Group #93068 — subgroups

Subgroups — #93068

Order 512.

155 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 155 conjugacy classes of subgroups — too many to draw. Only its 70 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	#12012	C₂	1	1	yes	maximal, characteristic
3	#8823	C₂	1	1	yes	maximal, characteristic
4	#2162	—	2	1	no	—
5	#2162	—	2	1	no	—
6	#1848	C₄	1	2	yes	—
7	#1567	—	2	1	no	—
8	#1567	—	2	1	no	—
9	C₂×C₈²	C₂²	1	1	yes	characteristic
10	#1370	—	2	1	no	—
11	#1370	C₄	1	2	yes	—
12	#1367	—	2	1	no	—
13	#1367	C₄	1	2	yes	—
14	C₂×#108	—	4	2	no	—
15	#423	—	2	2	no	—
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×C₈	D₄	1	1	yes	characteristic
19	C₄⋊C₁₆	—	2	2	no	—
20	C₄⋊C₁₆	—	2	2	no	—
21	C₂²⋊C₁₆	—	4	1	no	—
22	C₂²⋊C₁₆	—	4	1	no	—
23	C₄×C₁₆	—	4	1	no	—
24	C₄×C₁₆	—	4	1	no	—
25	#335	—	4	1	no	—
26	#335	—	4	1	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	SD₁₆	1	1	yes	characteristic
36	C₂²×C₈	SD₁₆	1	1	yes	characteristic
37	C₂²×C₈	Q₁₆	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	#108	—	8	4	no	—
43	C₄⋊C₈	—	4	2	no	—
44	C₄⋊C₈	—	4	2	no	—
45	C₄×C₈	—	2	1	no	—
46	C₄×C₈	C₄²	1	1	yes	characteristic
47	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₈	D₈	1	1	yes	characteristic
53	C₄×C₈	D₈	1	1	yes	characteristic
54	C₄×C₈	SD₁₆	1	1	yes	characteristic
55	C₄×C₈	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄	#100	1	1	yes	characteristic
57	C₂²×C₄	#103	1	1	yes	characteristic
58	C₂²×C₄	#103	1	1	yes	characteristic
59	M₁₆	—	8	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	—	4	1	no	—
68	C₂×C₈	—	8	2	no	—
69	C₂×C₈	#102	1	1	yes	characteristic
70	C₂×C₈	#102	1	1	yes	characteristic
71	C₂×C₈	#103	1	1	yes	characteristic
72	C₂×C₈	#103	1	1	yes	characteristic
73	C₂×C₈	#106	1	1	yes	characteristic
74	C₂×C₈	#106	1	1	yes	characteristic
75	C₂×C₈	#107	1	1	yes	characteristic
76	C₂×C₈	#107	1	1	yes	characteristic
77	C₄²	#95	1	1	yes	characteristic
78	C₄²	#101	1	1	yes	characteristic
79	C₄²	#102	1	1	yes	characteristic
80	C₄²	#102	1	1	yes	characteristic
81	C₁₆	—	8	2	no	—
82	C₁₆	—	8	2	no	—
83	C₂³	#333	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	#331	1	1	yes	characteristic
91	C₂×C₄	#332	1	1	yes	characteristic
92	C₂×C₄	#332	1	1	yes	characteristic
93	C₂×C₄	#340	1	1	yes	characteristic
94	C₂×C₄	#340	1	1	yes	characteristic
95	C₂×C₄	#343	1	1	yes	characteristic
96	C₈	—	2	2	no	—
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	2	1	no	—
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	16	2	no	—
105	C₈	#359	1	2	yes	—
106	C₈	#368	1	2	yes	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	#1296	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	#1357	1	1	yes	characteristic
115	C₄	#1358	1	1	yes	characteristic
116	C₄	#1381	1	1	yes	characteristic
117	C₄	#1382	1	1	yes	characteristic
118	C₄	#1388	1	1	yes	characteristic
119	C₄	#1389	1	1	yes	characteristic
120	C₂	—	4	1	no	—
121	C₂	#7269	1	1	yes	characteristic
122	C₂	#7276	1	1	yes	characteristic
123	C₂	#7277	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12012

C₂

yes

maximal, characteristic

#12012

C₂

yes

maximal, characteristic

#8823

C₂

yes

maximal, characteristic

#2162

—

#2162

—

#1848

C₄

yes

—

#1567

—

#1567

—

C₂×C₈²

C₂²

yes

characteristic

#1370

—

#1370

C₄

yes

—

#1367

—

#1367

C₄

yes

—

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

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

Q₁₆

yes

characteristic

C₂²×C₈

Q₁₆

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#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

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

C₂×C₈

—

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

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₂³

#333

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

#359

yes

—

106

C₈

#368

yes

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

#1296

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#1357

yes

characteristic

115

C₄

#1358

yes

characteristic

116

C₄

#1381

yes

characteristic

117

C₄

#1382

yes

characteristic

118

C₄

#1388

yes

characteristic

119

C₄

#1389

yes

characteristic

120

C₂

—

121

C₂

#7269

yes

characteristic

122

C₂

#7276

yes

characteristic

123

C₂

#7277

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1296
prime radical	C₂³	#333
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
#8823	2

subgroup

index

#12012

#8823

Subgroups — #93068

Conjugacy classes of subgroups

Maximal subgroups