Group #93069 — subgroups

Subgroups — #93069

Order 512.

161 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 82 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	#12012	C₂	1	1	yes	maximal, characteristic
2	C₂×#1370	C₂	1	1	yes	maximal, characteristic
3	#8823	C₂	1	1	yes	maximal, characteristic
4	#2162	—	2	1	no	—
5	C₂×C₄⋊C₁₆	—	2	1	no	—
6	#1848	—	2	1	no	—
7	#1567	—	2	1	no	—
8	#1567	C₄	1	2	yes	—
9	C₂×C₈²	C₂²	1	1	yes	characteristic
10	#1370	—	2	1	no	—
11	#1370	—	2	1	no	—
12	#1370	C₄	1	2	yes	—
13	#1367	C₄	1	2	yes	—
14	C₂²×C₁₆	—	4	1	no	—
15	C₂×#108	—	4	1	no	—
16	C₂×#108	—	4	1	no	—
17	#423	—	2	2	no	—
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₄⋊C₁₆	—	2	2	no	—
22	C₄⋊C₁₆	—	2	2	no	—
23	C₄⋊C₁₆	—	4	1	no	—
24	C₂²⋊C₁₆	—	4	1	no	—
25	C₄×C₁₆	—	4	1	no	—
26	#335	—	4	2	no	—
27	#335	—	4	1	no	—
28	#335	—	4	1	no	—
29	C₈²	C₂×C₄	1	1	yes	characteristic
30	C₈²	C₂×C₄	1	1	yes	characteristic
31	C₈²	D₄	1	1	yes	characteristic
32	C₈²	Q₈	1	1	yes	characteristic
33	C₂×M₁₆	—	4	2	no	—
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	D₈	1	1	yes	characteristic
36	C₂²×C₈	SD₁₆	1	1	yes	characteristic
37	C₂²×C₈	SD₁₆	1	1	yes	characteristic
38	C₂²×C₈	Q₁₆	1	1	yes	characteristic
39	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₁₆	—	4	2	no	—
41	C₂×C₁₆	—	4	2	no	—
42	C₂×C₁₆	—	8	1	no	—
43	C₂×C₁₆	—	8	1	no	—
44	#108	—	8	2	no	—
45	#108	—	8	1	no	—
46	#108	—	8	1	no	—
47	C₄⋊C₈	—	4	2	no	—
48	C₄⋊C₈	—	4	2	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	C₄²	1	1	yes	characteristic
51	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₈	D₈	1	1	yes	characteristic
57	C₄×C₈	SD₁₆	1	1	yes	characteristic
58	C₄×C₈	SD₁₆	1	1	yes	characteristic
59	C₄×C₈	Q₁₆	1	1	yes	characteristic
60	C₂²×C₄	#101	1	1	yes	characteristic
61	C₂²×C₄	#102	1	1	yes	characteristic
62	C₂²×C₄	#103	1	1	yes	characteristic
63	M₁₆	—	8	2	no	—
64	M₁₆	—	8	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	—	4	2	no	—
72	C₂×C₈	—	8	2	no	—
73	C₂×C₈	#102	1	1	yes	characteristic
74	C₂×C₈	#102	1	1	yes	characteristic
75	C₂×C₈	#103	1	1	yes	characteristic
76	C₂×C₈	#103	1	1	yes	characteristic
77	C₂×C₈	#106	1	1	yes	characteristic
78	C₂×C₈	#106	1	1	yes	characteristic
79	C₂×C₈	#107	1	1	yes	characteristic
80	C₂×C₈	#107	1	1	yes	characteristic
81	C₂×C₈	D₁₆	1	1	yes	characteristic
82	C₂×C₈	SD₃₂	1	1	yes	characteristic
83	C₂×C₈	SD₃₂	1	1	yes	characteristic
84	C₂×C₈	Q₃₂	1	1	yes	characteristic
85	C₄²	#95	1	1	yes	characteristic
86	C₄²	#100	1	1	yes	characteristic
87	C₄²	#102	1	1	yes	characteristic
88	C₄²	#103	1	1	yes	characteristic
89	C₁₆	—	8	2	no	—
90	C₁₆	—	8	2	no	—
91	C₂³	#332	1	1	yes	characteristic
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	#331	1	1	yes	characteristic
96	C₂×C₄	#332	1	1	yes	characteristic
97	C₂×C₄	#333	1	1	yes	characteristic
98	C₂×C₄	#340	1	1	yes	characteristic
99	C₂×C₄	#340	1	1	yes	characteristic
100	C₂×C₄	#343	1	1	yes	characteristic
101	C₂×C₄	#357	1	1	yes	characteristic
102	C₂×C₄	#358	1	1	yes	characteristic
103	C₂×C₄	#361	1	1	yes	characteristic
104	C₂×C₄	#362	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	2	1	no	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	4	1	no	—
112	C₈	—	16	2	no	—
113	C₈	#357	1	1	yes	characteristic
114	C₈	#358	1	1	yes	characteristic
115	C₈	#366	1	1	yes	characteristic
116	C₈	#367	1	1	yes	characteristic
117	C₈	#368	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#1296	1	1	yes	characteristic
121	C₂²	#1364	1	1	yes	characteristic
122	C₂²	#1365	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	#1361	1	1	yes	characteristic
127	C₄	#1362	1	1	yes	characteristic
128	C₄	#1380	1	1	yes	characteristic
129	C₄	#1381	1	1	yes	characteristic
130	C₄	#1387	1	1	yes	characteristic
131	C₄	#1389	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#7270	1	1	yes	characteristic
135	C₂	#7274	1	1	yes	characteristic
136	C₂	#7276	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12012

C₂

yes

maximal, characteristic

C₂×#1370

C₂

yes

maximal, characteristic

#8823

C₂

yes

maximal, characteristic

#2162

—

C₂×C₄⋊C₁₆

—

#1848

—

#1567

—

#1567

C₄

yes

—

C₂×C₈²

C₂²

yes

—

—

C₄

yes

—

#1367

C₄

yes

—

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#335

—

#335

—

#335

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

Q₁₆

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

D₁₆

yes

characteristic

C₂×C₈

SD₃₂

yes

characteristic

C₂×C₈

SD₃₂

yes

characteristic

C₂×C₈

Q₃₂

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₂³

#332

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#343

yes

characteristic

101

C₂×C₄

#357

yes

characteristic

102

C₂×C₄

#358

yes

characteristic

103

C₂×C₄

#361

yes

characteristic

104

C₂×C₄

#362

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

#357

yes

characteristic

114

C₈

#358

yes

characteristic

115

C₈

#366

yes

characteristic

116

C₈

#367

yes

characteristic

117

C₈

#368

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

#1296

yes

characteristic

121

C₂²

#1364

yes

characteristic

122

C₂²

#1365

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#1361

yes

characteristic

127

C₄

#1362

yes

characteristic

128

C₄

#1380

yes

characteristic

129

C₄

#1381

yes

characteristic

130

C₄

#1387

yes

characteristic

131

C₄

#1389

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#7270

yes

characteristic

135

C₂

#7274

yes

characteristic

136

C₂

#7276

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1296
prime radical	C₂³	#332
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#331

commutator

C₄×C₈

C₄²

Frattini, cosocle

C₂×C₈²

C₂²

C₂²

C₂³

G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#12012	2
C₂×#1370	2
#8823	2

subgroup

index

#12012

C₂×#1370

#8823

Subgroups — #93069

Conjugacy classes of subgroups

Maximal subgroups