Group #141730 — subgroups

Subgroups — #141730

Order 512.

368 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2247	C₂	1	1	yes	maximal, characteristic
2	#13212	C₂	1	2	yes	maximal
3	#12891	C₂	1	1	yes	maximal, characteristic
4	#11401	C₂	1	1	yes	maximal, characteristic
5	#9772	C₂	1	2	yes	maximal
6	C₂²×Q₃₂	—	2	2	no	—
7	C₂×#494	C₂²	1	2	yes	—
8	#2247	—	2	2	no	—
9	#2247	C₄	1	4	yes	—
10	C₂×#362	—	2	4	no	—
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1848	C₂²	1	2	yes	—
14	#1546	—	2	4	no	—
15	#1450	C₂²	1	1	yes	characteristic
16	#1381	—	2	2	no	—
17	#1312	—	2	2	no	—
18	C₂²×Q₁₆	—	2	2	no	—
19	C₂²×Q₁₆	—	2	2	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×Q₃₂	—	2	8	no	—
22	C₂×Q₃₂	—	4	8	no	—
23	C₂×M₃₂	D₄	1	2	yes	—
24	C₂²×C₁₆	D₄	1	2	yes	—
25	#494	—	2	2	no	—
26	#494	—	2	2	no	—
27	#494	—	2	2	no	—
28	#494	C₂×C₄	1	4	yes	—
29	#431	D₄	1	2	yes	—
30	C₂×#108	—	2	2	no	—
31	C₂×#108	—	2	2	no	—
32	#423	—	2	2	no	—
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	#362	—	4	8	no	—
35	#346	—	2	1	no	—
36	C₄×C₁₆	—	2	1	no	—
37	C₄×C₁₆	C₂×C₄	1	2	yes	—
38	C₄×C₁₆	D₄	1	2	yes	—
39	#332	—	4	8	no	—
40	#322	—	2	1	no	—
41	C₈²	—	2	1	no	—
42	C₂²×Q₈	—	4	2	no	—
43	C₂×Q₁₆	—	2	4	no	—
44	C₂×Q₁₆	—	2	4	no	—
45	C₂×Q₁₆	—	4	2	no	—
46	C₂×Q₁₆	—	4	2	no	—
47	C₂×Q₁₆	—	4	4	no	—
48	C₂×Q₁₆	—	4	4	no	—
49	C₂×Q₁₆	—	4	4	no	—
50	C₂×Q₁₆	—	4	4	no	—
51	C₂×M₁₆	—	4	2	no	—
52	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄⋊Q₈	—	2	4	no	—
55	C₄⋊Q₈	—	4	2	no	—
56	C₄⋊Q₈	—	4	4	no	—
57	C₂×C₄⋊C₄	—	4	4	no	—
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	Q₃₂	—	4	16	no	—
60	M₃₂	—	2	4	no	—
61	C₂×C₁₆	—	2	4	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
64	#108	—	4	4	no	—
65	#108	—	4	4	no	—
66	C₄⋊C₈	—	4	4	no	—
67	C₄⋊C₈	D₈	1	4	yes	—
68	C₂²⋊C₈	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₄×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄×C₈	#41	1	1	yes	characteristic
73	C₄×C₈	#41	1	1	yes	characteristic
74	C₂×Q₈	—	4	4	no	—
75	C₂×Q₈	—	8	2	no	—
76	C₂×Q₈	—	8	2	no	—
77	C₂×Q₈	—	8	2	no	—
78	C₂×Q₈	—	8	2	no	—
79	C₂×Q₈	—	8	2	no	—
80	C₂²×C₄	—	8	2	no	—
81	C₂²×C₄	#127	1	1	yes	characteristic
82	C₂²×C₄	#135	1	2	yes	—
83	Q₁₆	—	4	8	no	—
84	Q₁₆	—	4	8	no	—
85	Q₁₆	—	8	4	no	—
86	Q₁₆	—	8	4	no	—
87	M₁₆	—	8	2	no	—
88	M₁₆	—	8	2	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	8	2	no	—
91	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈	C₄×D₄	1	1	yes	characteristic
93	C₂×C₈	#121	1	1	yes	characteristic
94	C₂×C₈	#124	1	1	yes	characteristic
95	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
96	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
97	C₄⋊C₄	—	4	8	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄²	C₄×D₄	1	1	yes	characteristic
100	C₄²	#121	1	1	yes	characteristic
101	C₄²	C₂×D₈	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₁₆	—	4	2	no	—
104	C₂³	#495	1	1	yes	characteristic
105	Q₈	—	8	4	no	—
106	Q₈	—	8	4	no	—
107	Q₈	—	16	2	no	—
108	Q₈	—	16	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	8	4	no	—
111	C₂×C₄	—	16	2	no	—
112	C₂×C₄	—	16	2	no	—
113	C₂×C₄	#390	1	1	yes	characteristic
114	C₂×C₄	C₄×D₈	1	2	yes	—
115	C₂×C₄	#466	1	2	yes	—
116	C₂×C₄	#493	1	1	yes	characteristic
117	C₂×C₄	#496	1	2	yes	—
118	C₂×C₄	#496	1	2	yes	—
119	C₈	—	4	4	no	—
120	C₈	—	16	2	no	—
121	C₈	#418	1	2	yes	—
122	C₈	#442	1	2	yes	—
123	C₈	#469	1	2	yes	—
124	C₈	#490	1	2	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	#1717	1	2	yes	—
127	C₂²	#1970	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	16	4	no	—
130	C₄	#1721	1	2	yes	—
131	C₄	#1961	1	1	yes	characteristic
132	C₄	#1972	1	1	yes	characteristic
133	C₄	#2222	1	4	yes	—
134	C₂	—	2	2	no	—
135	C₂	#9832	1	1	yes	characteristic
136	C₂	#13244	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2247

C₂

yes

maximal, characteristic

#13212

C₂

yes

maximal

#12891

C₂

yes

maximal, characteristic

#11401

C₂

yes

maximal, characteristic

#9772

C₂

yes

maximal

C₂²×Q₃₂

—

C₂×#494

C₂²

yes

—

#2247

—

#2247

C₄

yes

—

C₂×#362

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1546

—

#1450

C₂²

yes

characteristic

#1381

—

#1312

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#431

D₄

yes

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂³

yes

characteristic

#362

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

#332

—

#322

—

C₈²

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

Q₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

D₈

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

#135

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₄

yes

characteristic

100

C₄²

#121

yes

characteristic

101

C₄²

C₂×D₈

yes

—

102

C₁₆

—

103

C₁₆

—

104

C₂³

#495

yes

characteristic

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#390

yes

characteristic

114

C₂×C₄

C₄×D₈

yes

—

115

C₂×C₄

#466

yes

—

116

C₂×C₄

#493

yes

characteristic

117

C₂×C₄

#496

yes

—

118

C₂×C₄

#496

yes

—

119

C₈

—

120

C₈

—

121

C₈

#418

yes

—

122

C₈

#442

yes

—

123

C₈

#469

yes

—

124

C₈

#490

yes

—

125

C₂²

—

126

C₂²

#1717

yes

—

127

C₂²

#1970

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

#1721

yes

—

131

C₄

#1961

yes

characteristic

132

C₄

#1972

yes

characteristic

133

C₄

#2222

yes

—

134

C₂

—

135

C₂

#9832

yes

characteristic

136

C₂

#13244

yes

—

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1970

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#495

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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
C₂×#2247	2
#13212	2
#12891	2
#11401	2
#9772	2

subgroup

index

Subgroups — #141730

Conjugacy classes of subgroups

Maximal subgroups