Group #11865 — subgroups

Subgroups — #11865

Order 256.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#3170	C₂	1	1	yes	maximal, characteristic
2	#2163	C₂	1	1	yes	maximal, characteristic
3	#1837	C₂	1	1	yes	maximal, characteristic
4	#1350	C₂	1	4	yes	maximal
5	#533	—	2	1	no	—
6	#500	C₄	1	1	yes	characteristic
7	#494	C₄	1	1	yes	characteristic
8	#487	—	2	1	no	—
9	#467	—	2	1	no	—
10	#467	C₂²	1	2	yes	—
11	C₄×Q₁₆	—	2	1	no	—
12	C₄×Q₁₆	C₄	1	2	yes	—
13	#423	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆	—	2	1	no	—
16	C₂²⋊C₁₆	C₂²	1	2	yes	—
17	#346	—	2	1	no	—
18	#341	—	2	1	no	—
19	#341	C₂²	1	2	yes	—
20	C₂×Q₁₆	—	2	1	no	—
21	C₂×Q₁₆	C₂×C₄	1	2	yes	—
22	C₂×M₁₆	—	2	2	no	—
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₈	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈	—	2	1	no	—
26	C₄⋊Q₈	—	2	1	no	—
27	#125	—	4	1	no	—
28	C₂²⋊Q₈	—	2	2	no	—
29	C₂²⋊Q₈	—	4	1	no	—
30	C₄×Q₈	—	2	2	no	—
31	C₄×Q₈	—	4	1	no	—
32	#117	—	4	1	no	—
33	C₂×C₄²	D₄	1	1	yes	characteristic
34	C₂×C₁₆	—	2	2	no	—
35	#107	—	2	1	no	—
36	#107	C₂×C₄	1	2	yes	—
37	C₄⋊C₈	—	4	1	no	—
38	C₄⋊C₈	—	4	1	no	—
39	#103	—	2	2	no	—
40	#103	—	2	2	no	—
41	C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈	D₄	1	1	yes	characteristic
44	C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×Q₈	—	2	2	no	—
46	C₂×Q₈	—	4	1	no	—
47	C₂²×C₄	SD₁₆	1	1	yes	characteristic
48	C₂²×C₄	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	Q₁₆	—	4	2	no	—
51	Q₁₆	—	4	2	no	—
52	M₁₆	—	4	2	no	—
53	M₁₆	—	4	2	no	—
54	C₂×C₈	—	4	1	no	—
55	C₂×C₈	—	4	1	no	—
56	C₂×C₈	—	4	2	no	—
57	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	4	1	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄⋊C₄	—	4	1	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₂²⋊C₄	—	4	2	no	—
68	C₄²	—	4	2	no	—
69	C₄²	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²	D₈	1	1	yes	characteristic
72	C₄²	D₈	1	1	yes	characteristic
73	C₁₆	—	4	2	no	—
74	C₂³	C₂×SD₁₆	1	1	yes	characteristic
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	Q₈	—	8	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	#102	1	1	yes	characteristic
85	C₂×C₄	#102	1	1	yes	characteristic
86	C₂×C₄	#102	1	1	yes	characteristic
87	C₂×C₄	#102	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄	C₂×D₈	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	2	1	no	—
92	C₈	—	2	1	no	—
93	C₈	—	2	1	no	—
94	C₈	—	8	2	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	C₂×#102	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	—	8	2	no	—
104	C₄	—	8	2	no	—
105	C₄	#362	1	2	yes	—
106	C₄	#410	1	1	yes	characteristic
107	C₄	#421	1	1	yes	characteristic
108	C₂	—	4	1	no	—
109	C₂	#1517	1	1	yes	characteristic
110	C₂	#2143	1	1	yes	characteristic
111	C₂	C₂×#362	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3170

C₂

yes

maximal, characteristic

#2163

C₂

yes

maximal, characteristic

#1837

C₂

yes

maximal, characteristic

#1350

C₂

yes

maximal

#533

—

#500

C₄

yes

characteristic

#494

C₄

yes

—

—

C₂²

yes

—

C₄×Q₁₆

—

C₄×Q₁₆

C₄

yes

—

#423

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₂²

yes

—

#346

—

#341

—

#341

C₂²

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₁₆

—

#107

—

#107

C₂×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

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

—

C₂²×C₄

SD₁₆

yes

characteristic

C₂²×C₄

SD₁₆

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

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

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

D₈

yes

characteristic

C₄²

D₈

yes

characteristic

C₁₆

—

C₂³

C₂×SD₁₆

yes

characteristic

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

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×#102

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#362

yes

—

106

C₄

#410

yes

characteristic

107

C₄

#421

yes

characteristic

108

C₂

—

109

C₂

#1517

yes

characteristic

110

C₂

#2143

yes

characteristic

111

C₂

C₂×#362

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#102
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#102

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

C₂×SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
#3170	2
#2163	2
#1837	2
#1350	2

subgroup

index

Subgroups — #11865

Conjugacy classes of subgroups

Maximal subgroups