Group #9617 — subgroups

Subgroups — #9617

Order 256.

185 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 185 conjugacy classes of subgroups — too many to draw. Only its 91 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	#1947	C₂	1	2	yes	maximal
2	#1844	C₂	1	1	yes	maximal, characteristic
3	#1770	C₂	1	1	yes	maximal, characteristic
4	C₂×#335	C₂	1	1	yes	maximal, characteristic
5	#1296	C₂	1	2	yes	maximal
6	C₂×#107	—	2	2	no	—
7	C₂×#106	—	2	1	no	—
8	C₂×#106	—	2	1	no	—
9	C₂×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#97	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	#340	—	2	2	no	—
16	#336	—	2	1	no	—
17	#335	C₄	1	4	yes	complemented
18	C₂²×C₈	—	2	1	no	—
19	C₂²×C₈	—	2	2	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄⋊C₄	D₄	1	2	yes	—
26	C₂×C₄²	—	2	1	no	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	#107	—	4	4	no	—
29	#106	—	4	2	no	—
30	#106	—	4	2	no	—
31	C₄⋊C₈	—	2	2	no	—
32	C₄⋊C₈	—	2	1	no	—
33	C₄⋊C₈	C₂×C₄	1	4	yes	—
34	C₄⋊C₈	Q₈	1	2	yes	—
35	#97	—	4	2	no	—
36	C₄×C₈	C₂×C₄	1	2	yes	—
37	C₄×C₈	Q₈	1	2	yes	—
38	#95	—	2	1	no	—
39	C₂²×C₄	—	2	2	no	—
40	C₂²×C₄	—	4	2	no	—
41	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄	#41	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	2	4	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	4	1	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	C₄⋊C₄	1	2	yes	—
53	C₂×C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₈	D₈	1	2	yes	—
55	C₂×C₈	Q₁₆	1	2	yes	—
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	—	8	2	no	—
63	C₄²	—	4	1	no	—
64	C₄²	—	4	1	no	—
65	C₄²	C₂²×C₄	1	1	yes	characteristic
66	C₄²	C₂×Q₈	1	1	yes	characteristic
67	C₄²	C₂×Q₈	1	1	yes	characteristic
68	C₄²	#41	1	1	yes	characteristic
69	C₂³	#121	1	1	yes	characteristic
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	—	8	2	no	—
76	C₂×C₄	—	8	2	no	—
77	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄	#125	1	1	yes	characteristic
82	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
87	C₂×C₄	#137	1	1	yes	characteristic
88	C₂×C₄	#137	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	4	no	—
91	C₈	—	4	2	no	—
92	C₈	#107	1	4	yes	complemented
93	C₂²	#389	1	1	yes	characteristic
94	C₂²	#459	1	1	yes	characteristic
95	C₂²	#462	1	1	yes	characteristic
96	C₂²	#468	1	2	yes	—
97	C₂²	#471	1	2	yes	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	8	2	no	—
101	C₄	—	16	2	no	—
102	C₄	C₂×#107	1	1	yes	characteristic
103	C₄	#428	1	1	yes	characteristic
104	C₄	#441	1	1	yes	characteristic
105	C₄	#442	1	1	yes	characteristic
106	C₄	#474	1	1	yes	characteristic
107	C₄	#477	1	1	yes	characteristic
108	C₄	#501	1	2	yes	—
109	C₂	#1685	1	2	yes	—
110	C₂	#1692	1	2	yes	—
111	C₂	#1943	1	1	yes	characteristic
112	C₂	#1947	1	1	yes	characteristic
113	C₂	#1951	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1947

C₂

yes

maximal

#1844

C₂

yes

maximal, characteristic

#1770

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

#1296

C₂

yes

—

—

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

Q₈

yes

—

#97

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

Q₈

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#125

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#137

yes

characteristic

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

C₈

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#471

yes

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

C₂×#107

yes

characteristic

103

C₄

#428

yes

characteristic

104

C₄

#441

yes

characteristic

105

C₄

#442

yes

characteristic

106

C₄

#474

yes

characteristic

107

C₄

#477

yes

characteristic

108

C₄

#501

yes

—

109

C₂

#1685

yes

—

110

C₂

#1692

yes

—

111

C₂

#1943

yes

characteristic

112

C₂

#1947

yes

characteristic

113

C₂

#1951

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
17. #335	C₄	semidirect
92. C₈	#107	semidirect

H ≅ G/N

type

17. #335

C₄

semidirect

92. C₈

#107

semidirect

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

subgroup	index
#1947	2
#1844	2
#1770	2
C₂×#335	2
#1296	2

subgroup

index

Subgroups — #9617

Conjugacy classes of subgroups

Complements

Maximal subgroups