Group #8625 — subgroups

Subgroups — #8625

Order 256.

171 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 67 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×#446	C₂	1	1	yes	maximal, characteristic
2	#1568	C₂	1	1	yes	maximal, characteristic
3	#1568	C₂	1	1	yes	maximal, characteristic
4	#1467	C₂	1	2	yes	maximal
5	#1290	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	#446	—	2	1	no	—
8	#446	—	2	1	no	—
9	#446	—	2	1	no	—
10	#446	—	2	1	no	—
11	#432	C₂²	1	2	yes	—
12	#432	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×#97	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	#324	—	2	4	no	—
22	C₂²×Q₈	Q₈	1	1	yes	characteristic
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈	—	2	1	no	—
27	C₄×Q₈	—	2	1	no	—
28	C₄×Q₈	—	2	1	no	—
29	C₄×Q₈	—	2	1	no	—
30	C₂×C₄⋊C₄	D₄	1	2	yes	—
31	C₂×C₄⋊C₄	Q₈	1	1	yes	characteristic
32	C₂×C₄²	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	#97	—	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	C₂×Q₈	—	4	2	no	—
53	C₂²×C₄	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	1	no	—
71	C₄⋊C₄	—	2	1	no	—
72	C₄²	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×D₄	1	1	yes	characteristic
75	C₄²	C₂×D₄	1	1	yes	characteristic
76	C₄²	#41	1	1	yes	characteristic
77	C₄²	#41	1	1	yes	characteristic
78	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
79	Q₈	—	4	2	no	—
80	Q₈	—	4	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄	#117	1	1	yes	characteristic
91	C₂×C₄	#121	1	1	yes	characteristic
92	C₂×C₄	#121	1	1	yes	characteristic
93	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
95	C₈	—	2	1	no	—
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	—	4	2	no	—
99	C₈	—	8	2	no	—
100	C₈	—	8	2	no	—
101	C₈	#101	1	2	yes	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	#380	1	1	yes	characteristic
105	C₂²	#430	1	2	yes	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	C₂×#101	1	1	yes	characteristic
114	C₄	#413	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#1801	1	1	yes	characteristic
117	C₂	#1811	1	1	yes	characteristic
118	C₂	#1812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#446

C₂

yes

maximal, characteristic

#1568

C₂

yes

maximal, characteristic

#1568

C₂

yes

maximal, characteristic

#1467

C₂

yes

maximal

#1290

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

—

—

—

—

C₂²

yes

—

#432

C₂²

yes

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

C₂²×Q₈

Q₈

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊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

—

C₂×Q₈

—

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

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

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

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

—

C₂×C₄

C₂×C₂²⋊C₄

yes

C₂×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

—

100

C₈

—

101

C₈

#101

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

#380

yes

characteristic

105

C₂²

#430

yes

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₂×#101

yes

characteristic

114

C₄

#413

yes

characteristic

115

C₂

—

116

C₂

#1801

yes

characteristic

117

C₂

#1811

yes

characteristic

118

C₂

#1812

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#380
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₂×C₄⋊C₄
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

C₂×C₄⋊C₄

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
C₂×#446	2
#1568	2
#1467	2
#1290	2

subgroup

index

Subgroups — #8625

Conjugacy classes of subgroups

Maximal subgroups