Group #8635 — subgroups

Subgroups — #8635

Order 256.

160 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 65 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	#2960	C₂	1	1	yes	maximal, characteristic
2	#1568	C₂	1	1	yes	maximal, characteristic
3	#1568	C₂	1	1	yes	maximal, characteristic
4	#1537	C₂	1	2	yes	maximal
5	#1307	C₂	1	2	yes	maximal
6	#533	C₂²	1	1	yes	characteristic
7	#446	—	2	1	no	—
8	#446	—	2	1	no	—
9	C₈×Q₈	—	2	1	no	—
10	C₈×Q₈	—	2	1	no	—
11	#432	C₂²	1	2	yes	—
12	#432	C₂²	1	2	yes	—
13	#424	C₄	1	2	yes	—
14	#423	C₂²	1	1	yes	characteristic
15	#405	C₄	1	2	yes	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#335	—	2	1	no	—
18	#335	—	2	1	no	—
19	#334	—	2	1	no	—
20	#334	—	2	1	no	—
21	#330	—	2	4	no	—
22	C₂×M₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	C₄⋊Q₈	Q₈	1	1	yes	characteristic
27	#125	D₄	1	2	yes	—
28	C₂²⋊Q₈	—	2	1	no	—
29	C₂²⋊Q₈	—	2	1	no	—
30	C₄×Q₈	—	2	1	no	—
31	C₄×Q₈	—	2	1	no	—
32	#117	C₂×C₄	1	2	yes	—
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	C₄⋊C₈	—	2	2	no	—
35	C₄⋊C₈	—	2	2	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	2	no	—
40	C₂²⋊C₈	—	2	2	no	—
41	C₂²⋊C₈	—	2	2	no	—
42	#97	—	2	2	no	—
43	#97	—	2	2	no	—
44	#97	—	2	2	no	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	D₄	1	1	yes	characteristic
47	C₄×C₈	D₄	1	1	yes	characteristic
48	C₄×C₈	D₄	1	1	yes	characteristic
49	C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×Q₈	—	2	1	no	—
51	C₂×Q₈	—	2	1	no	—
52	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄	#41	1	1	yes	characteristic
54	C₂²×C₄	#41	1	1	yes	characteristic
55	M₁₆	—	4	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	4	1	no	—
59	C₂×C₈	—	4	2	no	—
60	C₂×C₈	—	4	2	no	—
61	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	1	no	—
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
70	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
71	C₂²⋊C₄	—	2	2	no	—
72	C₄²	—	2	2	no	—
73	C₄²	C₂×D₄	1	1	yes	characteristic
74	C₄²	C₂×D₄	1	1	yes	characteristic
75	C₄²	C₂×D₄	1	1	yes	characteristic
76	C₄²	C₂×Q₈	1	1	yes	characteristic
77	C₂³	#117	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄	#121	1	1	yes	characteristic
89	C₂×C₄	#121	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
92	C₈	—	2	1	no	—
93	C₈	—	2	1	no	—
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	—	8	2	no	—
97	C₈	—	8	2	no	—
98	C₈	#101	1	2	yes	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	#380	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	4	2	no	—
109	C₄	—	4	2	no	—
110	C₄	C₂×#101	1	1	yes	characteristic
111	C₄	#413	1	1	yes	characteristic
112	C₂	—	4	1	no	—
113	C₂	#1801	1	1	yes	characteristic
114	C₂	#1814	1	1	yes	characteristic
115	C₂	#1814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2960

C₂

yes

maximal, characteristic

#1568

C₂

yes

maximal, characteristic

#1568

C₂

yes

maximal, characteristic

#1537

C₂

yes

maximal

#1307

C₂

yes

maximal

#533

C₂²

yes

characteristic

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

#432

C₂²

yes

—

#432

C₂²

yes

—

#424

C₄

yes

—

#423

C₂²

yes

characteristic

#405

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₄⋊Q₈

Q₈

yes

characteristic

C₄⋊Q₈

Q₈

yes

characteristic

#125

D₄

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₄⋊C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₂³

#117

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#101

yes

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

#380

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

C₂×#101

yes

characteristic

111

C₄

#413

yes

characteristic

112

C₂

—

113

C₂

#1801

yes

characteristic

114

C₂

#1814

yes

characteristic

115

C₂

#1814

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#380
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

C₂²

C₂²×C₄

C₂×C₄²

C₂³

prime radical

C₂³

#117

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#2960	2
#1568	2
#1537	2
#1307	2

subgroup

index

Subgroups — #8635

Conjugacy classes of subgroups

Maximal subgroups