Group #10646 — subgroups

Subgroups — #10646

Order 256.

118 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 41 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	#1571	C₂	1	1	yes	maximal, characteristic
2	#1559	C₂	1	1	yes	maximal, characteristic
3	#1552	C₂	1	1	yes	maximal, characteristic
4	#1497	C₂	1	1	yes	maximal, characteristic
5	#1307	C₂	1	1	yes	maximal, characteristic
6	#1290	C₂	1	1	yes	maximal, characteristic
7	#1287	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	#432	C₂²	1	1	yes	characteristic
10	#432	C₂²	1	1	yes	characteristic
11	#432	C₂²	1	1	yes	characteristic
12	#431	C₂²	1	1	yes	characteristic
13	#423	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	#335	—	2	2	no	—
16	#333	—	2	1	no	—
17	#332	—	2	1	no	—
18	#332	—	2	1	no	—
19	#332	—	2	1	no	—
20	Q₈⋊C₈	—	2	2	no	—
21	C₂×M₁₆	—	2	1	no	—
22	C₂²×C₈	—	2	1	no	—
23	C₄⋊Q₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈	D₄	1	1	yes	characteristic
25	#125	—	2	1	no	—
26	C₂²⋊Q₈	—	2	1	no	—
27	C₂²⋊Q₈	—	2	1	no	—
28	C₄×Q₈	—	2	1	no	—
29	C₄×Q₈	—	2	1	no	—
30	#117	—	2	1	no	—
31	C₂×C₄²	C₂³	1	1	yes	characteristic
32	C₄⋊C₈	—	2	1	no	—
33	C₄⋊C₈	—	2	1	no	—
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₂²⋊C₈	—	2	1	no	—
41	C₂²⋊C₈	—	2	1	no	—
42	C₂²⋊C₈	—	2	1	no	—
43	C₂²⋊C₈	—	2	1	no	—
44	#97	—	2	1	no	—
45	#97	—	2	1	no	—
46	#97	—	2	1	no	—
47	C₄×C₈	D₄	1	2	yes	—
48	C₂×Q₈	—	2	1	no	—
49	C₂×Q₈	—	2	1	no	—
50	C₂²×C₄	#41	1	1	yes	characteristic
51	C₂²×C₄	#41	1	1	yes	characteristic
52	C₂²×C₄	#41	1	1	yes	characteristic
53	M₁₆	—	8	1	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	4	1	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	—	4	1	no	—
58	C₂×C₈	—	4	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	—	8	1	no	—
61	C₄⋊C₄	—	2	1	no	—
62	C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₄	—	2	1	no	—
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	—	4	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₂²⋊C₄	—	4	1	no	—
68	C₄²	—	4	1	no	—
69	C₄²	C₂×D₄	1	1	yes	characteristic
70	C₄²	C₂×D₄	1	1	yes	characteristic
71	C₄²	#41	1	1	yes	characteristic
72	C₄²	#41	1	1	yes	characteristic
73	C₂³	#126	1	1	yes	characteristic
74	Q₈	—	4	1	no	—
75	Q₈	—	4	1	no	—
76	Q₈	—	8	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	#121	1	1	yes	characteristic
84	C₂×C₄	#123	1	1	yes	characteristic
85	C₂×C₄	#123	1	1	yes	characteristic
86	C₂×C₄	#123	1	1	yes	characteristic
87	C₂×C₄	#124	1	1	yes	characteristic
88	C₂×C₄	#126	1	1	yes	characteristic
89	C₈	—	4	2	no	—
90	C₈	—	8	1	no	—
91	C₈	—	8	1	no	—
92	C₈	—	8	1	no	—
93	C₈	—	8	1	no	—
94	C₈	—	8	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#397	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	8	1	no	—
105	C₄	—	8	1	no	—
106	C₄	#472	1	1	yes	characteristic
107	C₄	#473	1	1	yes	characteristic
108	C₂	—	4	1	no	—
109	C₂	#2078	1	1	yes	characteristic
110	C₂	#2080	1	1	yes	characteristic
111	C₂	#2082	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1571

C₂

yes

maximal, characteristic

#1559

C₂

yes

maximal, characteristic

#1552

C₂

yes

maximal, characteristic

#1497

C₂

yes

maximal, characteristic

#1307

C₂

yes

maximal, characteristic

#1290

C₂

yes

maximal, characteristic

#1287

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

#432

C₂²

yes

characteristic

#432

C₂²

yes

characteristic

#432

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

—

—

—

—

Q₈⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

D₄

yes

—

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

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#397

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#472

yes

characteristic

107

C₄

#473

yes

characteristic

108

C₂

—

109

C₂

#2078

yes

characteristic

110

C₂

#2080

yes

characteristic

111

C₂

#2082

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#397

commutator, Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂³

#126

LCS

G ▷ C₂×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
#1571	2
#1559	2
#1552	2
#1497	2
#1307	2
#1290	2
#1287	2

subgroup

index

Subgroups — #10646

Conjugacy classes of subgroups

Maximal subgroups