Group #94057 — subgroups

Subgroups — #94057

Order 512.

154 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 74 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×#1398	C₂	1	1	yes	maximal, characteristic
2	#12671	C₂	1	1	yes	maximal, characteristic
3	C₂×#1368	C₂	1	1	yes	maximal, characteristic
4	C₂×M₆₄	—	2	1	no	—
5	C₂×#368	C₄	1	1	yes	characteristic
6	#2158	C₄	1	1	yes	characteristic
7	C₂×#346	C₂²	1	1	yes	characteristic
8	#1772	—	2	1	no	—
9	C₂×#334	—	2	1	no	—
10	#1398	—	2	1	no	—
11	#1398	C₄	1	2	yes	complemented
12	#1382	—	2	1	no	—
13	#1368	—	2	1	no	—
14	#1368	C₄	1	2	yes	complemented
15	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₁₆	D₄	1	1	yes	characteristic
17	C₂×#108	—	2	1	no	—
18	#427	—	2	1	no	—
19	C₂×C₄⋊C₈	—	4	1	no	—
20	#405	—	4	1	no	—
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	M₆₄	—	2	2	no	—
23	M₆₄	—	2	2	no	—
24	C₂×C₃₂	—	2	2	no	—
25	#368	—	2	1	no	—
26	#368	C₈	1	2	yes	complemented
27	#367	—	2	1	no	—
28	#366	C₈	1	2	yes	complemented
29	#346	C₂×C₄	1	1	yes	characteristic
30	#346	C₂×C₄	1	1	yes	characteristic
31	#346	D₄	1	1	yes	characteristic
32	#346	Q₈	1	1	yes	characteristic
33	#339	—	4	1	no	—
34	#334	—	2	2	no	—
35	#334	—	4	1	no	—
36	C₂×M₁₆	—	4	1	no	—
37	C₂²×C₈	—	8	1	no	—
38	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₈	SD₁₆	1	1	yes	characteristic
40	#117	—	4	1	no	—
41	C₂×C₄²	D₈	1	1	yes	characteristic
42	C₂×C₁₆	—	2	1	no	—
43	C₂×C₁₆	—	2	1	no	—
44	C₂×C₁₆	C₄²	1	1	yes	characteristic
45	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
49	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
50	C₂×C₁₆	M₁₆	1	1	yes	characteristic
51	C₂×C₁₆	M₁₆	1	1	yes	characteristic
52	#108	—	2	2	no	—
53	#108	—	4	1	no	—
54	#107	—	2	2	no	—
55	#106	—	4	1	no	—
56	C₄⋊C₈	—	4	2	no	—
57	C₄⋊C₈	—	8	1	no	—
58	#97	—	8	1	no	—
59	C₄×C₈	—	8	1	no	—
60	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₄×C₈	SD₁₆	1	1	yes	characteristic
63	C₄×C₈	Q₁₆	1	1	yes	characteristic
64	C₃₂	—	2	4	no	—
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	#102	1	1	yes	characteristic
67	M₁₆	—	4	2	no	—
68	M₁₆	—	8	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	8	1	no	—
72	C₂×C₈	—	8	2	no	—
73	C₂×C₈	—	16	2	no	—
74	C₂×C₈	#95	1	1	yes	characteristic
75	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₈	#103	1	1	yes	characteristic
78	C₂×C₈	#103	1	1	yes	characteristic
79	C₂×C₈	#104	1	1	yes	characteristic
80	C₂×C₈	#104	1	1	yes	characteristic
81	C₂×C₈	#106	1	1	yes	characteristic
82	C₄⋊C₄	—	4	2	no	—
83	C₂²⋊C₄	—	8	1	no	—
84	C₄²	—	2	1	no	—
85	C₄²	—	8	1	no	—
86	C₄²	#102	1	1	yes	characteristic
87	C₄²	#107	1	1	yes	characteristic
88	C₁₆	—	2	1	no	—
89	C₁₆	—	2	1	no	—
90	C₁₆	C₄×C₈	1	1	yes	characteristic
91	C₁₆	#97	1	1	yes	characteristic
92	C₁₆	C₄⋊C₈	1	1	yes	characteristic
93	C₁₆	C₄⋊C₈	1	1	yes	characteristic
94	C₂³	#360	1	1	yes	characteristic
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
100	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
101	C₂×C₄	#340	1	1	yes	characteristic
102	C₂×C₄	#360	1	1	yes	characteristic
103	C₂×C₄	#361	1	1	yes	characteristic
104	C₂×C₄	#362	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	2	1	no	—
107	C₈	—	8	2	no	—
108	C₈	—	16	2	no	—
109	C₈	Q₈⋊C₈	1	1	yes	characteristic
110	C₈	Q₈⋊C₈	1	1	yes	characteristic
111	C₈	#336	1	1	yes	characteristic
112	C₈	#339	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	#1333	1	1	yes	characteristic
115	C₂²	#1334	1	1	yes	characteristic
116	C₂²	#1386	1	1	yes	characteristic
117	C₄	—	4	1	no	—
118	C₄	—	4	1	no	—
119	C₄	—	16	2	no	—
120	C₄	#1279	1	1	yes	characteristic
121	C₄	#1333	1	1	yes	characteristic
122	C₄	#1334	1	1	yes	characteristic
123	C₄	#1387	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#7178	1	1	yes	characteristic
127	C₂	#7389	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1398

C₂

yes

maximal, characteristic

#12671

C₂

yes

maximal, characteristic

C₂×#1368

C₂

yes

maximal, characteristic

C₂×M₆₄

—

C₂×#368

C₄

yes

characteristic

#2158

C₄

yes

characteristic

C₂×#346

C₂²

yes

—

—

—

C₄

yes

—

—

C₄

yes

complemented

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂×#108

—

#427

—

C₂×C₄⋊C₈

—

#405

—

C₂×C₄×C₈

D₄

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#368

—

#368

C₈

yes

complemented

#367

—

#366

C₈

yes

complemented

#346

C₂×C₄

yes

characteristic

#346

C₂×C₄

yes

characteristic

#346

D₄

yes

characteristic

#346

Q₈

yes

—

—

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

#117

—

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

C₂×C₈

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

characteristic

C₂×C₁₆

M₁₆

yes

characteristic

C₂×C₁₆

M₁₆

yes

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

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

—

C₂²×C₄

#102

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#95

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

C₄×C₈

yes

characteristic

C₁₆

#97

yes

characteristic

C₁₆

C₄⋊C₈

yes

characteristic

C₁₆

C₄⋊C₈

yes

characteristic

C₂³

#360

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄⋊C₈

yes

characteristic

100

C₂×C₄

D₄⋊C₈

yes

characteristic

101

C₂×C₄

#340

yes

characteristic

102

C₂×C₄

#360

yes

characteristic

103

C₂×C₄

#361

yes

characteristic

104

C₂×C₄

#362

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

Q₈⋊C₈

yes

characteristic

110

C₈

Q₈⋊C₈

yes

characteristic

111

C₈

#336

yes

characteristic

112

C₈

#339

yes

characteristic

113

C₂²

—

114

C₂²

#1333

yes

characteristic

115

C₂²

#1334

yes

characteristic

116

C₂²

#1386

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#1279

yes

characteristic

121

C₄

#1333

yes

characteristic

122

C₄

#1334

yes

characteristic

123

C₄

#1387

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

#7178

yes

characteristic

127

C₂

#7389

yes

—

	subgroup	G/H
center, socle	C₂²	#1333
commutator	C₁₆	C₄×C₈
Frattini, cosocle	C₂×#346	C₂²
prime radical	C₂³	#360
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1333

commutator

C₁₆

C₄×C₈

C₂²

C₂³

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G

LECS

G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
11. #1398	C₄	semidirect
14. #1368	C₄	semidirect
26. #368	C₈	semidirect
28. #366	C₈	semidirect

H ≅ G/N

type

11. #1398

C₄

semidirect

14. #1368

C₄

semidirect

26. #368

C₈

semidirect

28. #366

C₈

semidirect

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

subgroup	index
C₂×#1398	2
#12671	2
C₂×#1368	2

subgroup

index

C₂×#1398

#12671

C₂×#1368

Subgroups — #94057

Conjugacy classes of subgroups

Complements

Maximal subgroups