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

—

#334

—

#328

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

Q₁₆

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

#107

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

#123

yes

—

C₄²

C₂×Q₁₆

yes

characteristic

C₄²

#135

yes

characteristic

C₁₆

—

C₁₆

—

100

C₁₆

—

101

C₂³

#469

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#397

yes

characteristic

108

C₂×C₄

#460

yes

characteristic

109

C₂×C₄

#462

yes

characteristic

110

C₂×C₄

#467

yes

characteristic

111

C₂×C₄

#472

yes

characteristic

112

C₂×C₄

#473

yes

characteristic

113

C₂×C₄

#482

yes

—

114

C₂×C₄

#484

yes

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

#483

yes

characteristic

119

C₈

#484

yes

characteristic

120

C₈

#487

yes

characteristic

121

C₈

#489

yes

characteristic

122

C₂²

—

123

C₂²

#1688

yes

characteristic

124

C₂²

#1694

yes

characteristic

125

C₂²

#2076

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#1670

yes

characteristic

131

C₄

#1676

yes

characteristic

132

C₄

#2072

yes

characteristic

133

C₄

#2078

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#10618

yes

characteristic

137

C₂

#13500

yes

—

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2076

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#469

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ 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
#13266	2
#12907	2
#8677	2
#7279	2
#7155	2

subgroup

index

Subgroups — #146035

Conjugacy classes of subgroups

Maximal subgroups