Group #94358 — subgroups

Subgroups — #94358

Order 512.

150 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 80 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	#13627	C₂	1	1	yes	maximal, characteristic
2	#12909	C₂	1	1	yes	maximal, characteristic
3	C₂×#1371	C₂	1	1	yes	maximal, characteristic
4	C₂×M₆₄	—	2	1	no	—
5	C₂×#368	—	2	1	no	—
6	C₂×#368	C₄	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	#1848	—	2	1	no	—
9	C₂×#335	—	2	1	no	—
10	C₄⋊C₃₂	—	2	1	no	—
11	C₄⋊C₃₂	C₄	1	2	yes	—
12	#1371	—	2	1	no	—
13	#1371	C₄	1	2	yes	—
14	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₁₆	D₄	1	1	yes	characteristic
16	C₂×#108	—	2	2	no	—
17	C₂×#108	—	4	1	no	—
18	#423	—	4	1	no	—
19	C₂×C₄⋊C₈	—	4	1	no	—
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	M₆₄	—	4	1	no	—
22	M₆₄	—	4	1	no	—
23	C₂×C₃₂	—	2	2	no	—
24	#368	—	4	1	no	—
25	#368	—	4	1	no	—
26	#368	—	4	2	no	—
27	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₄×C₁₆	D₄	1	1	yes	characteristic
30	C₄×C₁₆	Q₈	1	1	yes	characteristic
31	#335	—	2	2	no	—
32	#335	—	4	1	no	—
33	C₂×M₁₆	—	4	2	no	—
34	C₂²×C₈	—	8	1	no	—
35	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₈	SD₁₆	1	1	yes	characteristic
37	C₂×C₄²	D₈	1	1	yes	characteristic
38	C₂×C₁₆	—	2	1	no	—
39	C₂×C₁₆	—	2	1	no	—
40	C₂×C₁₆	C₄²	1	1	yes	characteristic
41	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₁₆	D₈	1	1	yes	characteristic
45	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
46	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
47	C₂×C₁₆	Q₁₆	1	1	yes	characteristic
48	#108	—	4	2	no	—
49	#108	—	4	2	no	—
50	#108	—	8	2	no	—
51	C₄⋊C₈	—	4	2	no	—
52	C₄⋊C₈	—	8	1	no	—
53	C₄⋊C₈	—	8	1	no	—
54	C₄⋊C₈	—	8	1	no	—
55	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₈	SD₁₆	1	1	yes	characteristic
58	C₄×C₈	Q₁₆	1	1	yes	characteristic
59	C₃₂	—	4	2	no	—
60	C₂²×C₄	#102	1	1	yes	characteristic
61	C₂²×C₄	SD₃₂	1	1	yes	characteristic
62	C₂²×C₄	Q₃₂	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	—	8	2	no	—
68	C₂×C₈	—	8	2	no	—
69	C₂×C₈	—	16	2	no	—
70	C₂×C₈	#95	1	1	yes	characteristic
71	C₂×C₈	#100	1	1	yes	characteristic
72	C₂×C₈	#101	1	1	yes	characteristic
73	C₂×C₈	#102	1	1	yes	characteristic
74	C₂×C₈	#103	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	C₄²	#102	1	1	yes	characteristic
79	C₄²	#107	1	1	yes	characteristic
80	C₄²	D₁₆	1	1	yes	characteristic
81	C₄²	SD₃₂	1	1	yes	characteristic
82	C₁₆	—	2	1	no	—
83	C₁₆	—	2	1	no	—
84	C₁₆	#102	1	1	yes	characteristic
85	C₁₆	#103	1	1	yes	characteristic
86	C₁₆	#106	1	1	yes	characteristic
87	C₁₆	#107	1	1	yes	characteristic
88	C₂³	#358	1	1	yes	characteristic
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	#331	1	1	yes	characteristic
93	C₂×C₄	#332	1	1	yes	characteristic
94	C₂×C₄	#340	1	1	yes	characteristic
95	C₂×C₄	#357	1	1	yes	characteristic
96	C₂×C₄	#357	1	1	yes	characteristic
97	C₂×C₄	#358	1	1	yes	characteristic
98	C₂×C₄	#361	1	1	yes	characteristic
99	C₂×C₄	#362	1	1	yes	characteristic
100	C₂×C₄	#366	1	1	yes	characteristic
101	C₂×C₄	#367	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	16	2	no	—
105	C₈	—	16	2	no	—
106	C₈	#332	1	1	yes	characteristic
107	C₈	#333	1	1	yes	characteristic
108	C₈	#340	1	1	yes	characteristic
109	C₈	#343	1	1	yes	characteristic
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	#1362	1	1	yes	characteristic
113	C₂²	#1365	1	1	yes	characteristic
114	C₂²	#1380	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	#1296	1	1	yes	characteristic
119	C₄	#1361	1	1	yes	characteristic
120	C₄	#1364	1	1	yes	characteristic
121	C₄	#1387	1	1	yes	characteristic
122	C₄	#1419	1	1	yes	characteristic
123	C₄	#1420	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#7274	1	1	yes	characteristic
127	C₂	#7476	1	1	yes	characteristic
128	C₂	#7478	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13627

C₂

yes

maximal, characteristic

#12909

C₂

yes

maximal, characteristic

C₂×#1371

C₂

yes

maximal, characteristic

C₂×M₆₄

—

C₂×#368

—

C₂×#368

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

—

C₂×#335

—

C₄⋊C₃₂

—

C₄⋊C₃₂

C₄

yes

—

#1371

—

#1371

C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#368

—

#368

—

#368

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

Q₈

yes

characteristic

#335

—

#335

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

D₈

yes

characteristic

C₂×C₁₆

SD₁₆

yes

characteristic

C₂×C₁₆

SD₁₆

yes

characteristic

C₂×C₁₆

Q₁₆

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₃₂

—

C₂²×C₄

#102

yes

characteristic

C₂²×C₄

SD₃₂

yes

characteristic

C₂²×C₄

Q₃₂

yes

characteristic

M₁₆

—

M₁₆

—

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

C₄²

yes

C₄²

yes

C₄²

D₁₆

yes

characteristic

C₄²

SD₃₂

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

yes

C₁₆

yes

C₁₆

yes

C₁₆

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#366

yes

characteristic

101

C₂×C₄

#367

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

#332

yes

characteristic

107

C₈

#333

yes

characteristic

108

C₈

#340

yes

characteristic

109

C₈

#343

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

#1362

yes

characteristic

113

C₂²

#1365

yes

characteristic

114

C₂²

#1380

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#1296

yes

characteristic

119

C₄

#1361

yes

characteristic

120

C₄

#1364

yes

characteristic

121

C₄

#1387

yes

characteristic

122

C₄

#1419

yes

characteristic

123

C₄

#1420

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

#7274

yes

characteristic

127

C₂

#7476

yes

characteristic

128

C₂

#7478

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
prime radical	C₂³	#358
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1380

commutator

C₂×C₁₆

C₄²

Frattini, cosocle

C₂×C₄×C₁₆

C₂²

prime radical

C₂³

#358

LCS

G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13627	2
#12909	2
C₂×#1371	2

subgroup

index

#13627

#12909

C₂×#1371

Subgroups — #94358

Conjugacy classes of subgroups

Maximal subgroups