Group #8188 — subgroups

Subgroups — #8188

Order 256.

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

normal subgroups only

This group has 206 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	#2922	C₂	1	1	yes	maximal, characteristic
2	#1476	C₂	1	2	yes	maximal
3	#1452	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	C₄×#41	C₂²	1	1	yes	characteristic
7	#433	—	2	2	no	—
8	#431	—	2	2	no	—
9	#431	C₂²	1	2	yes	—
10	#424	C₄	1	2	yes	—
11	#408	C₄	1	2	yes	—
12	C₄×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	Q₈⋊C₈	C₄	1	4	yes	—
16	D₄⋊C₈	C₄	1	4	yes	—
17	#322	—	2	2	no	—
18	#322	—	2	2	no	—
19	C₂×#41	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	C₂×C₄	1	2	yes	—
23	C₄×Q₈	C₂×C₄	1	2	yes	—
24	C₄×D₄	—	2	2	no	—
25	C₄×D₄	C₂×C₄	1	2	yes	—
26	#117	—	2	1	no	—
27	#117	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²	—	2	1	no	—
29	C₂×C₄²	C₂³	1	1	yes	characteristic
30	C₄⋊C₈	—	2	4	no	—
31	C₄⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²⋊C₈	—	2	2	no	—
33	C₂²⋊C₈	—	2	4	no	—
34	#97	—	2	1	no	—
35	C₄×C₈	—	2	2	no	—
36	C₄×C₈	—	2	2	no	—
37	C₄×C₈	—	2	1	no	—
38	C₄×C₈	C₂×C₄	1	4	yes	—
39	C₄×C₈	D₄	1	2	yes	—
40	C₄×C₈	D₄	1	2	yes	—
41	#41	—	2	4	no	—
42	C₂×Q₈	C₄²	1	1	yes	characteristic
43	C₂×D₄	—	2	1	no	—
44	C₂×D₄	C₄²	1	1	yes	characteristic
45	C₂²×C₄	—	2	1	no	—
46	C₂²×C₄	—	2	2	no	—
47	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄	#41	1	2	yes	—
49	M₁₆	—	2	4	no	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	—	4	4	no	—
55	C₂×C₈	C₂²⋊C₄	1	4	yes	—
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	C₄²	1	2	yes	—
58	C₂²⋊C₄	—	2	1	no	—
59	C₂²⋊C₄	—	2	2	no	—
60	C₄²	—	2	1	no	—
61	C₄²	—	2	2	no	—
62	C₄²	C₂²×C₄	1	2	yes	—
63	C₄²	C₂×D₄	1	1	yes	characteristic
64	C₄²	C₂×D₄	1	1	yes	characteristic
65	C₂³	—	2	1	no	—
66	C₂³	#117	1	1	yes	characteristic
67	Q₈	—	2	2	no	—
68	D₄	—	2	2	no	—
69	D₄	—	4	2	no	—
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	4	no	—
75	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄	C₄×D₄	1	2	yes	—
78	C₂×C₄	C₄×D₄	1	2	yes	—
79	C₂×C₄	#131	1	4	yes	—
80	C₈	—	2	4	no	—
81	C₈	—	4	4	no	—
82	C₈	—	4	2	no	—
83	C₈	—	4	4	no	—
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	2	no	—
87	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²	#433	1	2	yes	—
89	C₄	—	2	2	no	—
90	C₄	—	4	1	no	—
91	C₄	—	4	2	no	—
92	C₄	#405	1	2	yes	—
93	C₄	#408	1	2	yes	—
94	C₄	#444	1	4	yes	—
95	C₂	—	2	2	no	—
96	C₂	—	8	1	no	—
97	C₂	#1750	1	1	yes	characteristic
98	C₂	#1765	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2922

C₂

yes

maximal, characteristic

#1476

C₂

yes

maximal

#1452

C₂

yes

maximal, characteristic, complemented

C₂×#322

C₂

yes

maximal, characteristic, complemented

#1278

C₂

yes

maximal, complemented

C₄×#41

C₂²

yes

—

—

C₂²

yes

—

#424

C₄

yes

—

#408

C₄

yes

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

Q₈⋊C₈

C₄

yes

—

D₄⋊C₈

C₄

yes

—

#322

—

#322

—

C₂×#41

C₂×C₄

yes

characteristic

C₂×M₁₆

C₂×C₄

yes

—

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

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

#41

—

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

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

—

D₄

—

D₄

—

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

#131

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₄×C₂²⋊C₄

yes

characteristic

C₂²

#433

yes

—

C₄

—

C₄

—

C₄

—

C₄

#405

yes

—

C₄

#408

yes

—

C₄

#444

yes

—

C₂

—

C₂

—

C₂

#1750

yes

characteristic

C₂

#1765

yes

—

	subgroup	G/H
center	C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₄×C₂²⋊C₄
prime radical	C₂×#41	C₂×C₄
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×C₂²⋊C₄

prime radical

C₂×#41

C₂×C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. #1452	C₂	semidirect
4. C₂×#322	C₂	semidirect
5. #1278	C₂	semidirect

H ≅ G/N

type

3. #1452

C₂

semidirect

4. C₂×#322

C₂

semidirect

5. #1278

C₂

semidirect

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

subgroup	index
#2922	2
#1476	2
#1452	2
C₂×#322	2
#1278	2

subgroup

index

Subgroups — #8188

Conjugacy classes of subgroups

Complements

Maximal subgroups