Group #67656 — subgroups

Subgroups — #67656

Order 384.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	#4925	C₂	1	1	yes	maximal, characteristic
3	C₄×#208	C₂	1	1	yes	maximal, characteristic
4	#1278	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	—	2	1	no	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×#208	C₄	1	2	yes	—
8	C₃×C₄×C₈	—	2	1	no	—
9	C₃×C₄×C₈	C₄	1	2	yes	—
10	#830	—	2	1	no	—
11	C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
12	#803	—	2	1	no	—
13	C₄×C₃⋊C₈	—	2	1	no	—
14	C₄×C₃⋊C₈	—	2	1	no	—
15	#431	—	3	1	no	—
16	C₄×M₁₆	—	3	1	no	—
17	C₂×C₄×C₈	D₃	1	1	yes	characteristic
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
21	C₂×C₃×C₈	—	2	4	no	—
22	C₂×C₃×C₈	—	2	2	no	—
23	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄²	D₄	1	1	yes	characteristic
26	C₃×C₄²	D₄	1	1	yes	characteristic
27	#208	—	2	2	no	—
28	#208	C₈	1	4	yes	complemented
29	C₂×C₃⋊C₈	—	2	2	no	—
30	C₂×C₃⋊C₈	—	2	2	no	—
31	C₂×M₁₆	—	3	2	no	—
32	C₂²×C₈	—	2	1	no	—
33	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
34	C₄⋊C₈	—	3	2	no	—
35	C₂²⋊C₈	—	6	1	no	—
36	#97	—	6	1	no	—
37	C₄×C₈	—	2	1	no	—
38	C₄×C₈	—	6	1	no	—
39	C₄×C₈	—	6	1	no	—
40	C₄×C₈	Dic₃	1	2	yes	—
41	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₃×C₄	—	2	2	no	—
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
51	C₂×C₃×C₄	M₁₆	1	2	yes	—
52	C₃×C₈	—	2	4	no	—
53	C₃⋊C₈	—	2	4	no	—
54	C₃⋊C₈	—	4	2	no	—
55	C₂²×C₄	Dic₆	1	1	yes	characteristic
56	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
57	C₂²×C₄	#67	1	1	yes	characteristic
58	M₁₆	—	3	4	no	—
59	M₁₆	—	6	2	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	6	2	no	—
63	C₂×C₈	—	6	2	no	—
64	C₄²	C₄×D₃	1	1	yes	characteristic
65	C₄²	D₁₂	1	1	yes	characteristic
66	C₄²	C₂×Dic₃	1	1	yes	characteristic
67	C₄²	#67	1	1	yes	characteristic
68	C₂²×C₃	—	2	1	no	—
69	C₂²×C₃	—	2	1	no	—
70	C₂²×C₃	#95	1	1	yes	characteristic
71	C₂²×C₃	C₄⋊C₈	1	2	yes	—
72	C₃×C₄	—	2	1	no	—
73	C₃×C₄	—	2	1	no	—
74	C₃×C₄	C₄×C₈	1	1	yes	characteristic
75	C₃×C₄	#97	1	1	yes	characteristic
76	C₃×C₄	C₂²⋊C₈	1	2	yes	—
77	C₃×C₄	#104	1	2	yes	—
78	C₃×C₄	#108	1	2	yes	—
79	C₂³	Dic₃⋊C₄	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	D₃×C₈	1	2	yes	—
83	C₂×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
84	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
85	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
86	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
87	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
88	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	#217	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	6	4	no	—
92	C₈	—	12	2	no	—
93	C₂×C₃	—	2	2	no	—
94	C₂×C₃	#336	1	1	yes	characteristic
95	C₂×C₃	#339	1	1	yes	characteristic
96	C₂×C₃	#341	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	Dic₃⋊C₈	1	2	yes	—
100	C₂²	C₃⋊#95	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	#805	1	2	yes	—
104	C₄	C₈×Dic₃	1	1	yes	characteristic
105	C₄	#815	1	1	yes	characteristic
106	C₄	#819	1	2	yes	—
107	C₄	(C₂×D₃)⋊C₈	1	2	yes	—
108	C₃	#1278	1	1	yes	characteristic, complemented, Sylow
109	C₂	—	2	2	no	—
110	C₂	#4523	1	1	yes	characteristic
111	C₂	#4550	1	1	yes	characteristic
112	C₂	#4552	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄×C₈

C₂

yes

maximal, characteristic

#4925

C₂

yes

maximal, characteristic

C₄×#208

C₂

yes

maximal, characteristic

#1278

—

maximal, Sylow

C₂²×C₃×C₈

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

C₄

yes

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₄

yes

—

#830

—

C₄⋊(C₃⋊C₈)

C₄

yes

—

#803

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

D₃

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

Q₈

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

#208

—

#208

C₈

yes

complemented

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₃

yes

—

C₂³×C₃

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄²

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

M₁₆

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₄

Dic₆

yes

characteristic

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

#67

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₃

yes

characteristic

C₄²

D₁₂

yes

characteristic

C₄²

C₂×Dic₃

yes

characteristic

C₄²

#67

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#95

yes

characteristic

C₂²×C₃

C₄⋊C₈

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×C₈

yes

characteristic

C₃×C₄

#97

yes

characteristic

C₃×C₄

C₂²⋊C₈

yes

—

C₃×C₄

#104

yes

—

C₃×C₄

#108

yes

—

C₂³

Dic₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃×C₈

yes

—

C₂×C₄

ΓL(1,𝔽₂₅)

yes

—

C₂×C₄

C₄×Dic₃

yes

characteristic

C₂×C₄

Dic₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₃

yes

characteristic

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×C₄

#217

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

C₂²

—

C₂²

—

C₂²

Dic₃⋊C₈

yes

—

100

C₂²

C₃⋊#95

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

#805

yes

—

104

C₄

C₈×Dic₃

yes

characteristic

105

C₄

#815

yes

characteristic

106

C₄

#819

yes

—

107

C₄

(C₂×D₃)⋊C₈

yes

—

108

C₃

#1278

yes

characteristic, complemented, Sylow

109

C₂

—

110

C₂

#4523

yes

characteristic

111

C₂

#4550

yes

characteristic

112

C₂

#4552

yes

characteristic

	subgroup	G/H
center	C₄²	D₁₂
commutator	C₃×C₄	C₄×C₈
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂²×C₃	#95
prime radical	C₂³×C₃	C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#1278
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

D₁₂

commutator

C₃×C₄

C₄×C₈

Frattini

C₂×C₄²

C₂×D₃

Fitting

C₂×C₃×C₄×C₈

C₂

socle

C₂²×C₃

#95

prime radical

C₂³×C₃

C₄⋊C₄

cosocle

C₂×C₃×C₄²

C₂²

hypercenter

C₂×C₄×C₈

D₃

hypocenter

C₃

#1278

LCS

G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
28. #208	C₈	semidirect
108. C₃	#1278	semidirect

H ≅ G/N

type

28. #208

C₈

semidirect

108. C₃

#1278

semidirect

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

subgroup	index
C₂×C₃×C₄×C₈	2
#4925	2
C₄×#208	2
#1278	3

subgroup

index

C₂×C₃×C₄×C₈

#4925

C₄×#208

#1278

Subgroups — #67656

Conjugacy classes of subgroups

Complements

Maximal subgroups