Group #135776 — subgroups

Subgroups — #135776

Order 512.

218 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 106 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	#12891	C₂	1	1	yes	maximal, characteristic
2	#12889	C₂	1	1	yes	maximal, characteristic
3	#12887	C₂	1	2	yes	maximal
4	#11399	C₂	1	1	yes	maximal, characteristic
5	#8790	C₂	1	2	yes	maximal
6	C₄×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	#1848	C₂²	1	2	yes	—
9	#1846	C₂²	1	2	yes	—
10	#1450	C₂²	1	1	yes	characteristic
11	#1383	—	2	4	no	—
12	#1381	—	2	2	no	—
13	#1380	—	2	2	no	—
14	#1311	C₄	1	2	yes	complemented
15	C₈×C₁₆	C₄	1	2	yes	complemented
16	#1286	—	2	4	no	—
17	C₂×M₃₂	D₄	1	2	yes	—
18	C₂²×C₁₆	D₄	1	2	yes	—
19	#431	D₄	1	2	yes	—
20	C₂×#108	—	2	2	no	—
21	C₂×#108	—	2	2	no	—
22	C₂×#107	—	2	2	no	—
23	C₂×#106	—	2	2	no	—
24	#423	—	2	2	no	—
25	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	#382	—	2	2	no	—
27	#346	Q₈	1	2	yes	—
28	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₄×C₁₆	C₂×C₄	1	2	yes	—
30	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₄×C₁₆	Q₈	1	2	yes	—
32	#322	Q₈	1	2	yes	—
33	C₈²	C₂×C₄	1	2	yes	—
34	C₂×M₁₆	—	4	2	no	—
35	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄	—	4	2	no	—
38	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
39	M₃₂	—	2	4	no	—
40	C₂×C₁₆	—	2	4	no	—
41	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
42	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
43	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
44	#108	—	4	4	no	—
45	#108	—	4	4	no	—
46	#107	—	4	4	no	—
47	#106	—	4	4	no	—
48	C₄⋊C₈	—	2	2	no	—
49	C₄⋊C₈	—	4	4	no	—
50	C₂²⋊C₈	—	2	2	no	—
51	C₄×C₈	C₄⋊C₄	1	4	yes	—
52	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
56	#95	—	4	4	no	—
57	C₂²×C₄	—	8	2	no	—
58	C₂²×C₄	#127	1	1	yes	characteristic
59	C₂²×C₄	#135	1	2	yes	—
60	M₁₆	—	8	2	no	—
61	M₁₆	—	8	2	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	8	2	no	—
64	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₈	#135	1	4	yes	—
69	C₂×C₈	#135	1	4	yes	—
70	C₄⋊C₄	—	8	2	no	—
71	C₄⋊C₄	—	8	2	no	—
72	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₄²	C₄⋊Q₈	1	1	yes	characteristic
74	C₄²	#135	1	2	yes	—
75	C₁₆	—	2	2	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	2	4	no	—
78	C₂³	#495	1	1	yes	characteristic
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	16	4	no	—
81	C₂×C₄	—	16	2	no	—
82	C₂×C₄	#384	1	1	yes	characteristic
83	C₂×C₄	#427	1	2	yes	—
84	C₂×C₄	#495	1	2	yes	—
85	C₂×C₄	#495	1	2	yes	—
86	C₂×C₄	#495	1	1	yes	characteristic
87	C₂×C₄	#499	1	2	yes	—
88	C₈	—	4	4	no	—
89	C₈	—	16	2	no	—
90	C₈	#427	1	2	yes	—
91	C₈	#427	1	2	yes	—
92	C₈	#499	1	2	yes	—
93	C₈	#499	1	2	yes	—
94	C₂²	—	2	2	no	—
95	C₂²	#1711	1	2	yes	—
96	C₂²	#1846	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	32	2	no	—
100	C₄	#1711	1	2	yes	—
101	C₄	#1846	1	1	yes	characteristic
102	C₄	#1846	1	1	yes	characteristic
103	C₂	—	2	2	no	—
104	C₂	#8817	1	1	yes	characteristic
105	C₂	#12911	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12891

C₂

yes

maximal, characteristic

#12889

C₂

yes

maximal, characteristic

#12887

C₂

yes

maximal

#11399

C₂

yes

maximal, characteristic

#8790

C₂

yes

maximal

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1846

C₂²

yes

—

#1450

C₂²

yes

—

—

—

C₄

yes

complemented

C₈×C₁₆

C₄

yes

complemented

#1286

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

#431

D₄

yes

—

—

—

—

—

—

C₂×C₄×C₈

C₂³

yes

characteristic

#382

—

#346

Q₈

yes

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

Q₈

yes

—

#322

Q₈

yes

—

C₈²

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

#135

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

#135

yes

—

C₂×C₈

#135

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₄⋊Q₈

yes

characteristic

C₄²

#135

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#495

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#384

yes

characteristic

C₂×C₄

#427

yes

—

C₂×C₄

#495

yes

—

C₂×C₄

#495

yes

—

C₂×C₄

#495

yes

characteristic

C₂×C₄

#499

yes

—

C₈

—

C₈

—

C₈

#427

yes

—

C₈

#427

yes

—

C₈

#499

yes

—

C₈

#499

yes

—

C₂²

—

C₂²

#1711

yes

—

C₂²

#1846

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

#1711

yes

—

101

C₄

#1846

yes

characteristic

102

C₄

#1846

yes

characteristic

103

C₂

—

104

C₂

#8817

yes

characteristic

105

C₂

#12911

yes

—

	subgroup	G/H
center, socle	C₂²	#1846
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1846

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#495

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
14. #1311	C₄	semidirect
15. C₈×C₁₆	C₄	semidirect

H ≅ G/N

type

14. #1311

C₄

semidirect

15. C₈×C₁₆

C₄

semidirect

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

subgroup	index
#12891	2
#12889	2
#12887	2
#11399	2
#8790	2

subgroup

index

Subgroups — #135776

Conjugacy classes of subgroups

Complements

Maximal subgroups