Group #94030 — subgroups

Subgroups — #94030

Order 512.

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

normal subgroups only

This group has 132 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	C₂×#1369	C₂	1	1	yes	maximal, characteristic
3	C₂×#1368	C₂	1	1	yes	maximal, characteristic
4	C₂×M₆₄	C₄	1	1	yes	characteristic
5	C₂×M₆₄	C₄	1	1	yes	characteristic
6	C₂×#364	—	2	1	no	—
7	C₂×#346	C₂²	1	1	yes	characteristic
8	C₂×#334	—	2	1	no	—
9	#1398	—	2	1	no	—
10	#1398	—	2	1	no	—
11	#1369	—	2	1	no	—
12	#1369	C₄	1	2	yes	—
13	#1368	—	2	1	no	—
14	#1368	C₄	1	2	yes	—
15	C₂×M₃₂	—	4	1	no	—
16	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₆	Q₈	1	1	yes	characteristic
18	C₂×C₄⋊C₈	—	4	1	no	—
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	M₆₄	—	2	1	no	—
21	M₆₄	—	2	1	no	—
22	M₆₄	—	2	1	no	—
23	M₆₄	C₈	1	2	yes	complemented
24	C₂×C₃₂	—	2	1	no	—
25	C₂×C₃₂	C₈	1	2	yes	complemented
26	#364	—	2	2	no	—
27	#364	—	4	1	no	—
28	#346	C₂×C₄	1	1	yes	characteristic
29	#346	C₂×C₄	1	1	yes	characteristic
30	#346	D₄	1	1	yes	characteristic
31	#346	D₄	1	1	yes	characteristic
32	#334	—	2	2	no	—
33	#334	—	4	1	no	—
34	C₂²×C₈	—	8	1	no	—
35	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₈	SD₁₆	1	1	yes	characteristic
37	C₂×C₄²	SD₁₆	1	1	yes	characteristic
38	M₃₂	—	4	2	no	—
39	M₃₂	—	8	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	2	1	no	—
42	C₂×C₁₆	—	8	1	no	—
43	C₂×C₁₆	C₄²	1	1	yes	characteristic
44	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
48	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
49	C₂×C₁₆	M₁₆	1	1	yes	characteristic
50	C₂×C₁₆	M₁₆	1	1	yes	characteristic
51	C₄⋊C₈	—	4	2	no	—
52	C₄⋊C₈	—	8	1	no	—
53	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₈	D₈	1	1	yes	characteristic
56	C₄×C₈	Q₁₆	1	1	yes	characteristic
57	C₃₂	—	2	2	no	—
58	C₃₂	—	2	2	no	—
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	#106	1	1	yes	characteristic
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	8	2	no	—
64	C₂×C₈	—	16	2	no	—
65	C₂×C₈	#95	1	1	yes	characteristic
66	C₂×C₈	#102	1	1	yes	characteristic
67	C₂×C₈	#103	1	1	yes	characteristic
68	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
70	C₂×C₈	#107	1	1	yes	characteristic
71	C₂×C₈	#108	1	1	yes	characteristic
72	C₂×C₈	#108	1	1	yes	characteristic
73	C₄²	—	2	1	no	—
74	C₄²	#102	1	1	yes	characteristic
75	C₄²	#103	1	1	yes	characteristic
76	C₁₆	—	2	1	no	—
77	C₁₆	—	2	1	no	—
78	C₁₆	—	8	2	no	—
79	C₁₆	C₄×C₈	1	1	yes	characteristic
80	C₁₆	#97	1	1	yes	characteristic
81	C₁₆	C₂²⋊C₈	1	1	yes	characteristic
82	C₁₆	C₂²⋊C₈	1	1	yes	characteristic
83	C₂³	#365	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	#334	1	1	yes	characteristic
88	C₂×C₄	#334	1	1	yes	characteristic
89	C₂×C₄	#340	1	1	yes	characteristic
90	C₂×C₄	#365	1	1	yes	characteristic
91	C₂×C₄	#365	1	1	yes	characteristic
92	C₂×C₄	#365	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	2	1	no	—
95	C₈	—	16	2	no	—
96	C₈	#335	1	1	yes	characteristic
97	C₈	#335	1	1	yes	characteristic
98	C₈	#336	1	1	yes	characteristic
99	C₈	#341	1	1	yes	characteristic
100	C₂²	—	2	2	no	—
101	C₂²	#1368	1	1	yes	characteristic
102	C₂²	#1368	1	1	yes	characteristic
103	C₂²	#1383	1	1	yes	characteristic
104	C₄	—	4	1	no	—
105	C₄	—	4	1	no	—
106	C₄	#1277	1	1	yes	characteristic
107	C₄	#1368	1	1	yes	characteristic
108	C₄	#1368	1	1	yes	characteristic
109	C₄	#1383	1	1	yes	characteristic
110	C₂	—	2	1	no	—
111	C₂	—	2	1	no	—
112	C₂	#7156	1	1	yes	characteristic
113	C₂	#7457	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1398

C₂

yes

maximal, characteristic

C₂×#1369

C₂

yes

maximal, characteristic

C₂×#1368

C₂

yes

maximal, characteristic

C₂×M₆₄

C₄

yes

characteristic

C₂×M₆₄

C₄

yes

characteristic

C₂×#364

—

C₂×#346

C₂²

yes

—

—

—

—

C₄

yes

—

#1368

—

#1368

C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

Q₈

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

M₆₄

—

M₆₄

—

M₆₄

—

M₆₄

C₈

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

C₈

yes

—

—

C₂×C₄

yes

characteristic

#346

C₂×C₄

yes

characteristic

#346

D₄

yes

characteristic

#346

D₄

yes

characteristic

#334

—

#334

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

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

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₃₂

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

#106

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₄×C₈

yes

characteristic

C₁₆

#97

yes

characteristic

C₁₆

C₂²⋊C₈

yes

characteristic

C₁₆

C₂²⋊C₈

yes

characteristic

C₂³

#365

yes

characteristic

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

yes

C₈

yes

C₈

yes

C₈

yes

100

C₂²

—

101

C₂²

#1368

yes

characteristic

102

C₂²

#1368

yes

characteristic

103

C₂²

#1383

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

#1277

yes

characteristic

107

C₄

#1368

yes

characteristic

108

C₄

#1368

yes

characteristic

109

C₄

#1383

yes

characteristic

110

C₂

—

111

C₂

—

112

C₂

#7156

yes

characteristic

113

C₂

#7457

yes

—

	subgroup	G/H
center, socle	C₂²	#1368
commutator	C₁₆	C₄×C₈
Frattini, cosocle	C₂×#346	C₂²
prime radical	C₂³	#365
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

#1368

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
23. M₆₄	C₈	semidirect
25. C₂×C₃₂	C₈	semidirect

H ≅ G/N

type

23. M₆₄

C₈

semidirect

25. C₂×C₃₂

C₈

semidirect

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

subgroup	index
C₂×#1398	2
C₂×#1369	2
C₂×#1368	2

subgroup

index

C₂×#1398

C₂×#1369

C₂×#1368

Subgroups — #94030

Conjugacy classes of subgroups

Complements

Maximal subgroups