Group #8689 — subgroups

Subgroups — #8689

Order 256.

173 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 173 conjugacy classes of subgroups — too many to draw. Only its 73 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	#1837	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	1	yes	maximal, characteristic
4	#1550	C₂	1	2	yes	maximal
5	#1292	C₂	1	2	yes	maximal
6	#533	C₂²	1	1	yes	characteristic
7	#446	—	2	1	no	—
8	#446	C₄	1	2	yes	—
9	C₈×Q₈	—	2	1	no	—
10	C₈×Q₈	C₄	1	2	yes	—
11	#424	—	2	1	no	—
12	#423	C₂²	1	2	yes	—
13	#423	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈	C₂²	1	2	yes	—
15	#405	—	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	#335	—	2	1	no	—
21	#333	—	2	2	no	—
22	#332	—	2	2	no	—
23	C₂×M₁₆	—	2	1	no	—
24	C₂×M₁₆	—	2	2	no	—
25	C₂²×C₈	—	2	2	no	—
26	C₂²×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₈	D₄	1	1	yes	characteristic
28	C₄⋊Q₈	D₄	1	1	yes	characteristic
29	C₄⋊Q₈	D₄	1	1	yes	characteristic
30	#125	—	2	1	no	—
31	C₂²⋊Q₈	—	2	1	no	—
32	C₂²⋊Q₈	Q₈	1	2	yes	—
33	C₄×Q₈	—	2	1	no	—
34	C₄×Q₈	C₂×C₄	1	2	yes	—
35	#117	—	2	1	no	—
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₄⋊C₈	C₂×C₄	1	2	yes	—
45	#97	—	2	2	no	—
46	C₄×C₈	—	2	2	no	—
47	C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈	D₄	1	1	yes	characteristic
50	C₄×C₈	D₄	1	1	yes	characteristic
51	C₂×Q₈	—	2	1	no	—
52	C₂×Q₈	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄	#41	1	1	yes	characteristic
56	M₁₆	—	4	2	no	—
57	M₁₆	—	4	2	no	—
58	M₁₆	—	4	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	8	2	no	—
65	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	1	no	—
71	C₄⋊C₄	—	2	1	no	—
72	C₄⋊C₄	—	2	1	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄	—	2	2	no	—
76	C₄²	—	2	2	no	—
77	C₄²	C₂²×C₄	1	1	yes	characteristic
78	C₄²	C₂×D₄	1	1	yes	characteristic
79	C₄²	C₂×D₄	1	1	yes	characteristic
80	C₄²	#41	1	1	yes	characteristic
81	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
82	Q₈	—	4	2	no	—
83	Q₈	—	4	1	no	—
84	Q₈	—	4	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	1	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	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	4	2	no	—
113	C₄	—	4	2	no	—
114	C₄	#416	1	1	yes	characteristic
115	C₄	#419	1	1	yes	characteristic
116	C₄	#464	1	2	yes	—
117	C₄	#472	1	1	yes	characteristic
118	C₄	#473	1	1	yes	characteristic
119	C₂	—	2	2	no	—
120	C₂	#1805	1	1	yes	characteristic
121	C₂	#1813	1	1	yes	characteristic
122	C₂	#1816	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2960

C₂

yes

maximal, characteristic

#1837

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

#1550

C₂

yes

maximal

#1292

C₂

yes

maximal

#533

C₂²

yes

characteristic

#446

—

#446

C₄

yes

—

C₈×Q₈

—

C₈×Q₈

C₄

yes

—

#424

—

#423

C₂²

yes

—

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

#405

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Q₈

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×C₄

yes

—

#117

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

#97

—

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

C₄⋊C₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

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

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₂²⋊C₄

—

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

—

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

#464

yes

—

117

C₄

#472

yes

characteristic

118

C₄

#473

yes

characteristic

119

C₂

—

120

C₂

#1805

yes

characteristic

121

C₂

#1813

yes

characteristic

122

C₂

#1816

yes

characteristic

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

subgroup

G/H

center, socle

C₂²

#380

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₄² ▷ 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
#1837	2
C₂×#335	2
#1550	2
#1292	2

subgroup

index

Subgroups — #8689

Conjugacy classes of subgroups

Maximal subgroups