Group #9624 — subgroups

Subgroups — #9624

Order 256.

149 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 149 conjugacy classes of subgroups — too many to draw. Only its 75 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	#1950	C₂	1	2	yes	maximal
2	#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
5	#1301	C₂	1	2	yes	maximal
6	#431	C₂²	1	1	yes	characteristic
7	C₂×#108	—	2	2	no	—
8	#423	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#341	—	2	2	no	—
13	#335	—	2	1	no	—
14	#335	—	2	1	no	—
15	#335	C₄	1	2	yes	—
16	#335	C₄	1	2	yes	—
17	#322	—	2	1	no	—
18	C₈²	—	2	1	no	—
19	C₂×M₁₆	—	2	2	no	—
20	C₂×M₁₆	D₄	1	1	yes	characteristic
21	C₂×M₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₈	—	2	2	no	—
23	C₂²×C₈	Q₈	1	1	yes	characteristic
24	C₂²×C₈	Q₈	1	1	yes	characteristic
25	C₂×C₄²	C₂³	1	1	yes	characteristic
26	#108	—	4	4	no	—
27	C₄⋊C₈	—	2	2	no	—
28	C₄⋊C₈	—	2	2	no	—
29	C₄⋊C₈	—	2	1	no	—
30	C₄⋊C₈	—	2	2	no	—
31	C₄⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²⋊C₈	Q₈	1	2	yes	—
33	#97	—	2	1	no	—
34	C₄×C₈	—	2	1	no	—
35	C₄×C₈	—	2	1	no	—
36	C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈	D₄	1	1	yes	characteristic
39	C₄×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
43	M₁₆	—	2	1	no	—
44	M₁₆	—	2	1	no	—
45	M₁₆	—	4	2	no	—
46	M₁₆	—	4	2	no	—
47	M₁₆	D₈	1	2	yes	—
48	M₁₆	Q₁₆	1	2	yes	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	4	1	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	4	2	no	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₄²	C₂²×C₄	1	1	yes	characteristic
63	C₄²	C₂×D₄	1	1	yes	characteristic
64	C₄²	#41	1	1	yes	characteristic
65	C₄²	#41	1	1	yes	characteristic
66	C₂³	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄	#121	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄	#125	1	1	yes	characteristic
76	C₂×C₄	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄	#135	1	1	yes	characteristic
79	C₂×C₄	#135	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	—	2	1	no	—
83	C₈	—	2	2	no	—
84	C₈	—	2	2	no	—
85	C₈	—	4	2	no	—
86	C₈	—	4	4	no	—
87	C₈	—	8	2	no	—
88	C₈	#108	1	2	yes	complemented
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#389	1	1	yes	characteristic
92	C₂²	#499	1	1	yes	characteristic
93	C₂²	#500	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	2	1	no	—
96	C₄	—	2	1	no	—
97	C₄	C₂×#108	1	1	yes	characteristic
98	C₄	#430	1	1	yes	characteristic
99	C₄	C₄×D₈	1	1	yes	characteristic
100	C₄	C₄×Q₁₆	1	1	yes	characteristic
101	C₄	#466	1	1	yes	characteristic
102	C₄	#467	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#1942	1	1	yes	characteristic
106	C₂	#1950	1	1	yes	characteristic
107	C₂	#1953	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1950

C₂

yes

maximal

#1567

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

C₈×M₁₆

C₂

yes

maximal, characteristic

#1301

C₂

yes

maximal

#431

C₂²

yes

characteristic

C₂×#108

—

#423

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

C₄

yes

—

#335

C₄

yes

—

#322

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Q₈

yes

characteristic

C₂²×C₈

Q₈

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

Q₈

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

D₈

yes

—

M₁₆

Q₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

C₄⋊Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

#121

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#125

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#108

yes

complemented

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

C₄×D₈

yes

characteristic

100

C₄

C₄×Q₁₆

yes

characteristic

101

C₄

#466

yes

characteristic

102

C₄

#467

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#1942

yes

characteristic

106

C₂

#1950

yes

characteristic

107

C₂

#1953

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#389
prime radical	C₂³	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#389

prime radical

C₂³

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
88. C₈	#108	semidirect

H ≅ G/N

type

88. C₈

#108

semidirect

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

subgroup	index
#1950	2
#1567	2
C₂×#335	2
C₈×M₁₆	2
#1301	2

subgroup

index

C₈×M₁₆

Subgroups — #9624

Conjugacy classes of subgroups

Complements

Maximal subgroups