Group #153380 — subgroups

Subgroups — #153380

Order 512.

195 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 84 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×#1424	C₂	1	2	yes	maximal
2	#13165	C₂	1	1	yes	maximal, characteristic
3	#12905	C₂	1	1	yes	maximal, characteristic
4	#7471	C₂	1	2	yes	maximal
5	#7465	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₃₂	C₂²	1	2	yes	—
7	C₂×#366	—	2	1	no	—
8	C₂×#366	C₂²	1	1	yes	characteristic
9	C₂×#366	C₂²	1	2	yes	—
10	C₂×#366	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1942	—	2	1	no	—
13	#1845	—	2	1	no	—
14	#1424	—	2	4	no	—
15	#1380	—	2	1	no	—
16	C₂²×C₁₆	D₄	1	1	yes	characteristic
17	C₂²×C₁₆	C₂³	1	1	yes	characteristic
18	C₂×#107	—	2	1	no	—
19	C₂×#107	—	2	2	no	—
20	C₂×#107	—	2	1	no	—
21	C₂×#107	—	4	1	no	—
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	#389	—	4	1	no	—
24	#384	—	4	1	no	—
25	C₂×C₃₂	—	2	2	no	—
26	C₂×C₃₂	—	2	2	no	—
27	C₂×C₃₂	Q₈	1	4	yes	—
28	#366	—	2	1	no	—
29	#366	—	2	2	no	—
30	#366	—	2	2	no	—
31	#366	—	2	2	no	—
32	#366	—	4	2	no	—
33	#366	D₄	1	2	yes	—
34	C₄×C₁₆	—	2	2	no	—
35	#340	—	4	1	no	—
36	C₂²×C₈	D₈	1	1	yes	characteristic
37	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄	—	4	1	no	—
39	C₂×C₄⋊C₄	—	4	1	no	—
40	C₂×C₄⋊C₄	—	4	2	no	—
41	C₂×C₄⋊C₄	—	8	1	no	—
42	C₂×C₄⋊C₄	—	8	1	no	—
43	C₂×C₄⋊C₄	—	8	1	no	—
44	C₂×C₄²	D₈	1	1	yes	characteristic
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	—	2	1	no	—
47	C₂×C₁₆	Q₁₆	1	2	yes	—
48	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
49	C₂×C₁₆	C₂×Q₈	1	2	yes	—
50	C₂×C₁₆	#41	1	2	yes	—
51	C₂×C₁₆	#41	1	1	yes	characteristic
52	#107	—	2	2	no	—
53	#107	—	4	1	no	—
54	#107	—	4	2	no	—
55	#107	—	4	1	no	—
56	#107	—	4	2	no	—
57	#107	—	4	1	no	—
58	#107	—	8	2	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	—	2	1	no	—
61	#95	—	8	1	no	—
62	C₃₂	—	2	4	no	—
63	C₂²×C₄	—	8	1	no	—
64	C₂²×C₄	—	8	1	no	—
65	C₂²×C₄	—	8	2	no	—
66	C₂²×C₄	D₁₆	1	1	yes	characteristic
67	C₂²×C₄	D₁₆	1	1	yes	characteristic
68	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	Q₃₂	1	2	yes	—
72	C₂×C₈	#121	1	1	yes	characteristic
73	C₂×C₈	#123	1	1	yes	characteristic
74	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×C₈	#137	1	1	yes	characteristic
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	8	1	no	—
80	C₄⋊C₄	—	8	2	no	—
81	C₄⋊C₄	—	8	1	no	—
82	C₄⋊C₄	—	8	2	no	—
83	C₄⋊C₄	—	8	1	no	—
84	C₄⋊C₄	—	16	2	no	—
85	C₄⋊C₄	—	16	2	no	—
86	C₄⋊C₄	—	16	2	no	—
87	C₄²	—	2	1	no	—
88	C₄²	—	2	1	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	C₂²⋊Q₈	1	2	yes	—
91	C₁₆	#125	1	2	yes	—
92	C₂³	C₂×D₁₆	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	16	1	no	—
99	C₂×C₄	—	16	1	no	—
100	C₂×C₄	—	16	2	no	—
101	C₂×C₄	—	16	1	no	—
102	C₂×C₄	—	16	2	no	—
103	C₂×C₄	—	16	1	no	—
104	C₂×C₄	—	16	2	no	—
105	C₂×C₄	—	16	1	no	—
106	C₂×C₄	SD₆₄	1	2	yes	—
107	C₂×C₄	SD₆₄	1	2	yes	—
108	C₂×C₄	#459	1	1	yes	characteristic
109	C₂×C₄	#480	1	1	yes	characteristic
110	C₂×C₄	#500	1	1	yes	characteristic
111	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
112	C₂×C₄	#509	1	1	yes	characteristic
113	C₂×C₄	#510	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	#398	1	1	yes	characteristic
116	C₈	#462	1	1	yes	characteristic
117	C₈	#484	1	1	yes	characteristic
118	C₈	#501	1	1	yes	characteristic
119	C₂²	#2206	1	1	yes	characteristic
120	C₂²	#2232	1	1	yes	characteristic
121	C₂²	#2252	1	1	yes	characteristic
122	C₂²	C₂×SD₆₄	1	1	yes	characteristic
123	C₂²	C₂×SD₆₄	1	1	yes	characteristic
124	C₂²	#2263	1	1	yes	characteristic
125	C₂²	#2264	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	2	no	—
128	C₄	—	16	2	no	—
129	C₄	—	32	2	no	—
130	C₄	—	32	1	no	—
131	C₄	#2083	1	1	yes	characteristic
132	C₄	#2209	1	1	yes	characteristic
133	C₄	#2236	1	1	yes	characteristic
134	C₄	#2255	1	1	yes	characteristic
135	C₂	#13505	1	1	yes	characteristic
136	C₂	#13686	1	1	yes	characteristic
137	C₂	#13687	1	1	yes	characteristic
138	C₂	#13712	1	1	yes	characteristic
139	C₂	#13714	1	1	yes	characteristic
140	C₂	#13733	1	1	yes	characteristic
141	C₂	#13734	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1424

C₂

yes

maximal

#13165

C₂

yes

maximal, characteristic

#12905

C₂

yes

maximal, characteristic

#7471

C₂

yes

maximal

#7465

C₂

yes

maximal, characteristic

C₂²×C₃₂

C₂²

yes

—

C₂×#366

—

C₂×#366

C₂²

yes

characteristic

C₂×#366

C₂²

yes

—

C₂×#366

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#389

—

#384

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

Q₈

yes

—

—

—

—

—

—

D₄

yes

—

C₄×C₁₆

—

#340

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

C₂²⋊Q₈

yes

—

C₁₆

#125

yes

—

C₂³

C₂×D₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

SD₆₄

yes

—

107

C₂×C₄

SD₆₄

yes

—

108

C₂×C₄

#459

yes

characteristic

109

C₂×C₄

#480

yes

characteristic

110

C₂×C₄

#500

yes

characteristic

111

C₂×C₄

C₂×Q₃₂

yes

characteristic

112

C₂×C₄

#509

yes

characteristic

113

C₂×C₄

#510

yes

characteristic

114

C₈

—

115

C₈

#398

yes

characteristic

116

C₈

#462

yes

characteristic

117

C₈

#484

yes

characteristic

118

C₈

#501

yes

characteristic

119

C₂²

#2206

yes

characteristic

120

C₂²

#2232

yes

characteristic

121

C₂²

#2252

yes

characteristic

122

C₂²

C₂×SD₆₄

yes

characteristic

123

C₂²

C₂×SD₆₄

yes

characteristic

124

C₂²

#2263

yes

characteristic

125

C₂²

#2264

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#2083

yes

characteristic

132

C₄

#2209

yes

characteristic

133

C₄

#2236

yes

characteristic

134

C₄

#2255

yes

characteristic

135

C₂

#13505

yes

characteristic

136

C₂

#13686

yes

characteristic

137

C₂

#13687

yes

characteristic

138

C₂

#13712

yes

characteristic

139

C₂

#13714

yes

characteristic

140

C₂

#13733

yes

characteristic

141

C₂

#13734

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₁₆

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#1424	2
#13165	2
#12905	2
#7471	2
#7465	2

subgroup

index

Subgroups — #153380

Conjugacy classes of subgroups

Maximal subgroups