Group #153312 — subgroups

Subgroups — #153312

Order 512.

254 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 66 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	#33981	C₂	1	1	yes	maximal, characteristic
2	#13628	C₂	1	1	yes	maximal, characteristic, complemented
3	#13615	C₂	1	2	yes	maximal, complemented
4	#13243	C₂	1	2	yes	maximal, complemented
5	#7475	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#508	C₂²	1	2	yes	—
7	#2264	—	2	2	no	—
8	#2263	—	2	2	no	—
9	#2262	—	2	2	no	—
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃₂	C₂²	1	1	yes	characteristic
12	#2158	C₂²	1	2	yes	—
13	#2145	—	2	2	no	—
14	#2108	C₂²	1	1	yes	characteristic
15	#1976	—	2	2	no	—
16	C₄⋊C₃₂	—	2	2	no	—
17	#1416	—	2	2	no	—
18	#1415	—	2	2	no	—
19	C₂×#135	—	2	2	no	—
20	#508	—	2	2	no	—
21	#508	—	2	2	no	—
22	#508	—	2	2	no	—
23	#508	—	2	2	no	—
24	C₂×Q₃₂	D₄	1	2	yes	—
25	C₂×SD₃₂	—	2	2	no	—
26	C₂×D₁₆	D₄	1	2	yes	—
27	C₂×M₃₂	D₄	1	1	yes	characteristic
28	C₂²×C₁₆	C₂³	1	1	yes	characteristic
29	#427	—	2	2	no	—
30	#421	—	4	2	no	—
31	#410	—	4	2	no	—
32	#405	D₄	1	1	yes	characteristic
33	Q₆₄	—	2	2	no	—
34	SD₆₄	—	2	4	no	—
35	D₃₂	—	2	2	no	—
36	M₆₄	—	2	2	no	—
37	C₂×C₃₂	—	2	2	no	—
38	C₂×C₃₂	—	2	1	no	—
39	C₂×C₃₂	D₄	1	2	yes	—
40	#367	—	2	2	no	—
41	#366	—	2	2	no	—
42	#360	—	4	4	no	—
43	#346	—	2	1	no	—
44	C₄×C₁₆	—	2	1	no	—
45	C₂×#41	—	4	2	no	—
46	#135	—	4	2	no	—
47	#135	—	4	2	no	—
48	#135	—	4	2	no	—
49	#135	—	4	2	no	—
50	C₂×Q₁₆	—	2	2	no	—
51	C₂×SD₁₆	—	4	2	no	—
52	C₂×D₈	—	2	2	no	—
53	C₂×M₁₆	D₈	1	1	yes	characteristic
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	#117	—	4	2	no	—
56	#117	D₈	1	1	yes	characteristic
57	Q₃₂	—	2	2	no	—
58	Q₃₂	—	2	2	no	—
59	SD₃₂	—	4	2	no	—
60	SD₃₂	—	4	2	no	—
61	D₁₆	—	2	2	no	—
62	D₁₆	—	2	2	no	—
63	M₃₂	—	2	2	no	—
64	C₂×C₁₆	—	2	1	no	—
65	C₂×C₁₆	C₂×D₄	1	2	yes	—
66	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
67	C₂×C₁₆	#41	1	1	yes	characteristic
68	C₂×C₁₆	#41	1	2	yes	—
69	#107	—	4	2	no	—
70	#106	—	4	2	no	—
71	#104	—	8	4	no	—
72	#99	—	8	4	no	—
73	#97	—	2	1	no	—
74	C₄×C₈	—	2	1	no	—
75	C₃₂	—	2	2	no	—
76	C₃₂	—	4	1	no	—
77	#41	—	8	2	no	—
78	#41	—	8	2	no	—
79	#41	—	8	2	no	—
80	#41	—	8	2	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×D₄	—	4	2	no	—
83	C₂×D₄	—	8	2	no	—
84	C₂²×C₄	—	8	2	no	—
85	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
86	Q₁₆	—	4	2	no	—
87	Q₁₆	—	4	2	no	—
88	SD₁₆	—	8	2	no	—
89	SD₁₆	—	8	2	no	—
90	D₈	—	4	2	no	—
91	D₈	—	4	2	no	—
92	M₁₆	—	2	2	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	#121	1	2	yes	—
95	C₂×C₈	#121	1	1	yes	characteristic
96	C₂×C₈	#135	1	1	yes	characteristic
97	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	8	2	no	—
100	C₂²⋊C₄	—	8	2	no	—
101	C₂²⋊C₄	D₁₆	1	2	yes	—
102	C₄²	—	2	1	no	—
103	C₄²	—	8	2	no	—
104	C₁₆	—	4	1	no	—
105	C₁₆	#120	1	1	yes	characteristic
106	C₁₆	#123	1	2	yes	—
107	C₁₆	#124	1	1	yes	characteristic
108	C₂³	—	8	2	no	—
109	C₂³	C₂×D₁₆	1	1	yes	characteristic
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	8	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	16	2	no	—
115	D₄	—	16	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	—	16	2	no	—
119	C₂×C₄	—	16	2	no	—
120	C₂×C₄	—	16	2	no	—
121	C₂×C₄	#459	1	2	yes	—
122	C₂×C₄	#466	1	1	yes	characteristic
123	C₂×C₄	#508	1	1	yes	characteristic
124	C₂×C₄	#509	1	2	yes	—
125	C₈	—	4	1	no	—
126	C₈	#394	1	1	yes	characteristic
127	C₈	#463	1	2	yes	—
128	C₈	#469	1	1	yes	characteristic
129	C₂²	—	4	1	no	—
130	C₂²	—	16	2	no	—
131	C₂²	—	32	2	no	—
132	C₂²	#2206	1	2	yes	—
133	C₂²	#2215	1	1	yes	characteristic
134	C₄	—	4	2	no	—
135	C₄	—	16	2	no	—
136	C₄	—	32	2	no	—
137	C₄	#2054	1	1	yes	characteristic
138	C₄	#2210	1	2	yes	—
139	C₄	#2218	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	2	no	—
142	C₂	—	32	2	no	—
143	C₂	#13448	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33981

C₂

yes

maximal, characteristic

#13628

C₂

yes

maximal, characteristic, complemented

#13615

C₂

yes

maximal, complemented

#13243

C₂

yes

maximal, complemented

#7475

C₂

yes

maximal, characteristic, complemented

C₂×#508

C₂²

yes

—

#2264

—

#2263

—

#2262

—

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

#2158

C₂²

yes

—

#2145

—

#2108

C₂²

yes

characteristic

#1976

—

C₄⋊C₃₂

—

—

—

—

—

—

—

—

C₂×Q₃₂

D₄

yes

—

C₂×SD₃₂

—

C₂×D₁₆

D₄

yes

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

D₄

yes

characteristic

Q₆₄

—

SD₆₄

—

D₃₂

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

C₄×C₁₆

—

C₂×#41

—

—

—

—

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

#117

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

#121

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

D₁₆

yes

—

102

C₄²

—

103

C₄²

—

104

C₁₆

—

105

C₁₆

#120

yes

characteristic

106

C₁₆

#123

yes

—

107

C₁₆

#124

yes

characteristic

108

C₂³

—

109

C₂³

C₂×D₁₆

yes

characteristic

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#459

yes

—

122

C₂×C₄

#466

yes

characteristic

123

C₂×C₄

#508

yes

characteristic

124

C₂×C₄

#509

yes

—

125

C₈

—

126

C₈

#394

yes

characteristic

127

C₈

#463

yes

—

128

C₈

#469

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#2206

yes

—

133

C₂²

#2215

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#2054

yes

characteristic

138

C₄

#2210

yes

—

139

C₄

#2218

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

—

143

C₂

#13448

yes

characteristic

	subgroup	G/H
center	C₄	#2218
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13448
prime radical	#33981	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#2218

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

C₂

C₂

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13628	C₂	semidirect
3. #13615	C₂	semidirect
4. #13243	C₂	semidirect
5. #7475	C₂	semidirect

H ≅ G/N

type

2. #13628

C₂

semidirect

3. #13615

C₂

semidirect

4. #13243

C₂

semidirect

5. #7475

C₂

semidirect

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

subgroup	index
#33981	2
#13628	2
#13615	2
#13243	2
#7475	2

subgroup

index

Subgroups — #153312

Conjugacy classes of subgroups

Complements

Maximal subgroups