Group #13270 — subgroups

Subgroups — #13270

Order 256.

158 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 69 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	#2960	C₂	1	1	yes	maximal, characteristic
2	#2153	C₂	1	1	yes	maximal, characteristic
3	#2110	C₂	1	1	yes	maximal, characteristic
4	#1378	C₂	1	2	yes	maximal
5	#1376	C₂	1	2	yes	maximal
6	#533	C₄	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	2	yes	—
8	C₂×M₃₂	C₂²	1	2	yes	—
9	#446	—	2	2	no	—
10	C₈×Q₈	—	2	2	no	—
11	#424	C₄	1	1	yes	characteristic
12	#424	C₄	1	1	yes	characteristic
13	#423	C₄	1	1	yes	characteristic
14	#405	C₂²	1	1	yes	characteristic
15	#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#364	—	2	2	no	—
18	#347	—	2	2	no	—
19	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈	—	2	1	no	—
24	#125	—	2	1	no	—
25	C₂²⋊Q₈	—	2	2	no	—
26	C₄×Q₈	—	2	2	no	—
27	#117	C₂×C₄	1	1	yes	characteristic
28	#117	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
30	M₃₂	—	2	2	no	—
31	M₃₂	—	2	2	no	—
32	M₃₂	—	2	2	no	—
33	M₃₂	Q₈	1	4	yes	complemented
34	C₂×C₁₆	—	2	2	no	—
35	C₂×C₁₆	—	2	2	no	—
36	C₄⋊C₈	—	2	2	no	—
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	#97	—	2	1	no	—
40	#97	D₄	1	2	yes	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₂×Q₈	—	2	2	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
48	M₁₆	—	2	2	no	—
49	M₁₆	—	2	2	no	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈	C₂×Q₈	1	2	yes	—
57	C₂×C₈	#41	1	2	yes	—
58	C₄⋊C₄	—	2	4	no	—
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	2	1	no	—
62	C₂²⋊C₄	—	2	1	no	—
63	C₂²⋊C₄	M₁₆	1	2	yes	—
64	C₄²	—	2	1	no	—
65	C₄²	—	2	1	no	—
66	C₄²	C₂²⋊C₄	1	2	yes	—
67	C₄²	M₁₆	1	1	yes	characteristic
68	C₄²	M₁₆	1	1	yes	characteristic
69	C₁₆	—	2	4	no	—
70	C₁₆	—	4	2	no	—
71	C₂³	C₂×M₁₆	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄	C₄×Q₈	1	2	yes	—
80	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄	#131	1	2	yes	—
82	C₈	—	2	2	no	—
83	C₈	—	4	1	no	—
84	C₈	—	4	1	no	—
85	C₈	C₂²⋊Q₈	1	2	yes	—
86	C₈	#124	1	1	yes	characteristic
87	C₈	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²	—	4	1	no	—
89	C₂²	#407	1	1	yes	characteristic
90	C₂²	#446	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	2	no	—
94	C₄	—	4	2	no	—
95	C₄	#391	1	1	yes	characteristic
96	C₄	#408	1	1	yes	characteristic
97	C₄	#446	1	2	yes	—
98	C₂	—	2	1	no	—
99	C₂	—	2	2	no	—
100	C₂	#1985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2960

C₂

yes

maximal, characteristic

#2153

C₂

yes

maximal, characteristic

#2110

C₂

yes

maximal, characteristic

#1378

C₂

yes

maximal

#1376

C₂

yes

maximal

#533

C₄

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₂×M₃₂

C₂²

yes

—

#446

—

C₈×Q₈

—

#424

C₄

yes

characteristic

#424

C₄

yes

characteristic

#423

C₄

yes

characteristic

#405

C₂²

yes

characteristic

#405

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#364

—

#347

—

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×C₄

yes

characteristic

#117

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

Q₈

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

D₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₄²

M₁₆

yes

characteristic

C₄²

M₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₂×M₁₆

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

—

C₈

C₂²⋊Q₈

yes

—

C₈

#124

yes

characteristic

C₈

C₄⋊Q₈

yes

characteristic

C₂²

—

C₂²

#407

yes

characteristic

C₂²

#446

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

—

100

C₂

#1985

yes

characteristic

	subgroup	G/H
center	C₈	#124
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1985
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

C₈

C₂²×C₄

C₂²×C₈

C₂³

socle

C₂

#1985

prime radical

C₂³

C₂×M₁₆

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
33. M₃₂	Q₈	semidirect

H ≅ G/N

type

33. M₃₂

Q₈

semidirect

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

subgroup	index
#2960	2
#2153	2
#2110	2
#1378	2
#1376	2

subgroup

index

Subgroups — #13270

Conjugacy classes of subgroups

Complements

Maximal subgroups