Group #144128 — subgroups

Subgroups — #144128

Order 512.

521 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 128 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	2	yes	maximal
2	C₂×#1713	C₂	1	1	yes	maximal, characteristic
3	#13203	C₂	1	2	yes	maximal
4	#9825	C₂	1	1	yes	maximal, characteristic
5	#7136	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	—	2	4	no	—
7	C₂×#500	—	2	2	no	—
8	C₂×#494	C₂²	1	1	yes	characteristic
9	C₂×#494	C₂²	1	2	yes	—
10	#2247	—	2	4	no	—
11	#2247	—	2	4	no	—
12	C₂×#358	—	2	4	no	—
13	C₂×C₄×C₁₆	C₂²	1	2	yes	—
14	#1990	—	2	2	no	—
15	#1845	C₂²	1	1	yes	characteristic
16	#1713	—	2	4	no	—
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆	—	2	2	no	—
19	C₂²×Q₁₆	—	2	2	no	—
20	C₂²×Q₁₆	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂×C₄⋊Q₈	—	2	1	no	—
23	C₂×Q₃₂	—	2	16	no	—
24	C₂×Q₃₂	—	4	8	no	—
25	C₂×Q₃₂	—	4	8	no	—
26	C₂²×C₁₆	D₄	1	4	yes	—
27	#500	—	2	8	no	—
28	#500	—	4	4	no	—
29	#494	—	2	2	no	—
30	#494	—	2	2	no	—
31	#494	—	2	2	no	—
32	#494	—	2	1	no	—
33	#494	—	2	2	no	—
34	#494	D₄	1	4	yes	—
35	#494	D₄	1	2	yes	—
36	C₂×#107	—	2	2	no	—
37	C₂×#107	—	4	4	no	—
38	C₂×C₄×C₈	D₄	1	2	yes	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#384	—	2	1	no	—
41	#358	—	4	8	no	—
42	C₄×C₁₆	—	2	2	no	—
43	C₄×C₁₆	—	2	2	no	—
44	#340	—	4	4	no	—
45	C₈²	—	2	2	no	—
46	C₂²×Q₈	—	4	2	no	—
47	C₂²×Q₈	—	4	1	no	—
48	C₂×Q₁₆	—	2	4	no	—
49	C₂×Q₁₆	—	2	4	no	—
50	C₂×Q₁₆	—	2	4	no	—
51	C₂×Q₁₆	—	4	2	no	—
52	C₂×Q₁₆	—	4	2	no	—
53	C₂×Q₁₆	—	4	2	no	—
54	C₂×Q₁₆	—	4	4	no	—
55	C₂×Q₁₆	—	4	4	no	—
56	C₂×Q₁₆	—	4	4	no	—
57	C₂×Q₁₆	—	4	4	no	—
58	C₂×Q₁₆	—	4	4	no	—
59	C₂×Q₁₆	—	4	4	no	—
60	C₂²×C₈	D₈	1	4	yes	—
61	C₂²×C₈	C₂×D₄	1	2	yes	—
62	C₄⋊Q₈	—	2	4	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	4	1	no	—
65	C₄⋊Q₈	—	4	2	no	—
66	C₄⋊Q₈	—	4	2	no	—
67	C₄⋊Q₈	—	4	4	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄⋊C₄	—	4	4	no	—
70	C₂×C₄⋊C₄	—	4	1	no	—
71	C₂×C₄⋊C₄	—	4	2	no	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	Q₃₂	—	4	32	no	—
74	C₂×C₁₆	—	2	4	no	—
75	C₂×C₁₆	—	2	4	no	—
76	C₂×C₁₆	D₈	1	8	yes	—
77	#107	—	4	16	no	—
78	#107	—	4	4	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	C₂×D₄	1	2	yes	—
82	C₄×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	C₂×Q₈	—	4	4	no	—
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	8	1	no	—
87	C₂×Q₈	—	8	2	no	—
88	C₂×Q₈	—	8	2	no	—
89	C₂×Q₈	—	8	2	no	—
90	C₂×Q₈	—	8	2	no	—
91	C₂×Q₈	—	8	2	no	—
92	C₂×Q₈	—	8	2	no	—
93	C₂×Q₈	—	8	2	no	—
94	C₂²×C₄	—	8	1	no	—
95	C₂²×C₄	—	8	2	no	—
96	C₂²×C₄	—	8	1	no	—
97	C₂²×C₄	#127	1	1	yes	characteristic
98	C₂²×C₄	C₂×D₈	1	2	yes	—
99	Q₁₆	—	4	8	no	—
100	Q₁₆	—	4	8	no	—
101	Q₁₆	—	4	8	no	—
102	Q₁₆	—	8	4	no	—
103	Q₁₆	—	8	4	no	—
104	Q₁₆	—	8	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	Q₃₂	1	8	yes	complemented
108	C₂×C₈	#120	1	2	yes	—
109	C₂×C₈	#121	1	2	yes	—
110	C₂×C₈	C₂×D₈	1	4	yes	—
111	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
112	C₄⋊C₄	—	4	8	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	8	2	no	—
115	C₄⋊C₄	—	8	4	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄⋊C₄	—	8	4	no	—
118	C₄²	#120	1	1	yes	characteristic
119	C₄²	#121	1	1	yes	characteristic
120	C₄²	AGL(1,ℤ/8)	1	2	yes	—
121	C₁₆	—	2	8	no	—
122	C₂³	#493	1	1	yes	characteristic
123	Q₈	—	8	4	no	—
124	Q₈	—	8	4	no	—
125	Q₈	—	8	4	no	—
126	Q₈	—	16	2	no	—
127	Q₈	—	16	2	no	—
128	Q₈	—	16	2	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	—	16	1	no	—
132	C₂×C₄	—	16	2	no	—
133	C₂×C₄	—	16	1	no	—
134	C₂×C₄	—	16	2	no	—
135	C₂×C₄	—	16	1	no	—
136	C₂×C₄	—	16	2	no	—
137	C₂×C₄	#392	1	1	yes	characteristic
138	C₂×C₄	#447	1	2	yes	—
139	C₂×C₄	#459	1	2	yes	—
140	C₂×C₄	#493	1	2	yes	—
141	C₂×C₄	#496	1	1	yes	characteristic
142	C₂×C₄	#496	1	2	yes	—
143	C₂×C₄	C₂×Q₃₂	1	4	yes	—
144	C₂×C₄	#510	1	4	yes	—
145	C₈	—	2	8	no	—
146	C₈	#447	1	4	yes	—
147	C₈	#459	1	4	yes	—
148	C₂²	#1712	1	1	yes	characteristic
149	C₂²	#1721	1	1	yes	characteristic
150	C₂²	#2011	1	1	yes	characteristic
151	C₂²	#2247	1	2	yes	—
152	C₂²	#2251	1	2	yes	—
153	C₄	—	16	4	no	—
154	C₄	—	16	2	no	—
155	C₄	—	32	1	no	—
156	C₄	#1721	1	2	yes	—
157	C₄	#2011	1	2	yes	—
158	C₄	#2188	1	4	yes	—
159	C₄	#2209	1	4	yes	—
160	C₂	#10304	1	1	yes	characteristic
161	C₂	#12516	1	1	yes	characteristic
162	C₂	#12520	1	1	yes	characteristic
163	C₂	#12528	1	2	yes	—
164	C₂	#13330	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2247

C₂

yes

maximal

C₂×#1713

C₂

yes

maximal, characteristic

#13203

C₂

yes

maximal

#9825

C₂

yes

maximal, characteristic

#7136

C₂

yes

maximal, characteristic

C₂²×Q₃₂

—

C₂×#500

—

C₂×#494

C₂²

yes

characteristic

C₂×#494

C₂²

yes

—

#2247

—

#2247

—

C₂×#358

—

C₂×C₄×C₁₆

C₂²

yes

—

#1990

—

#1845

C₂²

yes

characteristic

#1713

—

C₂×C₈²

C₂²

yes

characteristic

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂²×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

D₄

yes

—

#494

D₄

yes

—

C₂×#107

—

C₂×#107

—

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

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

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

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

#107

—

#107

—

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

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

102

Q₁₆

—

103

Q₁₆

—

104

Q₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

Q₃₂

yes

complemented

108

C₂×C₈

#120

yes

—

109

C₂×C₈

#121

yes

—

110

C₂×C₈

C₂×D₈

yes

—

111

C₂×C₈

AGL(1,ℤ/8)

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

#120

yes

characteristic

119

C₄²

#121

yes

characteristic

120

C₄²

AGL(1,ℤ/8)

yes

—

121

C₁₆

—

122

C₂³

#493

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#392

yes

characteristic

138

C₂×C₄

#447

yes

—

139

C₂×C₄

#459

yes

—

140

C₂×C₄

#493

yes

—

141

C₂×C₄

#496

yes

characteristic

142

C₂×C₄

#496

yes

—

143

C₂×C₄

C₂×Q₃₂

yes

—

144

C₂×C₄

#510

yes

—

145

C₈

—

146

C₈

#447

yes

—

147

C₈

#459

yes

—

148

C₂²

#1712

yes

characteristic

149

C₂²

#1721

yes

characteristic

150

C₂²

#2011

yes

characteristic

151

C₂²

#2247

yes

—

152

C₂²

#2251

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#1721

yes

—

157

C₄

#2011

yes

—

158

C₄

#2188

yes

—

159

C₄

#2209

yes

—

160

C₂

#10304

yes

characteristic

161

C₂

#12516

yes

characteristic

162

C₂

#12520

yes

characteristic

163

C₂

#12528

yes

—

164

C₂

#13330

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

N	H ≅ G/N	type
107. C₂×C₈	Q₃₂	semidirect

H ≅ G/N

type

107. C₂×C₈

Q₃₂

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#2247	2
C₂×#1713	2
#13203	2
#9825	2
#7136	2

subgroup

index

Subgroups — #144128

Conjugacy classes of subgroups

Complements

Maximal subgroups