Group #9694 — subgroups

Subgroups — #9694

Order 256.

285 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 119 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	#2868	C₂	1	1	yes	maximal, characteristic
2	#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#332	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈	—	2	2	no	—
9	C₂×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈	—	2	1	no	—
12	#384	—	2	2	no	—
13	#384	C₄	1	4	yes	—
14	C₄³	C₂²	1	1	yes	characteristic
15	#336	—	2	1	no	—
16	#332	—	2	8	no	—
17	C₂²×Q₈	—	2	2	no	—
18	C₂²×C₈	—	2	2	no	—
19	C₂²×C₈	—	2	2	no	—
20	C₄⋊Q₈	—	2	4	no	—
21	C₄⋊Q₈	—	2	4	no	—
22	C₄⋊Q₈	—	4	4	no	—
23	C₄⋊Q₈	—	4	2	no	—
24	C₄⋊Q₈	—	4	2	no	—
25	C₂×C₄⋊C₄	—	2	4	no	—
26	C₂×C₄⋊C₄	—	2	4	no	—
27	C₂×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₄²	C₂×C₄	1	2	yes	—
30	C₂×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₄²	D₄	1	1	yes	characteristic
32	C₂×C₄²	D₄	1	2	yes	—
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	C₄⋊C₈	—	2	4	no	—
35	C₄⋊C₈	—	2	2	no	—
36	C₄⋊C₈	—	4	4	no	—
37	C₄⋊C₈	D₄	1	4	yes	—
38	C₄×C₈	—	4	2	no	—
39	C₂×Q₈	—	4	4	no	—
40	C₂×Q₈	—	4	2	no	—
41	C₂×Q₈	—	4	2	no	—
42	C₂×Q₈	—	4	4	no	—
43	C₂²×C₄	—	2	4	no	—
44	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	—	4	2	no	—
52	C₂×C₈	—	4	2	no	—
53	C₂×C₈	—	4	4	no	—
54	C₄⋊C₄	—	2	8	no	—
55	C₄⋊C₄	—	4	8	no	—
56	C₄⋊C₄	—	4	8	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄²	—	2	4	no	—
60	C₄²	—	2	2	no	—
61	C₄²	D₈	1	2	yes	—
62	C₄²	SD₁₆	1	2	yes	—
63	C₄²	SD₁₆	1	4	yes	—
64	C₄²	Q₁₆	1	4	yes	—
65	C₄²	C₂×D₄	1	2	yes	—
66	C₄²	#41	1	2	yes	—
67	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	Q₈	—	8	4	no	—
69	Q₈	—	8	4	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	4	4	no	—
74	C₂×C₄	—	4	4	no	—
75	C₂×C₄	—	4	4	no	—
76	C₂×C₄	#101	1	4	yes	—
77	C₂×C₄	#102	1	4	yes	—
78	C₂×C₄	#103	1	4	yes	—
79	C₂×C₄	#103	1	4	yes	—
80	C₂×C₄	C₄×D₄	1	2	yes	—
81	C₂×C₄	#121	1	2	yes	—
82	C₂×C₄	#124	1	1	yes	characteristic
83	C₂×C₄	#127	1	1	yes	characteristic
84	C₂×C₄	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
89	C₈	—	4	4	no	—
90	C₈	—	8	2	no	—
91	C₂²	#390	1	1	yes	characteristic
92	C₂²	C₂×#101	1	2	yes	—
93	C₂²	C₂×#102	1	1	yes	characteristic
94	C₂²	C₂×#103	1	2	yes	—
95	C₂²	#418	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	2	2	no	—
98	C₄	—	8	4	no	—
99	C₄	#332	1	8	yes	complemented
100	C₄	#459	1	2	yes	—
101	C₄	#460	1	2	yes	—
102	C₄	#487	1	2	yes	—
103	C₄	#492	1	1	yes	characteristic
104	C₄	#494	1	1	yes	characteristic
105	C₂	C₂×#332	1	2	yes	—
106	C₂	#1549	1	2	yes	—
107	C₂	#1956	1	1	yes	characteristic
108	C₂	#1966	1	1	yes	characteristic
109	C₂	#1982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2868

C₂

yes

maximal, characteristic

#1842

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

—

#384

—

#384

C₄

yes

—

C₄³

C₂²

yes

characteristic

#336

—

#332

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

—

C₂²×C₄

C₂²⋊C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

D₈

yes

—

C₄²

SD₁₆

yes

—

C₄²

SD₁₆

yes

—

C₄²

Q₁₆

yes

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

C₂×C₂²⋊C₄

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#101

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

#103

yes

—

C₂×C₄

#103

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

#121

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₈

—

C₈

—

C₂²

#390

yes

characteristic

C₂²

C₂×#101

yes

—

C₂²

C₂×#102

yes

characteristic

C₂²

C₂×#103

yes

—

C₂²

#418

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#332

yes

complemented

100

C₄

#459

yes

—

101

C₄

#460

yes

—

102

C₄

#487

yes

—

103

C₄

#492

yes

characteristic

104

C₄

#494

yes

characteristic

105

C₂

C₂×#332

yes

—

106

C₂

#1549

yes

—

107

C₂

#1956

yes

characteristic

108

C₂

#1966

yes

characteristic

109

C₂

#1982

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×C₂²⋊C₄

commutator

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
99. C₄	#332	semidirect

H ≅ G/N

type

99. C₄

#332

semidirect

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

subgroup	index
#2868	2
#1842	2
C₄×C₄⋊C₈	2
C₂×#332	2

subgroup

index

#2868

#1842

C₄×C₄⋊C₈

C₂×#332

Subgroups — #9694

Conjugacy classes of subgroups

Complements

Maximal subgroups