Group #12832 — subgroups

Subgroups — #12832

Order 256.

179 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 77 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×#504	C₂	1	1	yes	maximal, characteristic
2	#2155	C₂	1	2	yes	maximal, complemented
3	#1813	C₂	1	1	yes	maximal, characteristic
4	#1812	C₂	1	1	yes	maximal, characteristic
5	#1382	C₂	1	2	yes	maximal, complemented
6	C₂×#131	C₂²	1	1	yes	characteristic
7	#504	—	2	4	no	—
8	C₂×M₃₂	C₄	1	2	yes	complemented
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆	C₄	1	2	yes	complemented
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×#108	C₂²	1	2	yes	—
13	#427	C₂²	1	2	yes	—
14	#413	—	2	1	no	—
15	#410	—	2	1	no	—
16	#365	—	2	4	no	—
17	#338	—	2	1	no	—
18	H(3,ℤ/4)	—	2	1	no	—
19	C₂×#41	D₄	1	1	yes	characteristic
20	#131	—	2	2	no	—
21	#131	—	2	2	no	—
22	C₂×M₁₆	—	2	2	no	—
23	C₂×M₁₆	D₄	1	1	yes	characteristic
24	C₂×M₁₆	Q₈	1	2	yes	—
25	C₂²×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈	C₂³	1	1	yes	characteristic
27	#117	—	2	2	no	—
28	M₃₂	—	2	4	no	—
29	M₃₂	D₄	1	4	yes	—
30	C₂×C₁₆	—	2	2	no	—
31	C₂×C₁₆	—	2	4	no	—
32	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆	C₂×C₄	1	2	yes	—
35	#108	—	2	2	no	—
36	#108	—	2	2	no	—
37	#107	—	2	2	no	—
38	#106	—	2	2	no	—
39	#101	—	4	1	no	—
40	#100	—	4	1	no	—
41	#99	—	4	2	no	—
42	#41	—	2	2	no	—
43	#41	—	2	2	no	—
44	C₂×Q₈	SD₁₆	1	1	yes	characteristic
45	C₂×D₄	SD₁₆	1	2	yes	—
46	C₂×D₄	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
49	M₁₆	—	2	2	no	—
50	M₁₆	—	2	4	no	—
51	M₁₆	—	4	2	no	—
52	M₁₆	—	4	2	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	C₄⋊C₄	1	2	yes	—
57	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈	C₂×D₄	1	2	yes	—
59	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈	#41	1	2	yes	—
61	C₄⋊C₄	—	4	2	no	—
62	C₂²⋊C₄	—	4	2	no	—
63	C₄²	—	4	2	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	—	2	1	no	—
66	C₁₆	C₂²⋊C₄	1	4	yes	—
67	C₂³	#106	1	2	yes	—
68	C₂³	C₂×SD₁₆	1	1	yes	characteristic
69	Q₈	—	2	2	no	—
70	D₄	—	2	4	no	—
71	D₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	#106	1	2	yes	—
76	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄	#121	1	2	yes	—
78	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄	#135	1	2	yes	—
80	C₈	—	2	2	no	—
81	C₈	—	8	2	no	—
82	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₈	#117	1	1	yes	characteristic
84	C₈	C₂²⋊Q₈	1	2	yes	—
85	C₂²	—	2	2	no	—
86	C₂²	—	4	1	no	—
87	C₂²	—	4	2	no	—
88	C₂²	C₂×#106	1	1	yes	characteristic
89	C₂²	#465	1	2	yes	—
90	C₄	—	2	2	no	—
91	C₄	—	16	2	no	—
92	C₄	#380	1	1	yes	characteristic
93	C₄	#427	1	1	yes	characteristic
94	C₄	#465	1	2	yes	—
95	C₂	—	2	2	no	—
96	C₂	—	2	1	no	—
97	C₂	—	4	2	no	—
98	C₂	#1807	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#504

C₂

yes

maximal, characteristic

#2155

C₂

yes

maximal, complemented

#1813

C₂

yes

maximal, characteristic

#1812

C₂

yes

maximal, characteristic

#1382

C₂

yes

maximal, complemented

C₂×#131

C₂²

yes

characteristic

#504

—

C₂×M₃₂

C₄

yes

complemented

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₄

yes

complemented

C₂²×C₁₆

C₂²

yes

characteristic

C₂×#108

C₂²

yes

—

#427

C₂²

yes

—

—

—

—

—

—

C₂×#41

D₄

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

Q₈

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂³

yes

characteristic

#117

—

M₃₂

—

M₃₂

D₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₆

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

—

C₂×Q₈

SD₁₆

yes

characteristic

C₂×D₄

SD₁₆

yes

—

C₂×D₄

SD₁₆

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

C₂²⋊C₄

yes

—

C₂³

#106

yes

—

C₂³

C₂×SD₁₆

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#106

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

#121

yes

—

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

#135

yes

—

C₈

—

C₈

—

C₈

C₂×C₂²⋊C₄

yes

characteristic

C₈

#117

yes

characteristic

C₈

C₂²⋊Q₈

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×#106

yes

characteristic

C₂²

#465

yes

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

—

C₂

—

C₂

#1807

yes

characteristic

	subgroup	G/H
center	C₄	#427
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1807
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#427

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂

#1807

prime radical

C₂×#41

D₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2155	C₂	semidirect
5. #1382	C₂	semidirect
8. C₂×M₃₂	C₄	semidirect
10. C₂²×C₁₆	C₄	semidirect

H ≅ G/N

type

2. #2155

C₂

semidirect

5. #1382

C₂

semidirect

8. C₂×M₃₂

C₄

semidirect

10. C₂²×C₁₆

C₄

semidirect

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

subgroup	index
C₂×#504	2
#2155	2
#1813	2
#1812	2
#1382	2

subgroup

index

Subgroups — #12832

Conjugacy classes of subgroups

Complements

Maximal subgroups