Group #8836 — subgroups

Subgroups — #8836

Order 256.

205 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 85 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	#2982	C₂	1	1	yes	maximal, characteristic
2	#1984	C₂	1	1	yes	maximal, characteristic
3	C₄×#101	C₂	1	1	yes	maximal, characteristic
4	#1565	C₂	1	2	yes	maximal
5	#1278	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	C₈×Q₈	—	2	2	no	—
8	C₈×Q₈	—	2	2	no	—
9	#431	—	2	2	no	—
10	#431	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#101	C₄	1	1	yes	characteristic
13	C₂×#101	C₄	1	1	yes	characteristic
14	C₄×M₁₆	C₂²	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#343	C₄	1	2	yes	—
17	#336	—	2	2	no	—
18	#335	—	2	2	no	—
19	#334	—	2	2	no	—
20	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×M₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₈	—	2	1	no	—
24	C₂²×C₈	—	2	1	no	—
25	C₄×Q₈	—	2	2	no	—
26	C₄×Q₈	—	2	2	no	—
27	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄	Q₈	1	2	yes	—
29	C₂×C₄²	D₄	1	2	yes	—
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	C₄⋊C₈	—	2	2	no	—
32	C₄⋊C₈	—	2	2	no	—
33	C₄⋊C₈	—	2	2	no	—
34	C₄⋊C₈	D₄	1	2	yes	—
35	C₄⋊C₈	D₄	1	2	yes	—
36	#101	—	2	2	no	—
37	#101	C₈	1	4	yes	complemented
38	C₂²⋊C₈	—	2	2	no	—
39	#97	—	2	2	no	—
40	C₄×C₈	—	2	4	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	—	2	2	no	—
43	C₄×C₈	—	2	2	no	—
44	C₂×Q₈	—	2	2	no	—
45	C₂×Q₈	—	2	2	no	—
46	C₂×Q₈	C₂×C₈	1	2	yes	—
47	C₂×Q₈	M₁₆	1	2	yes	—
48	C₂²×C₄	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
53	M₁₆	—	2	2	no	—
54	M₁₆	—	2	2	no	—
55	M₁₆	—	2	2	no	—
56	M₁₆	C₂×C₈	1	4	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	2	no	—
62	C₂×C₈	—	4	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	4	1	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₄²	—	2	2	no	—
69	C₄²	—	2	2	no	—
70	C₄²	C₂×D₄	1	2	yes	—
71	C₄²	#41	1	2	yes	—
72	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
73	Q₈	—	2	4	no	—
74	Q₈	—	2	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	C₄⋊C₈	1	4	yes	—
83	C₂×C₄	C₄×D₄	1	2	yes	—
84	C₂×C₄	#120	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄	#123	1	1	yes	characteristic
87	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄	#131	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	—	4	2	no	—
92	C₈	—	4	2	no	—
93	C₈	—	4	2	no	—
94	C₈	—	4	2	no	—
95	C₈	—	8	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#385	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²	#424	1	1	yes	characteristic
101	C₄	—	2	4	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	2	no	—
105	C₄	C₈×D₄	1	2	yes	—
106	C₄	#435	1	2	yes	—
107	C₄	#454	1	1	yes	characteristic
108	C₄	#456	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	#1850	1	1	yes	characteristic
112	C₂	#1857	1	1	yes	characteristic
113	C₂	#1858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2982

C₂

yes

maximal, characteristic

#1984

C₂

yes

maximal, characteristic

C₄×#101

C₂

yes

maximal, characteristic

#1565

C₂

yes

maximal

#1278

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#431

—

#431

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#101

C₄

yes

characteristic

C₂×#101

C₄

yes

characteristic

C₄×M₁₆

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#343

C₄

yes

—

#336

—

#335

—

#334

—

C₂²×Q₈

C₂×C₄

yes

characteristic

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄

Q₈

yes

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₄⋊C₈

D₄

yes

—

#101

—

#101

C₈

yes

complemented

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₈

yes

—

C₂×Q₈

M₁₆

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×C₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

C₂×C₄⋊C₄

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

#120

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

—

C₂×C₄

#123

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

#131

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#385

yes

characteristic

C₂²

C₂×C₄⋊C₈

yes

characteristic

100

C₂²

#424

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₈×D₄

yes

—

106

C₄

#435

yes

—

107

C₄

#454

yes

characteristic

108

C₄

#456

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

#1850

yes

characteristic

112

C₂

#1857

yes

characteristic

113

C₂

#1858

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#385
prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#120

commutator

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#385

prime radical

C₂³

C₂×C₄⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
37. #101	C₈	semidirect

H ≅ G/N

type

37. #101

C₈

semidirect

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

subgroup	index
#2982	2
#1984	2
C₄×#101	2
#1565	2
#1278	2

subgroup

index

Subgroups — #8836

Conjugacy classes of subgroups

Complements

Maximal subgroups