Group #70446 — subgroups

Subgroups — #70446

Order 384.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×Dic₃⋊C₈	C₂	1	4	yes	maximal
3	#4932	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#1843	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃	C₂²	1	2	yes	—
9	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
10	C₃×C₄⋊C₈	—	2	2	no	—
11	C₃⋊#95	C₄	1	2	yes	—
12	C₃⋊#95	C₄	1	2	yes	—
13	Dic₃⋊C₈	—	2	8	no	—
14	C₄⋊(C₃⋊C₈)	—	2	2	no	—
15	C₂×C₄⋊C₈	—	3	4	no	—
16	C₂×C₄⋊C₈	—	3	1	no	—
17	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic
18	#376	—	3	1	no	—
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₈	—	2	4	no	—
24	C₂×C₃×C₈	—	2	2	no	—
25	C₃×C₄²	D₄	1	2	yes	—
26	C₃×C₄²	Q₈	1	2	yes	—
27	C₄×Dic₃	D₄	1	4	yes	—
28	C₄×Dic₃	Q₈	1	4	yes	—
29	C₂×C₃⋊C₈	—	2	2	no	—
30	C₂×C₃⋊C₈	—	2	4	no	—
31	C₂²×C₈	—	3	2	no	—
32	C₂²×C₈	C₂×D₃	1	2	yes	—
33	C₂×C₄²	—	3	2	no	—
34	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
35	C₄⋊C₈	—	2	2	no	—
36	C₄⋊C₈	—	6	8	no	—
37	C₄⋊C₈	—	6	2	no	—
38	#95	—	3	2	no	—
39	#95	—	3	2	no	—
40	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄	—	2	2	no	—
42	C₂×C₃×C₄	—	2	2	no	—
43	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
44	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
45	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
46	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
49	C₂×Dic₃	—	2	4	no	—
50	C₂×Dic₃	—	2	4	no	—
51	C₂×Dic₃	C₄⋊C₄	1	8	yes	—
52	C₃×C₈	—	4	2	no	—
53	C₃⋊C₈	—	4	2	no	—
54	C₂²×C₄	—	3	4	no	—
55	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
56	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
57	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	6	4	no	—
61	C₂×C₈	—	6	2	no	—
62	C₄²	—	3	4	no	—
63	C₄²	—	3	4	no	—
64	C₄²	Dic₆	1	2	yes	—
65	C₄²	#67	1	2	yes	—
66	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
67	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₃	#131	1	2	yes	—
69	C₂²×C₃	#131	1	2	yes	—
70	C₃×C₄	—	2	2	no	—
71	C₃×C₄	—	2	2	no	—
72	C₃×C₄	#127	1	1	yes	characteristic
73	C₃×C₄	C₄⋊Q₈	1	2	yes	—
74	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
75	Dic₃	—	2	8	no	—
76	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	3	8	no	—
80	C₂×C₄	—	6	4	no	—
81	C₂×C₄	—	6	4	no	—
82	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
83	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
84	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
85	C₂×C₄	D₃×D₄	1	2	yes	—
86	C₂×C₄	D₃×Q₈	1	2	yes	—
87	C₂×C₄	C₂×#67	1	1	yes	characteristic
88	C₈	—	4	2	no	—
89	C₈	—	12	2	no	—
90	C₂×C₃	#384	1	1	yes	characteristic
91	C₂×C₃	#424	1	4	yes	—
92	C₂×C₃	#424	1	1	yes	characteristic
93	C₂×C₃	#424	1	1	yes	characteristic
94	C₂²	D₃×C₄⋊C₄	1	2	yes	—
95	C₂²	#901	1	2	yes	—
96	C₂²	#907	1	2	yes	—
97	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	6	8	no	—
101	C₄	#888	1	2	yes	—
102	C₄	#940	1	1	yes	characteristic
103	C₄	#944	1	1	yes	characteristic
104	C₃	#1843	1	1	yes	characteristic, complemented, Sylow
105	C₂	#4839	1	4	yes	—
106	C₂	#4975	1	1	yes	characteristic
107	C₂	#5102	1	1	yes	characteristic
108	C₂	#5119	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×Dic₃⋊C₈

C₂

yes

maximal

#4932

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#1843

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

C₃×C₄⋊C₈

—

C₃⋊#95

C₄

yes

—

C₃⋊#95

C₄

yes

—

Dic₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₃

yes

characteristic

#376

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

D₄

yes

—

C₃×C₄²

Q₈

yes

—

C₄×Dic₃

D₄

yes

—

C₄×Dic₃

Q₈

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#95

—

#95

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₄⋊C₄

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

Dic₆

yes

—

C₄²

#67

yes

—

C₂²×C₃

C₂×C₄⋊C₄

yes

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

#131

yes

—

C₂²×C₃

#131

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#127

yes

characteristic

C₃×C₄

C₄⋊Q₈

yes

—

C₃×C₄

C₄⋊Q₈

yes

characteristic

Dic₃

—

C₂³

C₂×C₄×D₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₃⋊C₄

yes

—

C₂×C₄

Dic₃⋊C₄

yes

—

C₂×C₄

C₂×Dic₆

yes

characteristic

C₂×C₄

D₃×D₄

yes

—

C₂×C₄

D₃×Q₈

yes

—

C₂×C₄

C₂×#67

yes

characteristic

C₈

—

C₈

—

C₂×C₃

#384

yes

characteristic

C₂×C₃

#424

yes

—

C₂×C₃

yes

C₂×C₃

yes

C₂²

D₃×C₄⋊C₄

yes

—

C₂²

#901

yes

—

C₂²

#907

yes

—

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

#888

yes

—

102

C₄

#940

yes

characteristic

103

C₄

#944

yes

characteristic

104

C₃

#1843

yes

characteristic, complemented, Sylow

105

C₂

#4839

yes

—

106

C₂

#4975

yes

characteristic

107

C₂

#5102

yes

characteristic

108

C₂

#5119

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator, socle, prime radical	C₂³×C₃	C₂²×C₄
Fitting	C₂×C₃×C₄⋊C₈	C₂
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#1843
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃

commutator, socle, prime radical

C₂³×C₃

C₂²×C₄

Fitting

C₂×C₃×C₄⋊C₈

C₂

cosocle

C₂²×C₃×C₄

C₂³

hypercenter

C₂×C₄⋊C₈

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
104. C₃	#1843	semidirect

H ≅ G/N

type

104. C₃

#1843

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
C₂×Dic₃⋊C₈	2
#4932	2
C₂×C₄⋊(C₃⋊C₈)	2
#1843	3

subgroup

index

C₂×C₃×C₄⋊C₈

C₂×Dic₃⋊C₈

#4932

C₂×C₄⋊(C₃⋊C₈)

#1843

Subgroups — #70446

Conjugacy classes of subgroups

Complements

Maximal subgroups