Group #9600 — subgroups

Subgroups — #9600

Order 256.

189 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 87 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	#1939	C₂	1	2	yes	maximal
2	C₄×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×#322	C₂	1	1	yes	maximal, characteristic
5	#1279	C₂	1	2	yes	maximal
6	C₂×#106	C₄	1	2	yes	—
7	C₂×C₄⋊C₈	—	2	2	no	—
8	C₂×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄	C₂²	1	2	yes	—
13	#340	C₄	1	1	yes	characteristic
14	#340	C₄	1	1	yes	characteristic
15	#336	—	2	2	no	—
16	#335	—	2	2	no	—
17	#322	—	2	2	no	—
18	C₂²×C₈	—	2	1	no	—
19	C₂²×C₈	—	2	2	no	—
20	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
26	C₂×C₄²	—	2	2	no	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	#106	—	2	4	no	—
29	C₄⋊C₈	—	2	2	no	—
30	C₄⋊C₈	—	2	2	no	—
31	C₄⋊C₈	—	4	4	no	—
32	C₄×C₈	—	2	1	no	—
33	C₄×C₈	—	2	1	no	—
34	C₄×C₈	—	2	1	no	—
35	C₄×C₈	D₄	1	2	yes	—
36	C₄×C₈	Q₈	1	2	yes	—
37	C₄×C₈	Q₈	1	2	yes	—
38	#95	—	2	2	no	—
39	C₂²×C₄	—	2	2	no	—
40	C₂²×C₄	—	2	2	no	—
41	C₂²×C₄	C₂²×C₄	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	2	no	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	4	2	no	—
54	C₂×C₈	—	4	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	C₄⋊C₄	1	2	yes	—
57	C₂×C₈	C₄⋊C₄	1	2	yes	—
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	M₁₆	1	4	yes	—
62	C₄²	—	4	2	no	—
63	C₄²	—	4	2	no	—
64	C₄²	C₂×D₄	1	1	yes	characteristic
65	C₄²	C₂×Q₈	1	1	yes	characteristic
66	C₄²	C₂×Q₈	1	1	yes	characteristic
67	C₄²	#41	1	1	yes	characteristic
68	C₂³	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄	#121	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×M₁₆	1	2	yes	—
84	C₂×C₄	#131	1	2	yes	—
85	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
86	C₂×C₄	#137	1	2	yes	—
87	C₈	—	2	4	no	—
88	C₈	—	2	2	no	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	4	1	no	—
92	C₈	—	8	2	no	—
93	C₂²	#389	1	1	yes	characteristic
94	C₂²	#435	1	2	yes	—
95	C₂²	#440	1	2	yes	—
96	C₂²	#443	1	1	yes	characteristic
97	C₂²	#443	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	4	4	no	—
101	C₄	—	8	2	no	—
102	C₄	#423	1	1	yes	characteristic
103	C₄	#424	1	1	yes	characteristic
104	C₄	#446	1	2	yes	—
105	C₄	#469	1	1	yes	characteristic
106	C₄	#470	1	1	yes	characteristic
107	C₄	#501	1	2	yes	—
108	C₂	#1585	1	2	yes	—
109	C₂	#1588	1	2	yes	—
110	C₂	#1940	1	1	yes	characteristic
111	C₂	#1944	1	1	yes	characteristic
112	C₂	#1945	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1939

C₂

yes

maximal

C₄×#106

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

#1279

C₂

yes

maximal

C₂×#106

C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

#340

C₄

yes

characteristic

#340

C₄

yes

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

yes

—

C₄×C₈

Q₈

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄

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

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

#389

yes

characteristic

C₂²

#435

yes

—

C₂²

#440

yes

—

C₂²

yes

C₂²

yes

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#423

yes

characteristic

103

C₄

#424

yes

characteristic

104

C₄

#446

yes

—

105

C₄

#469

yes

characteristic

106

C₄

#470

yes

characteristic

107

C₄

#501

yes

—

108

C₂

#1585

yes

—

109

C₂

#1588

yes

—

110

C₂

#1940

yes

characteristic

111

C₂

#1944

yes

characteristic

112

C₂

#1945

yes

characteristic

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

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#1939	2
C₄×#106	2
C₂×#335	2
C₂×#322	2
#1279	2

subgroup

index

Subgroups — #9600

Conjugacy classes of subgroups

Maximal subgroups