Group #13263 — subgroups

Subgroups — #13263

Order 256.

199 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#1374	C₂	1	4	yes	maximal
5	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₆	C₂²	1	2	yes	—
8	C₈×Q₈	—	2	2	no	—
9	C₈×Q₈	—	2	2	no	—
10	C₂×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	—	2	2	no	—
15	C₄×C₁₆	—	2	2	no	—
16	C₂²×Q₈	C₈	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	C₄×Q₈	—	2	2	no	—
21	C₄×Q₈	—	2	2	no	—
22	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
23	C₂×C₄⋊C₄	C₈	1	2	yes	—
24	C₂×C₄²	C₂×C₄	1	2	yes	—
25	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	—	2	2	no	—
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	—	2	4	no	—
30	C₂×C₁₆	Q₈	1	4	yes	complemented
31	C₄⋊C₈	—	2	4	no	—
32	C₄⋊C₈	—	2	2	no	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	D₄	1	2	yes	—
36	C₄×C₈	D₄	1	2	yes	—
37	C₂×Q₈	—	2	2	no	—
38	C₂×Q₈	—	2	2	no	—
39	C₂×Q₈	C₁₆	1	4	yes	complemented
40	C₂²×C₄	C₂×C₈	1	2	yes	—
41	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄	C₂×C₈	1	2	yes	—
43	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈	C₂×Q₈	1	2	yes	—
52	C₂×C₈	#41	1	2	yes	—
53	C₄⋊C₄	—	2	4	no	—
54	C₄⋊C₄	—	2	2	no	—
55	C₄²	—	2	2	no	—
56	C₄²	—	2	2	no	—
57	C₄²	C₂²⋊C₄	1	2	yes	—
58	C₄²	M₁₆	1	2	yes	—
59	C₁₆	—	2	4	no	—
60	C₁₆	—	4	2	no	—
61	C₂³	C₂²×C₈	1	1	yes	characteristic
62	Q₈	—	2	8	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄	C₂×C₁₆	1	4	yes	—
71	C₂×C₄	C₂×C₁₆	1	2	yes	—
72	C₂×C₄	M₃₂	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄	C₄×Q₈	1	2	yes	—
75	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄	#131	1	2	yes	—
77	C₈	—	2	2	no	—
78	C₈	—	4	2	no	—
79	C₈	C₂²⋊Q₈	1	2	yes	—
80	C₈	#124	1	1	yes	characteristic
81	C₈	C₄⋊Q₈	1	1	yes	characteristic
82	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂²	C₈×Q₈	1	2	yes	—
84	C₂²	C₂²×C₁₆	1	1	yes	characteristic
85	C₂²	C₂×M₃₂	1	1	yes	characteristic
86	C₂²	#504	1	2	yes	—
87	C₄	—	2	4	no	—
88	C₄	—	2	2	no	—
89	C₄	—	4	2	no	—
90	C₄	C₂²⋊C₁₆	1	4	yes	—
91	C₄	#391	1	1	yes	characteristic
92	C₄	#408	1	1	yes	characteristic
93	C₄	#446	1	2	yes	—
94	C₂	#1984	1	1	yes	characteristic
95	C₂	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
96	C₂	#2113	1	1	yes	characteristic
97	C₂	Q₈×C₁₆	1	2	yes	—
98	C₂	#2183	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic

#1374

C₂

yes

maximal

C₂×C₄×Q₈

C₄

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

C₄×C₁₆

—

C₂²×Q₈

C₈

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₈

yes

characteristic

C₂×C₄⋊C₄

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₈

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₁₆

yes

complemented

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₄²

M₁₆

yes

—

C₁₆

—

C₁₆

—

C₂³

C₂²×C₈

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₂×C₁₆

yes

—

C₂×C₄

C₂×C₁₆

yes

—

C₂×C₄

M₃₂

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

#131

yes

—

C₈

—

C₈

—

C₈

C₂²⋊Q₈

yes

—

C₈

#124

yes

characteristic

C₈

C₄⋊Q₈

yes

characteristic

C₂²

C₂×C₂²⋊C₈

yes

characteristic

C₂²

C₈×Q₈

yes

—

C₂²

C₂²×C₁₆

yes

characteristic

C₂²

C₂×M₃₂

yes

characteristic

C₂²

#504

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₂²⋊C₁₆

yes

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

#1984

yes

characteristic

C₂

C₂×C₂²⋊C₁₆

yes

characteristic

C₂

#2113

yes

characteristic

C₂

Q₈×C₁₆

yes

—

C₂

#2183

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂³
commutator	C₂²	C₂²×C₁₆
socle, prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂²×C₈

C₂³

commutator

C₂²

C₂²×C₁₆

socle, prime radical

C₂³

C₂²×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
30. C₂×C₁₆	Q₈	semidirect
39. C₂×Q₈	C₁₆	semidirect

H ≅ G/N

type

30. C₂×C₁₆

Q₈

semidirect

39. C₂×Q₈

C₁₆

semidirect

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

subgroup	index
C₂×C₈×Q₈	2
C₂×C₄⋊C₁₆	2
C₂×C₄×C₁₆	2
#1374	2

subgroup

index

C₂×C₈×Q₈

C₂×C₄⋊C₁₆

C₂×C₄×C₁₆

#1374

Subgroups — #13263

Conjugacy classes of subgroups

Complements

Maximal subgroups