Group #8185 — subgroups

Subgroups — #8185

Order 256.

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

normal subgroups only

This group has 257 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×#131	C₂	1	1	yes	maximal, characteristic
2	#1476	C₂	1	2	yes	maximal
3	#1450	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	#1278	C₂	1	2	yes	maximal, complemented
6	C₂×#131	C₄	1	2	yes	—
7	C₄×#41	C₂²	1	1	yes	characteristic
8	#431	C₂²	1	2	yes	—
9	#405	—	2	2	no	—
10	#405	C₄	1	2	yes	—
11	C₄×M₁₆	—	2	1	no	—
12	C₄×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	Q₈⋊C₈	C₄	1	4	yes	—
17	D₄⋊C₈	C₄	1	4	yes	—
18	#322	—	2	2	no	—
19	#322	—	2	1	no	—
20	C₈²	—	2	1	no	—
21	C₂×#41	C₂×C₄	1	1	yes	characteristic
22	#131	—	2	4	no	—
23	#131	C₈	1	8	yes	complemented
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₈	—	2	2	no	—
27	C₂²×C₈	—	2	2	no	—
28	C₂²×C₈	C₂×C₄	1	2	yes	—
29	C₄×Q₈	C₂×C₄	1	2	yes	—
30	C₄×D₄	—	2	2	no	—
31	C₄×D₄	C₂×C₄	1	2	yes	—
32	#117	—	2	1	no	—
33	#117	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²	—	2	1	no	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	C₄⋊C₈	C₂×C₄	1	4	yes	—
37	C₂²⋊C₈	—	2	2	no	—
38	#97	—	2	2	no	—
39	#97	—	2	1	no	—
40	C₄×C₈	—	2	2	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	—	2	2	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	C₂×C₄	1	4	yes	—
45	C₄×C₈	D₄	1	2	yes	—
46	C₄×C₈	D₄	1	1	yes	characteristic
47	C₄×C₈	D₄	1	1	yes	characteristic
48	#41	—	2	2	no	—
49	#41	C₂×C₈	1	4	yes	—
50	C₂×Q₈	C₄²	1	1	yes	characteristic
51	C₂×D₄	—	2	1	no	—
52	C₂×D₄	C₄²	1	1	yes	characteristic
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄	#41	1	2	yes	—
57	M₁₆	—	2	2	no	—
58	M₁₆	—	2	8	no	—
59	M₁₆	C₂×C₈	1	4	yes	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	8	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	4	4	no	—
65	C₂×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈	C₂×C₈	1	4	yes	—
68	C₂×C₈	M₁₆	1	4	yes	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	C₄²	1	2	yes	—
71	C₂²⋊C₄	—	2	1	no	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	1	no	—
74	C₄²	—	2	2	no	—
75	C₄²	C₂²×C₄	1	2	yes	—
76	C₄²	C₂×D₄	1	1	yes	characteristic
77	C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂³	—	2	1	no	—
79	C₂³	#117	1	1	yes	characteristic
80	Q₈	—	2	1	no	—
81	Q₈	C₄×C₈	1	2	yes	—
82	D₄	—	2	4	no	—
83	D₄	—	2	1	no	—
84	D₄	C₄×C₈	1	2	yes	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₄×D₄	1	2	yes	—
93	C₂×C₄	C₄×D₄	1	2	yes	—
94	C₂×C₄	C₂²×C₈	1	2	yes	—
95	C₂×C₄	C₂×M₁₆	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	4	4	no	—
98	C₈	—	4	4	no	—
99	C₈	C₂²⋊C₈	1	4	yes	—
100	C₈	C₂²⋊C₈	1	4	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	4	2	no	—
104	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²	#431	1	2	yes	—
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	4	2	no	—
109	C₄	C₂×C₄×C₈	1	1	yes	characteristic
110	C₄	C₄×M₁₆	1	1	yes	characteristic
111	C₄	C₂×C₂²⋊C₈	1	2	yes	—
112	C₄	#444	1	4	yes	—
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	C₄×C₂²⋊C₈	1	1	yes	characteristic
116	C₂	#1765	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#131

C₂

yes

maximal, characteristic

#1476

C₂

yes

maximal

#1450

C₂

yes

maximal, characteristic, complemented

C₂×#322

C₂

yes

maximal, characteristic, complemented

#1278

C₂

yes

maximal, complemented

C₂×#131

C₄

yes

—

C₄×#41

C₂²

yes

characteristic

#431

C₂²

yes

—

#405

—

#405

C₄

yes

—

C₄×M₁₆

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

Q₈⋊C₈

C₄

yes

—

D₄⋊C₈

C₄

yes

—

#322

—

#322

—

C₈²

—

C₂×#41

C₂×C₄

yes

characteristic

#131

—

#131

C₈

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₄×Q₈

C₂×C₄

yes

—

C₄×D₄

—

C₄×D₄

C₂×C₄

yes

—

#117

—

#117

C₂×C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

#41

—

#41

C₂×C₈

yes

—

C₂×Q₈

C₄²

yes

characteristic

C₂×D₄

—

C₂×D₄

C₄²

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

#41

yes

—

M₁₆

—

M₁₆

—

M₁₆

C₂×C₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₄²

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₂³

—

C₂³

#117

yes

characteristic

Q₈

—

Q₈

C₄×C₈

yes

—

D₄

—

D₄

—

D₄

C₄×C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₈

—

C₈

—

C₈

—

C₈

C₂²⋊C₈

yes

—

100

C₈

C₂²⋊C₈

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₄×C₂²⋊C₄

yes

characteristic

105

C₂²

#431

yes

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

C₂×C₄×C₈

yes

characteristic

110

C₄

C₄×M₁₆

yes

characteristic

111

C₄

C₂×C₂²⋊C₈

yes

—

112

C₄

#444

yes

—

113

C₂

—

114

C₂

—

115

C₂

C₄×C₂²⋊C₈

yes

characteristic

116

C₂

#1765

yes

—

	subgroup	G/H
center	C₄²	C₂×D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₄×C₂²⋊C₄
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₄²

C₂×D₄

commutator

C₄

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

C₄×C₂²⋊C₄

prime radical

C₂×#41

C₂×C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. #1450	C₂	semidirect
4. C₂×#322	C₂	semidirect
5. #1278	C₂	semidirect
23. #131	C₈	semidirect

H ≅ G/N

type

3. #1450

C₂

semidirect

4. C₂×#322

C₂

semidirect

5. #1278

C₂

semidirect

23. #131

C₈

semidirect

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

subgroup	index
C₄×#131	2
#1476	2
#1450	2
C₂×#322	2
#1278	2

subgroup

index

Subgroups — #8185

Conjugacy classes of subgroups

Complements

Maximal subgroups