Group #11495 — subgroups

Subgroups — #11495

Order 256.

157 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 157 conjugacy classes of subgroups — too many to draw. Only its 69 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	#2163	C₂	1	1	yes	maximal, characteristic
3	#2162	C₂	1	1	yes	maximal, characteristic
4	#1322	C₂	1	4	yes	maximal
5	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
6	#446	—	2	1	no	—
7	#446	—	2	1	no	—
8	#446	—	2	1	no	—
9	#446	—	2	1	no	—
10	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×#97	C₄	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	—	2	1	no	—
15	C₄⋊C₁₆	—	2	1	no	—
16	C₂²⋊C₁₆	C₂²	1	2	yes	—
17	C₂²⋊C₁₆	C₂²	1	2	yes	—
18	#346	—	2	1	no	—
19	C₄×C₁₆	—	2	1	no	—
20	C₂²×Q₈	C₈	1	1	yes	characteristic
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	C₄×Q₈	—	2	1	no	—
25	C₄×Q₈	—	2	1	no	—
26	C₄×Q₈	—	2	1	no	—
27	C₄×Q₈	—	2	1	no	—
28	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
29	C₂×C₄⋊C₄	C₈	1	2	yes	—
30	C₂×C₄²	C₂×C₄	1	2	yes	—
31	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆	—	2	2	no	—
33	C₂×C₁₆	—	2	2	no	—
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	#97	—	2	2	no	—
39	#97	—	2	2	no	—
40	C₄×C₈	D₄	1	1	yes	characteristic
41	C₄×C₈	D₄	1	1	yes	characteristic
42	C₄×C₈	D₄	1	1	yes	characteristic
43	C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×Q₈	—	2	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	4	2	no	—
47	C₂²×C₄	C₂×C₈	1	2	yes	—
48	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄	C₂×C₈	1	2	yes	—
51	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	4	1	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	2	1	no	—
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄²	—	2	2	no	—
65	C₄²	—	2	2	no	—
66	C₄²	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²	M₁₆	1	1	yes	characteristic
69	C₄²	M₁₆	1	1	yes	characteristic
70	C₁₆	—	4	2	no	—
71	C₁₆	—	4	2	no	—
72	C₂³	C₂²×C₈	1	1	yes	characteristic
73	Q₈	—	4	2	no	—
74	Q₈	—	4	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	2	1	no	—
92	C₈	—	4	2	no	—
93	C₈	#101	1	2	yes	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²	#504	1	2	yes	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	4	2	no	—
103	C₄	—	4	2	no	—
104	C₄	#349	1	2	yes	—
105	C₄	#410	1	1	yes	characteristic
106	C₄	C₂×#101	1	1	yes	characteristic
107	C₂	—	2	2	no	—
108	C₂	#1461	1	1	yes	characteristic
109	C₂	#2113	1	1	yes	characteristic
110	C₂	C₂×#349	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#446

C₂

yes

maximal, characteristic

#2163

C₂

yes

maximal, characteristic

#2162

C₂

yes

maximal, characteristic

#1322

C₂

yes

maximal

C₂×C₄×Q₈

C₄

yes

—

—

—

—

C₂×C₄⋊C₈

C₄

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#97

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

#346

—

C₄×C₁₆

—

C₂²×Q₈

C₈

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₈

yes

characteristic

C₂×C₄⋊C₄

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

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

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

M₁₆

yes

characteristic

C₄²

M₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₂²×C₈

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#101

yes

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊C₈

yes

characteristic

C₂²

#504

yes

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#349

yes

—

105

C₄

#410

yes

characteristic

106

C₄

C₂×#101

yes

characteristic

107

C₂

—

108

C₂

#1461

yes

characteristic

109

C₂

#2113

yes

characteristic

110

C₂

C₂×#349

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊C₄
commutator, prime radical	C₂³	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₂²⋊C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊C₄

commutator, prime radical

C₂³

C₂²×C₈

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂²

C₂×C₂²⋊C₈

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#446	2
#2163	2
#2162	2
#1322	2

subgroup

index

Subgroups — #11495

Conjugacy classes of subgroups

Maximal subgroups