Group #8661 — subgroups

Subgroups — #8661

Order 256.

189 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 77 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#1837	C₂	1	1	yes	maximal, characteristic
3	#1564	C₂	1	1	yes	maximal, characteristic
4	#1487	C₂	1	2	yes	maximal
5	#1301	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	C₈×Q₈	—	2	1	no	—
8	C₈×Q₈	—	2	1	no	—
9	C₈×Q₈	C₄	1	2	yes	—
10	C₈×Q₈	C₄	1	2	yes	—
11	#431	C₂²	1	2	yes	—
12	#423	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈	—	2	1	no	—
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	—	2	1	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#341	—	2	1	no	—
18	#341	—	2	1	no	—
19	#335	—	2	1	no	—
20	#334	—	2	1	no	—
21	Q₈⋊C₈	—	2	4	no	—
22	C₂²×Q₈	Q₈	1	1	yes	characteristic
23	C₂×M₁₆	—	2	2	no	—
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₈	D₄	1	1	yes	characteristic
27	C₄×Q₈	—	2	1	no	—
28	C₄×Q₈	—	2	1	no	—
29	C₄×Q₈	C₂×C₄	1	2	yes	—
30	C₄×Q₈	D₄	1	2	yes	—
31	C₂×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄⋊C₄	Q₈	1	1	yes	characteristic
33	C₂×C₄²	—	2	1	no	—
34	C₂×C₄²	C₂³	1	1	yes	characteristic
35	C₄⋊C₈	—	2	2	no	—
36	C₄⋊C₈	—	2	2	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	2	no	—
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈	—	2	2	no	—
43	C₄×C₈	—	2	2	no	—
44	C₄×C₈	—	2	2	no	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈	D₄	1	1	yes	characteristic
49	C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	—	2	1	no	—
52	C₂×Q₈	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄	#41	1	1	yes	characteristic
58	M₁₆	—	4	2	no	—
59	M₁₆	—	4	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	1	no	—
73	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
74	C₄²	—	2	2	no	—
75	C₄²	—	2	2	no	—
76	C₄²	C₂²×C₄	1	1	yes	characteristic
77	C₄²	C₂×D₄	1	1	yes	characteristic
78	C₄²	C₂×D₄	1	1	yes	characteristic
79	C₄²	#41	1	1	yes	characteristic
80	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
81	Q₈	—	2	4	no	—
82	Q₈	—	2	2	no	—
83	Q₈	#108	1	4	yes	—
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄	#117	1	1	yes	characteristic
93	C₂×C₄	#121	1	1	yes	characteristic
94	C₂×C₄	#121	1	1	yes	characteristic
95	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄	#135	1	2	yes	—
97	C₂×C₄	#137	1	2	yes	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	2	1	no	—
101	C₈	—	2	1	no	—
102	C₈	—	4	2	no	—
103	C₈	—	8	2	no	—
104	C₈	—	8	2	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	#380	1	1	yes	characteristic
108	C₂²	#479	1	2	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	#416	1	1	yes	characteristic
115	C₄	#419	1	1	yes	characteristic
116	C₄	C₂×#108	1	2	yes	—
117	C₄	#464	1	2	yes	—
118	C₂	—	2	2	no	—
119	C₂	#1805	1	1	yes	characteristic
120	C₂	#1810	1	1	yes	characteristic
121	C₂	#1814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

#1837

C₂

yes

maximal, characteristic

#1564

C₂

yes

maximal, characteristic

#1487

C₂

yes

maximal

#1301

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

C₄

yes

—

C₈×Q₈

C₄

yes

—

#431

C₂²

yes

—

#423

C₂²

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

Q₈⋊C₈

—

C₂²×Q₈

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

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

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₂²⋊Q₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

#108

yes

—

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

yes

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#135

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

#380

yes

characteristic

108

C₂²

#479

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#416

yes

characteristic

115

C₄

#419

yes

characteristic

116

C₄

C₂×#108

yes

—

117

C₄

#464

yes

—

118

C₂

—

119

C₂

#1805

yes

characteristic

120

C₂

#1810

yes

characteristic

121

C₂

#1814

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#380
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#380

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ 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
C₂×C₈×Q₈	2
#1837	2
#1564	2
#1487	2
#1301	2

subgroup

index

C₂×C₈×Q₈

Subgroups — #8661

Conjugacy classes of subgroups

Maximal subgroups