Group #8296 — subgroups

Subgroups — #8296

Order 256.

171 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 95 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×#108	C₂	1	1	yes	maximal, characteristic
2	#1564	C₂	1	1	yes	maximal, characteristic
3	#1451	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#322	C₂	1	1	yes	maximal, characteristic
6	#1278	C₂	1	1	yes	maximal, characteristic
7	#1278	C₂	1	1	yes	maximal, characteristic
8	#431	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	C₂×#108	C₄	1	2	yes	—
11	C₄×M₁₆	C₂²	1	1	yes	characteristic
12	C₄×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#341	C₄	1	2	yes	—
17	#335	C₄	1	2	yes	—
18	#334	C₄	1	2	yes	—
19	#322	—	2	1	no	—
20	#322	—	2	1	no	—
21	#322	—	2	1	no	—
22	#322	C₄	1	2	yes	—
23	#322	C₄	1	2	yes	—
24	C₈²	—	2	1	no	—
25	C₂×M₁₆	C₂×C₄	1	2	yes	—
26	C₂×M₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	Q₈	1	2	yes	—
31	C₂×C₄²	C₂³	1	1	yes	characteristic
32	#108	—	2	4	no	—
33	C₄⋊C₈	C₂×C₄	1	2	yes	—
34	C₄⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²⋊C₈	—	2	1	no	—
36	C₂²⋊C₈	—	2	1	no	—
37	#97	—	2	1	no	—
38	#97	—	2	1	no	—
39	C₄×C₈	—	2	1	no	—
40	C₄×C₈	—	2	1	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₈	D₄	1	1	yes	characteristic
51	C₄×C₈	D₄	1	1	yes	characteristic
52	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄	#41	1	1	yes	characteristic
55	M₁₆	—	2	2	no	—
56	M₁₆	—	2	2	no	—
57	M₁₆	—	2	2	no	—
58	M₁₆	M₁₆	1	4	yes	—
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	C₄²	1	1	yes	characteristic
69	C₂×C₈	C₄²	1	1	yes	characteristic
70	C₂×C₈	C₄²	1	1	yes	characteristic
71	C₂×C₈	C₄²	1	1	yes	characteristic
72	C₂×C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₈	C₄⋊C₄	1	2	yes	—
74	C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₄²	C₂×D₄	1	1	yes	characteristic
77	C₄²	#41	1	1	yes	characteristic
78	C₂³	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄	#117	1	1	yes	characteristic
84	C₂×C₄	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄	C₂×M₁₆	1	2	yes	—
88	C₂×C₄	#131	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	—	2	1	no	—
91	C₈	—	2	1	no	—
92	C₈	—	2	1	no	—
93	C₈	—	2	1	no	—
94	C₈	—	2	2	no	—
95	C₈	—	2	2	no	—
96	C₈	—	2	4	no	—
97	C₈	—	4	2	no	—
98	C₈	—	4	2	no	—
99	C₈	—	4	2	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₂²	#446	1	2	yes	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	C₄×M₁₆	1	1	yes	characteristic
107	C₄	#405	1	1	yes	characteristic
108	C₄	#430	1	2	yes	—
109	C₄	#435	1	2	yes	—
110	C₄	#443	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	#1771	1	1	yes	characteristic
113	C₂	#1772	1	1	yes	characteristic
114	C₂	#1778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#108

C₂

yes

maximal, characteristic

#1564

C₂

yes

maximal, characteristic

#1451

C₂

yes

maximal, characteristic

C₈×M₁₆

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

#1278

C₂

yes

maximal, characteristic

#1278

C₂

yes

maximal, characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

C₂×#108

C₄

yes

—

C₄×M₁₆

C₂²

yes

characteristic

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#341

C₄

yes

—

#335

C₄

yes

—

#334

C₄

yes

—

—

—

—

C₄

yes

—

#322

C₄

yes

—

C₈²

—

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

Q₈

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

#131

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₄×C₄⋊C₄

yes

characteristic

103

C₂²

#446

yes

—

104

C₄

—

105

C₄

—

106

C₄

C₄×M₁₆

yes

characteristic

107

C₄

#405

yes

characteristic

108

C₄

#430

yes

—

109

C₄

#435

yes

—

110

C₄

#443

yes

—

111

C₂

—

112

C₂

#1771

yes

characteristic

113

C₂

#1772

yes

characteristic

114

C₂

#1778

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₄×C₄⋊C₄
prime radical	C₂³	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₄²

C₂×D₄

commutator

C₂×C₄

C₂×C₄²

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

C₄×C₄⋊C₄

prime radical

C₂³

C₄×Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄² ◁ 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
C₄×#108	2
#1564	2
#1451	2
C₈×M₁₆	2
C₂×#322	2
#1278	2

subgroup

index

C₈×M₁₆

Subgroups — #8296

Conjugacy classes of subgroups

Maximal subgroups