Group #99281 — subgroups

Subgroups — #99281

Order 512.

256 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 114 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	#8785	C₂	1	1	yes	maximal, characteristic
2	#8785	C₂	1	1	yes	maximal, characteristic
3	#7841	C₂	1	1	yes	maximal, characteristic
4	#7018	C₂	1	4	yes	maximal
5	#1831	C₂²	1	2	yes	—
6	#1831	C₂²	1	2	yes	—
7	#1744	C₂²	1	2	yes	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#322	C₄	1	1	yes	characteristic
10	C₂×#322	C₄	1	1	yes	characteristic
11	C₂×C₈²	C₄	1	2	yes	—
12	#1277	—	2	2	no	—
13	#1277	—	2	2	no	—
14	C₂×C₄⋊C₈	—	2	2	no	—
15	C₂×C₄⋊C₈	—	2	2	no	—
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	D₄	1	2	yes	—
21	C₂×C₄×C₈	Q₈	1	2	yes	—
22	C₄³	C₂³	1	1	yes	characteristic
23	#336	—	2	2	no	—
24	#336	—	2	2	no	—
25	#336	—	2	4	no	—
26	#336	—	2	4	no	—
27	#322	—	2	2	no	—
28	#322	C₈	1	4	yes	complemented
29	C₈²	—	2	4	no	—
30	C₂²×C₈	—	2	2	no	—
31	C₂²×C₈	—	2	2	no	—
32	C₂²×C₈	—	4	2	no	—
33	C₂²×C₈	—	4	2	no	—
34	C₂²×C₈	C₄⋊C₄	1	2	yes	—
35	C₂²×C₈	C₄⋊C₄	1	2	yes	—
36	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄²	#41	1	2	yes	—
40	C₂×C₄²	#41	1	2	yes	—
41	C₄⋊C₈	—	4	4	no	—
42	C₄⋊C₈	—	4	4	no	—
43	C₄×C₈	—	2	4	no	—
44	C₄×C₈	—	2	2	no	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	—	2	2	no	—
47	C₄×C₈	—	2	4	no	—
48	C₄×C₈	C₂×C₈	1	4	yes	—
49	C₄×C₈	C₂×C₈	1	2	yes	—
50	C₄×C₈	M₁₆	1	4	yes	—
51	C₄×C₈	M₁₆	1	2	yes	—
52	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄	#117	1	2	yes	—
54	C₂²×C₄	#124	1	1	yes	characteristic
55	C₂²×C₄	#124	1	1	yes	characteristic
56	C₂²×C₄	#125	1	2	yes	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	—	8	4	no	—
68	C₂×C₈	—	8	4	no	—
69	C₂×C₈	—	8	2	no	—
70	C₂×C₈	—	8	2	no	—
71	C₂×C₈	C₄⋊C₈	1	4	yes	—
72	C₂×C₈	#108	1	4	yes	—
73	C₄²	—	2	2	no	—
74	C₄²	—	4	2	no	—
75	C₄²	—	4	2	no	—
76	C₄²	C₂²×C₈	1	1	yes	characteristic
77	C₄²	C₂×M₁₆	1	1	yes	characteristic
78	C₄²	C₂×M₁₆	1	2	yes	—
79	C₄²	#135	1	4	yes	—
80	C₂³	#382	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂×C₄	#423	1	1	yes	characteristic
90	C₂×C₄	#427	1	2	yes	—
91	C₂×C₄	C₂×#108	1	2	yes	—
92	C₂×C₄	#431	1	2	yes	—
93	C₂×C₄	#432	1	2	yes	—
94	C₂×C₄	#488	1	2	yes	—
95	C₂×C₄	#488	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	4	no	—
98	C₈	—	4	4	no	—
99	C₈	—	4	4	no	—
100	C₈	—	16	2	no	—
101	C₈	—	16	2	no	—
102	C₂²	#1831	1	1	yes	characteristic
103	C₂²	#1833	1	2	yes	—
104	C₂²	#1833	1	2	yes	—
105	C₂²	#1835	1	1	yes	characteristic
106	C₂²	#1837	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	2	2	no	—
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	—	4	2	no	—
112	C₄	#1564	1	2	yes	—
113	C₄	#1568	1	2	yes	—
114	C₂	#8781	1	2	yes	—
115	C₂	#8785	1	1	yes	characteristic
116	C₂	#8791	1	1	yes	characteristic
117	C₂	#8792	1	1	yes	characteristic
118	C₂	#8796	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#8785

C₂

yes

maximal, characteristic

#8785

C₂

yes

maximal, characteristic

#7841

C₂

yes

maximal, characteristic

#7018

C₂

yes

maximal

#1831

C₂²

yes

—

#1831

C₂²

yes

—

#1744

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×C₈²

C₄

yes

—

#1277

—

#1277

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₄³

C₂³

yes

—

—

—

—

—

C₈

yes

complemented

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

C₂×C₄²

#41

yes

—

C₂×C₄²

#41

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#117

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₄²

C₂×M₁₆

yes

—

C₄²

#135

yes

—

C₂³

#382

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

C₂×#108

yes

—

C₂×C₄

#431

yes

—

C₂×C₄

#432

yes

—

C₂×C₄

#488

yes

—

C₂×C₄

#488

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₂²

#1831

yes

characteristic

103

C₂²

#1833

yes

—

104

C₂²

#1833

yes

—

105

C₂²

#1835

yes

characteristic

106

C₂²

#1837

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#1564

yes

—

113

C₄

#1568

yes

—

114

C₂

#8781

yes

—

115

C₂

#8785

yes

characteristic

116

C₂

#8791

yes

characteristic

117

C₂

#8792

yes

characteristic

118

C₂

#8796

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#382

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₄³

C₂³

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
28. #322	C₈	semidirect

H ≅ G/N

type

28. #322

C₈

semidirect

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

subgroup	index
#8785	2
#7841	2
#7018	2

subgroup

index

#8785

#7841

#7018

Subgroups — #99281

Conjugacy classes of subgroups

Complements

Maximal subgroups