Group #144158 — subgroups

Subgroups — #144158

Order 512.

409 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 112 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×#2241	C₂	1	2	yes	maximal
2	C₂×#1710	C₂	1	1	yes	maximal, characteristic
3	#13282	C₂	1	2	yes	maximal
4	#9825	C₂	1	1	yes	maximal, characteristic
5	#7136	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	2	yes	—
7	C₂×#494	C₂²	1	1	yes	characteristic
8	C₂×#487	—	2	2	no	—
9	#2241	—	2	4	no	—
10	#2241	—	2	4	no	—
11	C₂×#358	—	2	4	no	—
12	C₂×C₄×C₁₆	C₂²	1	2	yes	—
13	#1990	—	2	2	no	—
14	#1845	C₂²	1	1	yes	characteristic
15	#1710	—	2	4	no	—
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	#1380	—	2	4	no	—
18	C₂²×Q₁₆	—	2	2	no	—
19	C₂×C₄⋊Q₈	—	2	2	no	—
20	C₂×C₄⋊Q₈	—	2	1	no	—
21	C₂²×C₁₆	D₄	1	4	yes	—
22	#500	—	2	2	no	—
23	#500	—	2	2	no	—
24	#500	—	2	2	no	—
25	#500	D₄	1	4	yes	—
26	#494	—	2	2	no	—
27	#494	—	2	1	no	—
28	#494	D₄	1	2	yes	—
29	#487	—	2	8	no	—
30	#487	—	4	4	no	—
31	C₂×#107	—	2	2	no	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#103	—	4	4	no	—
35	C₂×C₄×C₈	D₄	1	2	yes	—
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	#384	—	2	1	no	—
38	#358	—	2	16	no	—
39	C₄×C₁₆	—	2	2	no	—
40	C₄×C₁₆	—	2	2	no	—
41	#340	—	4	4	no	—
42	C₈²	—	2	2	no	—
43	C₂²×Q₈	—	4	1	no	—
44	C₂²×Q₈	—	4	2	no	—
45	C₂×Q₁₆	—	2	4	no	—
46	C₂×Q₁₆	—	4	2	no	—
47	C₂×Q₁₆	—	4	4	no	—
48	C₂×Q₁₆	—	4	4	no	—
49	C₂²×C₈	D₈	1	4	yes	—
50	C₂²×C₈	C₂×D₄	1	2	yes	—
51	C₄⋊Q₈	—	2	4	no	—
52	C₄⋊Q₈	—	2	2	no	—
53	C₄⋊Q₈	—	4	1	no	—
54	C₄⋊Q₈	—	4	2	no	—
55	C₄⋊Q₈	—	4	2	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₂×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄⋊C₄	—	4	1	no	—
62	C₂×C₄⋊C₄	—	4	2	no	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₁₆	—	2	4	no	—
65	C₂×C₁₆	—	2	4	no	—
66	C₂×C₁₆	—	2	4	no	—
67	#107	—	2	4	no	—
68	#107	—	2	4	no	—
69	#107	—	4	2	no	—
70	#107	—	4	2	no	—
71	#107	—	4	4	no	—
72	#103	—	4	16	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	C₂×D₄	1	2	yes	—
76	C₄×C₈	C₂×D₄	1	1	yes	characteristic
77	C₄×C₈	#41	1	1	yes	characteristic
78	C₂×Q₈	—	4	2	no	—
79	C₂×Q₈	—	4	4	no	—
80	C₂×Q₈	—	8	2	no	—
81	C₂×Q₈	—	8	1	no	—
82	C₂×Q₈	—	8	2	no	—
83	C₂×Q₈	—	8	2	no	—
84	C₂×Q₈	—	8	4	no	—
85	C₂×Q₈	—	8	4	no	—
86	C₂²×C₄	—	8	1	no	—
87	C₂²×C₄	—	8	2	no	—
88	C₂²×C₄	—	8	1	no	—
89	C₂²×C₄	#127	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₈	1	2	yes	—
91	Q₁₆	—	4	8	no	—
92	Q₁₆	—	8	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	#120	1	2	yes	—
97	C₂×C₈	#121	1	2	yes	—
98	C₂×C₈	#135	1	4	yes	—
99	C₂×C₈	#137	1	4	yes	—
100	C₄⋊C₄	—	4	4	no	—
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	8	2	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₄⋊C₄	—	8	2	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄⋊C₄	—	8	4	no	—
108	C₄²	#120	1	1	yes	characteristic
109	C₄²	#121	1	1	yes	characteristic
110	C₄²	AGL(1,ℤ/8)	1	2	yes	—
111	C₁₆	—	4	4	no	—
112	C₂³	#493	1	1	yes	characteristic
113	Q₈	—	8	8	no	—
114	Q₈	—	8	4	no	—
115	Q₈	—	16	4	no	—
116	Q₈	—	16	2	no	—
117	C₂×C₄	—	8	4	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	—	16	1	no	—
120	C₂×C₄	—	16	1	no	—
121	C₂×C₄	—	16	2	no	—
122	C₂×C₄	—	16	2	no	—
123	C₂×C₄	—	16	1	no	—
124	C₂×C₄	—	16	2	no	—
125	C₂×C₄	#392	1	1	yes	characteristic
126	C₂×C₄	#447	1	2	yes	—
127	C₂×C₄	#459	1	2	yes	—
128	C₂×C₄	#495	1	2	yes	—
129	C₂×C₄	#496	1	1	yes	characteristic
130	C₂×C₄	#497	1	2	yes	—
131	C₂×C₄	#508	1	4	yes	—
132	C₂×C₄	#510	1	4	yes	—
133	C₈	—	4	4	no	—
134	C₈	#452	1	4	yes	—
135	C₈	#464	1	4	yes	—
136	C₂²	#1709	1	1	yes	characteristic
137	C₂²	#1722	1	1	yes	characteristic
138	C₂²	#2011	1	1	yes	characteristic
139	C₂²	#2249	1	2	yes	—
140	C₂²	#2251	1	2	yes	—
141	C₄	—	16	4	no	—
142	C₄	—	16	2	no	—
143	C₄	—	32	1	no	—
144	C₄	#1720	1	2	yes	—
145	C₄	#2015	1	2	yes	—
146	C₄	#2189	1	4	yes	—
147	C₄	#2211	1	4	yes	—
148	C₂	#10306	1	1	yes	characteristic
149	C₂	#12511	1	1	yes	characteristic
150	C₂	#12514	1	1	yes	characteristic
151	C₂	#12533	1	2	yes	—
152	C₂	#13329	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2241

C₂

yes

maximal

C₂×#1710

C₂

yes

maximal, characteristic

#13282

C₂

yes

maximal

#9825

C₂

yes

maximal, characteristic

#7136

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

—

C₂×#494

C₂²

yes

—

—

—

—

C₂×C₄×C₁₆

C₂²

yes

—

#1990

—

#1845

C₂²

yes

characteristic

#1710

—

C₂×C₈²

C₂²

yes

characteristic

#1380

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

—

—

—

—

D₄

yes

—

#494

—

#494

—

#494

D₄

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

C₈²

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#120

yes

—

C₂×C₈

#121

yes

—

C₂×C₈

#135

yes

—

C₂×C₈

#137

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

#120

yes

characteristic

109

C₄²

#121

yes

characteristic

110

C₄²

AGL(1,ℤ/8)

yes

—

111

C₁₆

—

112

C₂³

#493

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#392

yes

characteristic

126

C₂×C₄

#447

yes

—

127

C₂×C₄

#459

yes

—

128

C₂×C₄

#495

yes

—

129

C₂×C₄

#496

yes

characteristic

130

C₂×C₄

#497

yes

—

131

C₂×C₄

#508

yes

—

132

C₂×C₄

#510

yes

—

133

C₈

—

134

C₈

#452

yes

—

135

C₈

#464

yes

—

136

C₂²

#1709

yes

characteristic

137

C₂²

#1722

yes

characteristic

138

C₂²

#2011

yes

characteristic

139

C₂²

#2249

yes

—

140

C₂²

#2251

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#1720

yes

—

145

C₄

#2015

yes

—

146

C₄

#2189

yes

—

147

C₄

#2211

yes

—

148

C₂

#10306

yes

characteristic

149

C₂

#12511

yes

characteristic

150

C₂

#12514

yes

characteristic

151

C₂

#12533

yes

—

152

C₂

#13329

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#493

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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×#2241	2
C₂×#1710	2
#13282	2
#9825	2
#7136	2

subgroup

index

Subgroups — #144158

Conjugacy classes of subgroups

Maximal subgroups