Group #9637 — subgroups

Subgroups — #9637

Order 256.

303 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#1842	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈	—	2	2	no	—
8	C₂×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈	—	2	2	no	—
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	#391	C₄	1	2	yes	—
12	#389	—	2	2	no	—
13	#384	C₄	1	1	yes	characteristic
14	C₄×C₄⋊C₄	—	2	2	no	—
15	C₄³	C₂²	1	1	yes	characteristic
16	Q₈⋊C₈	—	2	8	no	—
17	C₂²×Q₈	C₂×C₄	1	2	yes	—
18	C₂²×C₈	—	2	2	no	—
19	C₂²×C₈	—	2	2	no	—
20	C₄⋊Q₈	—	2	2	no	—
21	C₄⋊Q₈	C₈	1	4	yes	complemented
22	C₄×Q₈	—	2	4	no	—
23	C₄×Q₈	—	2	4	no	—
24	C₂×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₂×C₄²	—	2	2	no	—
29	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²	D₄	1	2	yes	—
32	C₂×C₄²	D₄	1	2	yes	—
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	C₄⋊C₈	—	2	4	no	—
35	C₄⋊C₈	—	4	4	no	—
36	C₄×C₈	—	2	8	no	—
37	C₄×C₈	—	2	2	no	—
38	C₄×C₈	D₄	1	4	yes	—
39	#95	—	2	4	no	—
40	C₂×Q₈	—	2	2	no	—
41	C₂×Q₈	—	2	4	no	—
42	C₂×Q₈	—	2	4	no	—
43	C₂×Q₈	C₂×C₈	1	4	yes	—
44	C₂²×C₄	—	2	2	no	—
45	C₂²×C₄	—	2	2	no	—
46	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈	—	2	8	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	4	2	no	—
54	C₂×C₈	—	4	4	no	—
55	C₄⋊C₄	—	2	8	no	—
56	C₄⋊C₄	—	2	4	no	—
57	C₄⋊C₄	—	4	4	no	—
58	C₄⋊C₄	—	4	4	no	—
59	C₄²	—	2	4	no	—
60	C₄²	—	2	2	no	—
61	C₄²	—	4	2	no	—
62	C₄²	—	4	2	no	—
63	C₄²	C₂×C₈	1	2	yes	—
64	C₄²	M₁₆	1	2	yes	—
65	C₄²	SD₁₆	1	4	yes	—
66	C₄²	Q₁₆	1	4	yes	—
67	C₄²	C₂×D₄	1	2	yes	—
68	C₄²	#41	1	2	yes	—
69	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	Q₈	—	2	8	no	—
71	Q₈	—	4	4	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	4	4	no	—
80	C₂×C₄	C₂²⋊C₈	1	4	yes	—
81	C₂×C₄	#103	1	4	yes	—
82	C₂×C₄	#103	1	4	yes	—
83	C₂×C₄	#104	1	4	yes	—
84	C₂×C₄	C₄×D₄	1	2	yes	—
85	C₂×C₄	#121	1	2	yes	—
86	C₂×C₄	#124	1	1	yes	characteristic
87	C₂×C₄	#127	1	1	yes	characteristic
88	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄	#131	1	2	yes	—
91	C₂×C₄	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄	C₂×Q₁₆	1	2	yes	—
93	C₈	—	2	8	no	—
94	C₈	—	8	2	no	—
95	C₂²	#390	1	1	yes	characteristic
96	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²	#408	1	1	yes	characteristic
98	C₂²	C₂×#103	1	2	yes	—
99	C₂²	C₂×#104	1	2	yes	—
100	C₄	—	2	2	no	—
101	C₄	—	2	4	no	—
102	C₄	—	4	4	no	—
103	C₄	—	8	2	no	—
104	C₄	Q₈⋊C₈	1	8	yes	complemented
105	C₄	C₈×D₄	1	2	yes	—
106	C₄	#436	1	2	yes	—
107	C₄	#487	1	2	yes	—
108	C₄	#492	1	1	yes	characteristic
109	C₄	#494	1	1	yes	characteristic
110	C₂	C₂×Q₈⋊C₈	1	2	yes	—
111	C₂	#1493	1	2	yes	—
112	C₂	#1954	1	1	yes	characteristic
113	C₂	#1956	1	1	yes	characteristic
114	C₂	#1962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊Q₈

C₂

yes

maximal, characteristic

#1842

C₂

yes

maximal, characteristic

C₄²×C₈

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₈

C₂

yes

maximal

C₂×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

—

#391

C₄

yes

—

#389

—

#384

C₄

yes

characteristic

C₄×C₄⋊C₄

—

C₄³

C₂²

yes

characteristic

Q₈⋊C₈

—

C₂²×Q₈

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₈

yes

complemented

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₈

yes

—

C₂²×C₄

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₈

yes

—

C₄²

M₁₆

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

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

#103

yes

—

C₂×C₄

#103

yes

—

C₂×C₄

#104

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

#121

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

#131

yes

—

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₈

—

C₈

—

C₂²

#390

yes

characteristic

C₂²

C₂×C₂²⋊C₈

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×#104

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

Q₈⋊C₈

yes

complemented

105

C₄

C₈×D₄

yes

—

106

C₄

#436

yes

—

107

C₄

#487

yes

—

108

C₄

#492

yes

characteristic

109

C₄

#494

yes

characteristic

110

C₂

C₂×Q₈⋊C₈

yes

—

111

C₂

#1493

yes

—

112

C₂

#1954

yes

characteristic

113

C₂

#1956

yes

characteristic

114

C₂

#1962

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₂×C₂²⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
21. C₄⋊Q₈	C₈	semidirect
104. C₄	Q₈⋊C₈	semidirect

H ≅ G/N

type

21. C₄⋊Q₈

C₈

semidirect

104. C₄

Q₈⋊C₈

semidirect

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

subgroup	index
C₄×C₄⋊Q₈	2
#1842	2
C₄²×C₈	2
C₂×Q₈⋊C₈	2

subgroup

index

C₄×C₄⋊Q₈

#1842

C₄²×C₈

C₂×Q₈⋊C₈

Subgroups — #9637

Conjugacy classes of subgroups

Complements

Maximal subgroups