Group #141723 — subgroups

Subgroups — #141723

Order 512.

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

normal subgroups only

This group has 288 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×#2252	C₂	1	1	yes	maximal, characteristic
2	#13290	C₂	1	2	yes	maximal
3	#12891	C₂	1	1	yes	maximal, characteristic
4	#11401	C₂	1	1	yes	maximal, characteristic
5	#9771	C₂	1	2	yes	maximal
6	C₂×#500	C₂²	1	2	yes	—
7	#2252	—	2	2	no	—
8	#2252	C₄	1	4	yes	—
9	C₂×#366	—	2	2	no	—
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1848	C₂²	1	2	yes	—
13	#1549	—	2	2	no	—
14	#1548	—	2	2	no	—
15	#1450	C₂²	1	1	yes	characteristic
16	#1387	—	2	4	no	—
17	#1381	—	2	2	no	—
18	#1312	—	2	2	no	—
19	C₂×C₄⋊Q₈	—	2	2	no	—
20	C₂×M₃₂	D₄	1	2	yes	—
21	C₂²×C₁₆	D₄	1	2	yes	—
22	#500	—	2	2	no	—
23	#500	—	2	2	no	—
24	#500	—	2	2	no	—
25	#500	C₂×C₄	1	4	yes	—
26	#431	D₄	1	1	yes	characteristic
27	#431	D₄	1	1	yes	characteristic
28	C₂×#108	—	2	2	no	—
29	C₂×#108	—	2	2	no	—
30	C₂×#107	—	2	2	no	—
31	C₂×#107	—	2	2	no	—
32	#423	—	2	2	no	—
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	#366	—	2	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	#333	—	4	4	no	—
40	#332	—	4	4	no	—
41	#322	—	2	1	no	—
42	C₈²	—	2	1	no	—
43	C₂²×Q₈	—	4	2	no	—
44	C₂×M₁₆	—	4	2	no	—
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
47	C₄⋊Q₈	—	2	4	no	—
48	C₄⋊Q₈	—	4	2	no	—
49	C₄⋊Q₈	—	4	2	no	—
50	C₄⋊Q₈	—	4	2	no	—
51	C₂×C₄⋊C₄	—	4	2	no	—
52	C₂×C₄⋊C₄	—	4	2	no	—
53	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
54	M₃₂	—	2	4	no	—
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
58	#108	—	4	4	no	—
59	#108	—	4	4	no	—
60	#107	—	2	4	no	—
61	#107	—	2	4	no	—
62	#107	—	4	2	no	—
63	#107	—	4	2	no	—
64	C₄⋊C₈	—	4	2	no	—
65	C₄⋊C₈	—	4	2	no	—
66	C₄⋊C₈	D₈	1	2	yes	—
67	C₄⋊C₈	D₈	1	2	yes	—
68	C₂²⋊C₈	—	2	1	no	—
69	C₂²⋊C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₄×C₈	C₂×D₄	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	C₄×C₈	#41	1	1	yes	characteristic
76	C₂×Q₈	—	4	4	no	—
77	C₂×Q₈	—	8	2	no	—
78	C₂×Q₈	—	16	4	no	—
79	C₂²×C₄	—	8	2	no	—
80	C₂²×C₄	#127	1	1	yes	characteristic
81	C₂²×C₄	#135	1	1	yes	characteristic
82	C₂²×C₄	#135	1	1	yes	characteristic
83	M₁₆	—	8	2	no	—
84	M₁₆	—	8	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	8	2	no	—
88	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈	C₄×D₄	1	1	yes	characteristic
90	C₂×C₈	#121	1	1	yes	characteristic
91	C₂×C₈	#124	1	1	yes	characteristic
92	C₂×C₈	#137	1	4	yes	—
93	C₂×C₈	#137	1	4	yes	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	4	4	no	—
96	C₄⋊C₄	—	8	2	no	—
97	C₄⋊C₄	—	8	2	no	—
98	C₄²	C₄×D₄	1	1	yes	characteristic
99	C₄²	#121	1	1	yes	characteristic
100	C₄²	C₂×D₈	1	1	yes	characteristic
101	C₄²	C₂×D₈	1	1	yes	characteristic
102	C₁₆	—	2	4	no	—
103	C₁₆	—	4	2	no	—
104	C₂³	#495	1	1	yes	characteristic
105	Q₈	—	16	4	no	—
106	Q₈	—	16	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	—	16	4	no	—
111	C₂×C₄	#390	1	1	yes	characteristic
112	C₂×C₄	C₄×D₈	1	1	yes	characteristic
113	C₂×C₄	C₄×D₈	1	1	yes	characteristic
114	C₂×C₄	#466	1	1	yes	characteristic
115	C₂×C₄	#466	1	1	yes	characteristic
116	C₂×C₄	#493	1	1	yes	characteristic
117	C₂×C₄	#497	1	2	yes	—
118	C₂×C₄	#497	1	2	yes	—
119	C₈	—	4	2	no	—
120	C₈	—	4	2	no	—
121	C₈	—	16	2	no	—
122	C₈	#419	1	2	yes	—
123	C₈	#441	1	2	yes	—
124	C₈	#470	1	2	yes	—
125	C₈	#491	1	2	yes	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	#1718	1	2	yes	—
129	C₂²	#1970	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	16	4	no	—
133	C₄	#1722	1	2	yes	—
134	C₄	#1960	1	1	yes	characteristic
135	C₄	#1971	1	1	yes	characteristic
136	C₄	#2221	1	2	yes	—
137	C₄	#2222	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	#9831	1	1	yes	characteristic
140	C₂	#13244	1	1	yes	characteristic
141	C₂	#13245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

#13290

C₂

yes

maximal

#12891

C₂

yes

maximal, characteristic

#11401

C₂

yes

maximal, characteristic

#9771

C₂

yes

maximal

C₂×#500

C₂²

yes

—

#2252

—

#2252

C₄

yes

—

C₂×#366

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1549

—

#1548

—

#1450

C₂²

yes

—

—

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#431

D₄

yes

characteristic

#431

D₄

yes

—

—

—

—

—

C₂×C₄×C₈

C₂³

yes

characteristic

#366

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

#333

—

#332

—

#322

—

C₈²

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₈

yes

—

C₄⋊C₈

D₈

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×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

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

C₂×C₈

—

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

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₄

yes

characteristic

C₄²

#121

yes

characteristic

100

C₄²

C₂×D₈

yes

characteristic

101

C₄²

C₂×D₈

yes

characteristic

102

C₁₆

—

103

C₁₆

—

104

C₂³

#495

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#390

yes

characteristic

112

C₂×C₄

C₄×D₈

yes

characteristic

113

C₂×C₄

C₄×D₈

yes

characteristic

114

C₂×C₄

#466

yes

characteristic

115

C₂×C₄

#466

yes

characteristic

116

C₂×C₄

#493

yes

characteristic

117

C₂×C₄

#497

yes

—

118

C₂×C₄

#497

yes

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

#419

yes

—

123

C₈

#441

yes

—

124

C₈

#470

yes

—

125

C₈

#491

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

#1718

yes

—

129

C₂²

#1970

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#1722

yes

—

134

C₄

#1960

yes

characteristic

135

C₄

#1971

yes

characteristic

136

C₄

#2221

yes

—

137

C₄

#2222

yes

—

138

C₂

—

139

C₂

#9831

yes

characteristic

140

C₂

#13244

yes

characteristic

141

C₂

#13245

yes

characteristic

	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×#2252	2
#13290	2
#12891	2
#11401	2
#9771	2

subgroup

index

Subgroups — #141723

Conjugacy classes of subgroups

Maximal subgroups