Group #70444 — subgroups

Subgroups — #70444

Order 384.

216 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 101 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	#4932	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#4550	C₂	1	4	yes	maximal
5	#1832	—	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	C₄⋊(C₃⋊C₈)	—	2	2	no	—
14	C₂×C₄⋊C₈	—	3	1	no	—
15	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic
16	#376	—	3	1	no	—
17	#336	—	3	4	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃	C₂×C₄	1	4	yes	—
22	C₂×C₃×C₈	—	2	2	no	—
23	C₂×C₃×C₈	—	2	4	no	—
24	C₃×C₄²	D₄	1	2	yes	—
25	C₃×C₄²	Q₈	1	2	yes	—
26	C₄×Dic₃	—	2	2	no	—
27	C₄×Dic₃	—	2	2	no	—
28	C₂×C₃⋊C₈	—	2	2	no	—
29	C₂×C₃⋊C₈	—	2	4	no	—
30	C₂²×C₈	—	3	2	no	—
31	C₂²×C₈	C₂×D₃	1	2	yes	—
32	C₂×C₄²	—	3	2	no	—
33	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
34	C₄⋊C₈	—	2	2	no	—
35	C₄⋊C₈	—	6	2	no	—
36	#95	—	3	2	no	—
37	#95	—	3	2	no	—
38	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄	—	2	2	no	—
40	C₂×C₃×C₄	—	2	2	no	—
41	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
42	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
43	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₃×C₄	#41	1	2	yes	—
46	C₂×C₃×C₄	#41	1	2	yes	—
47	C₂×Dic₃	—	2	4	no	—
48	C₂×Dic₃	—	2	4	no	—
49	C₂×Dic₃	—	2	4	no	—
50	C₃×C₈	—	4	2	no	—
51	C₃⋊C₈	—	4	2	no	—
52	C₂²×C₄	—	3	4	no	—
53	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
54	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
55	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₈	—	2	4	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	6	4	no	—
59	C₂×C₈	—	6	2	no	—
60	C₄²	—	6	2	no	—
61	C₄²	—	6	2	no	—
62	C₄²	Dic₆	1	2	yes	—
63	C₄²	#67	1	2	yes	—
64	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₃	#117	1	2	yes	—
66	C₂²×C₃	#131	1	2	yes	—
67	C₂²×C₃	#131	1	2	yes	—
68	C₃×C₄	—	2	2	no	—
69	C₃×C₄	—	2	2	no	—
70	C₃×C₄	#124	1	1	yes	characteristic
71	C₃×C₄	#124	1	1	yes	characteristic
72	C₃×C₄	#125	1	2	yes	—
73	Dic₃	—	4	4	no	—
74	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	6	4	no	—
78	C₂×C₄	—	6	4	no	—
79	C₂×C₄	—	6	4	no	—
80	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
81	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
82	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
83	C₂×C₄	#237	1	2	yes	—
84	C₂×C₄	#239	1	2	yes	—
85	C₂×C₄	C₂×#67	1	1	yes	characteristic
86	C₈	—	4	2	no	—
87	C₈	—	12	2	no	—
88	C₂×C₃	#382	1	1	yes	characteristic
89	C₂×C₃	#424	1	1	yes	characteristic
90	C₂×C₃	#424	1	1	yes	characteristic
91	C₂×C₃	#433	1	4	yes	—
92	C₂²	#892	1	2	yes	—
93	C₂²	#901	1	2	yes	—
94	C₂²	#907	1	2	yes	—
95	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	2	no	—
98	C₄	—	12	4	no	—
99	C₄	#890	1	2	yes	—
100	C₄	#936	1	1	yes	characteristic
101	C₄	#947	1	1	yes	characteristic
102	C₃	#1832	1	1	yes	characteristic, complemented, Sylow
103	C₂	#4840	1	4	yes	—
104	C₂	#4977	1	1	yes	characteristic
105	C₂	#5102	1	1	yes	characteristic
106	C₂	#5119	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊C₈

C₂

yes

maximal, characteristic

#4932

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#4550

C₂

yes

maximal

#1832

—

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

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₃

yes

characteristic

#376

—

#336

—

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

—

C₄×Dic₃

—

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

—

#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

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

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

characteristic

C₂²×C₃

#117

yes

—

C₂²×C₃

#131

yes

—

C₂²×C₃

#131

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

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

#237

yes

—

C₂×C₄

#239

yes

—

C₂×C₄

C₂×#67

yes

characteristic

C₈

—

C₈

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

—

C₂²

#892

yes

—

C₂²

#901

yes

—

C₂²

#907

yes

—

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#890

yes

—

100

C₄

#936

yes

characteristic

101

C₄

#947

yes

characteristic

102

C₃

#1832

yes

characteristic, complemented, Sylow

103

C₂

#4840

yes

—

104

C₂

#4977

yes

characteristic

105

C₂

#5102

yes

characteristic

106

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	#1832
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
102. C₃	#1832	semidirect

H ≅ G/N

type

102. C₃

#1832

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
#4932	2
C₂×C₄⋊(C₃⋊C₈)	2
#4550	2
#1832	3

subgroup

index

C₂×C₃×C₄⋊C₈

#4932

C₂×C₄⋊(C₃⋊C₈)

#4550

#1832

Subgroups — #70444

Conjugacy classes of subgroups

Complements

Maximal subgroups