Group #12876 — subgroups

Subgroups — #12876

Order 256.

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

normal subgroups only

This group has 185 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	C₂×#430	C₂	1	1	yes	maximal, characteristic
2	C₂×#349	C₂	1	1	yes	maximal, characteristic
3	#1811	C₂	1	1	yes	maximal, characteristic
4	#1388	C₂	1	2	yes	maximal, complemented
5	#1385	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆	—	2	1	no	—
7	C₂²×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×M₃₂	C₂²	1	2	yes	—
9	#430	—	2	2	no	—
10	#430	C₄	1	4	yes	—
11	C₂×#108	—	2	1	no	—
12	C₂×#108	C₂²	1	1	yes	characteristic
13	C₂×#108	C₂²	1	1	yes	characteristic
14	C₂×#108	C₂²	1	2	yes	—
15	#407	—	2	1	no	—
16	#349	—	2	2	no	—
17	#341	—	2	1	no	—
18	C₂³×C₄	D₄	1	1	yes	characteristic
19	C₂×M₁₆	—	2	2	no	—
20	C₂×M₁₆	—	2	2	no	—
21	C₂×M₁₆	—	2	4	no	—
22	C₂×M₁₆	—	2	2	no	—
23	C₂×M₁₆	—	2	1	no	—
24	C₂×M₁₆	—	2	1	no	—
25	C₂×M₁₆	—	4	2	no	—
26	C₂×M₁₆	—	4	1	no	—
27	C₂×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×M₁₆	D₄	1	2	yes	—
29	C₂²×C₈	—	4	1	no	—
30	C₂²×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	M₃₂	—	2	2	no	—
33	M₃₂	—	2	2	no	—
34	C₂×C₁₆	—	2	2	no	—
35	#108	—	2	4	no	—
36	#108	—	2	2	no	—
37	#108	—	2	2	no	—
38	#108	—	2	2	no	—
39	#108	C₂×C₄	1	4	yes	—
40	C₂²⋊C₈	—	4	2	no	—
41	C₂⁴	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	—	2	1	no	—
46	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄	SD₁₆	1	2	yes	—
48	C₂²×C₄	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	M₁₆	—	2	4	no	—
51	M₁₆	—	2	4	no	—
52	M₁₆	—	2	4	no	—
53	M₁₆	—	4	4	no	—
54	M₁₆	—	4	2	no	—
55	M₁₆	—	4	1	no	—
56	M₁₆	—	4	1	no	—
57	M₁₆	—	4	1	no	—
58	M₁₆	—	4	1	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	4	4	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈	#41	1	2	yes	—
69	C₂×C₈	#41	1	2	yes	—
70	C₁₆	—	4	2	no	—
71	C₂³	—	2	1	no	—
72	C₂³	—	2	1	no	—
73	C₂³	—	8	1	no	—
74	C₂³	#103	1	2	yes	—
75	C₂³	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	#103	1	2	yes	—
83	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄	#123	1	1	yes	characteristic
85	C₂×C₄	#123	1	1	yes	characteristic
86	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	4	4	no	—
91	C₈	—	8	1	no	—
92	C₈	—	8	1	no	—
93	C₈	#117	1	2	yes	—
94	C₈	#123	1	2	yes	—
95	C₂²	—	2	2	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	2	no	—
98	C₂²	—	4	2	no	—
99	C₂²	—	8	1	no	—
100	C₂²	C₂×#103	1	1	yes	characteristic
101	C₂²	#484	1	1	yes	characteristic
102	C₂²	#484	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	#381	1	1	yes	characteristic
105	C₄	#418	1	1	yes	characteristic
106	C₄	#481	1	1	yes	characteristic
107	C₄	#481	1	1	yes	characteristic
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	—	4	2	no	—
112	C₂	#1824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#430

C₂

yes

maximal, characteristic

C₂×#349

C₂

yes

maximal, characteristic

#1811

C₂

yes

maximal, characteristic

#1388

C₂

yes

maximal, complemented

#1385

C₂

yes

maximal, complemented

C₂²×M₁₆

—

C₂²×M₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

—

#430

—

#430

C₄

yes

—

C₂×#108

—

C₂×#108

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

C₂×#108

C₂²

yes

—

#407

—

#349

—

#341

—

C₂³×C₄

D₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

D₄

yes

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂⁴

Q₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

—

C₂²×C₄

SD₁₆

yes

—

C₂²×C₄

Q₁₆

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#103

yes

—

C₂³

C₂×Q₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#103

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×SD₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#117

yes

—

C₈

#123

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂×#103

yes

characteristic

101

C₂²

#484

yes

characteristic

102

C₂²

#484

yes

characteristic

103

C₄

—

104

C₄

#381

yes

characteristic

105

C₄

#418

yes

characteristic

106

C₄

#481

yes

characteristic

107

C₄

#481

yes

characteristic

108

C₂

—

109

C₂

—

110

C₂

—

111

C₂

—

112

C₂

#1824

yes

characteristic

	subgroup	G/H
center	C₄	#481
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1824
prime radical	C₂⁴	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#481

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂

#1824

prime radical

C₂⁴

Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
4. #1388	C₂	semidirect
5. #1385	C₂	semidirect

H ≅ G/N

type

4. #1388

C₂

semidirect

5. #1385

C₂

semidirect

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

subgroup	index
C₂×#430	2
C₂×#349	2
#1811	2
#1388	2
#1385	2

subgroup

index

Subgroups — #12876

Conjugacy classes of subgroups

Complements

Maximal subgroups