Group #14515892 — subgroups

Subgroups — 3₋¹⁺²⋊(Q₈⋊C₉) #14515892

Order 1944.

161 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 46 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	C₃×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃²×Q₈⋊C₉	C₃	1	1	yes	maximal, characteristic
3	#10619584	C₃	1	1	yes	maximal, characteristic
4	#10619584	C₃	1	1	yes	maximal, characteristic
5	C₂×3₋¹⁺²⋊C₉	—	4	1	no	maximal
6	C₃×C₄×3₋¹⁺²	—	3	1	no	—
7	3₋¹⁺²⋊C₉	—	4	1	no	Sylow
8	C₃³×Q₈	C₃²	1	1	yes	characteristic
9	Q₈×3₋¹⁺²	—	3	2	no	—
10	Q₈×3₋¹⁺²	C₉	1	3	yes	complemented
11	C₃×Q₈×C₉	—	3	1	no	—
12	C₃×Q₈⋊C₉	—	3	1	no	—
13	C₃×Q₈⋊C₉	—	3	3	no	—
14	C₃×Q₈⋊C₉	—	3	1	no	—
15	C₃×Q₈⋊C₉	—	3	1	no	—
16	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
17	C₂×C₃²×C₉	—	4	1	no	—
18	C₂×C₃²⋊C₉	—	4	1	no	—
19	C₂×C₃²⋊C₉	—	4	1	no	—
20	C₃³×C₄	—	3	1	no	—
21	C₄×3₋¹⁺²	—	3	6	no	—
22	C₄×3₋¹⁺²	—	3	3	no	—
23	C₃×C₄×C₉	—	3	3	no	—
24	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic
25	C₃²×C₉	—	4	1	no	—
26	C₃²⋊C₉	—	4	1	no	—
27	C₃²⋊C₉	—	4	1	no	—
28	C₃²×Q₈	—	3	1	no	—
29	C₃²×Q₈	—	3	1	no	—
30	C₃²×Q₈	—	3	1	no	—
31	C₃²×Q₈	C₃×C₉	1	1	yes	characteristic
32	C₃²×Q₈	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃²×Q₈	3₋¹⁺²	1	2	yes	—
34	Q₈×C₉	—	3	3	no	—
35	Q₈⋊C₉	—	3	3	no	—
36	Q₈⋊C₉	—	9	1	no	—
37	Q₈⋊C₉	—	9	1	no	—
38	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
39	C₂×3₋¹⁺²	—	3	2	no	—
40	C₂×3₋¹⁺²	C₂²⋊C₉	1	3	yes	—
41	C₂×C₃×C₉	—	3	1	no	—
42	C₂×C₃×C₉	—	12	1	no	—
43	C₂×C₃×C₉	—	12	3	no	—
44	C₂×C₃×C₉	—	12	1	no	—
45	C₂×C₃×C₉	—	12	1	no	—
46	C₃²×C₄	—	3	1	no	—
47	C₃²×C₄	—	3	1	no	—
48	C₃²×C₄	—	3	2	no	—
49	C₃²×C₄	—	9	1	no	—
50	C₃²×C₄	—	9	1	no	—
51	C₃²×C₄	—	9	1	no	—
52	C₄×C₉	—	3	9	no	—
53	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
54	3₋¹⁺²	—	3	2	no	—
55	3₋¹⁺²	Q₈⋊C₉	1	3	yes	complemented
56	C₃×C₉	—	3	1	no	—
57	C₃×C₉	—	12	1	no	—
58	C₃×C₉	—	12	3	no	—
59	C₃×C₉	—	12	1	no	—
60	C₃×C₉	—	12	1	no	—
61	C₃×Q₈	—	3	1	no	—
62	C₃×Q₈	—	3	2	no	—
63	C₃×Q₈	C₃²⋊C₉	1	1	yes	characteristic
64	C₃×Q₈	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
65	C₃×Q₈	#731	1	1	yes	characteristic
66	C₃×Q₈	#733	1	1	yes	characteristic
67	C₂×C₃²	—	3	1	no	—
68	C₂×C₃²	—	3	1	no	—
69	C₂×C₃²	—	3	1	no	—
70	C₂×C₃²	C₃×C₂²⋊C₉	1	1	yes	characteristic
71	C₂×C₃²	#1094	1	2	yes	—
72	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
73	C₂×C₉	—	3	3	no	—
74	C₂×C₉	—	12	3	no	—
75	C₂×C₉	—	36	1	no	—
76	C₂×C₉	—	36	1	no	—
77	C₃×C₄	—	3	1	no	—
78	C₃×C₄	—	3	1	no	—
79	C₃×C₄	—	3	1	no	—
80	C₃×C₄	—	3	1	no	—
81	C₃×C₄	—	9	1	no	—
82	C₃×C₄	—	9	2	no	—
83	C₃²	—	3	1	no	—
84	C₃²	—	3	1	no	—
85	C₃²	—	3	1	no	—
86	C₃²	C₃×Q₈⋊C₉	1	1	yes	characteristic
87	C₃²	#6204	1	2	yes	—
88	C₃²	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
89	C₉	—	3	3	no	—
90	C₉	—	12	3	no	—
91	C₉	—	36	1	no	—
92	C₉	—	36	1	no	—
93	Q₈	3₋¹⁺²⋊C₉	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₃	—	3	1	no	—
95	C₂×C₃	—	3	2	no	—
96	C₂×C₃	#66364	1	1	yes	characteristic
97	C₂×C₃	#66365	1	1	yes	characteristic
98	C₂×C₃	#66366	1	1	yes	characteristic
99	C₂×C₃	C₃²⋊(C₂²⋊C₉)	1	1	yes	characteristic
100	C₄	—	3	1	no	—
101	C₃	—	3	1	no	—
102	C₃	—	3	2	no	—
103	C₃	#10619581	1	1	yes	characteristic
104	C₃	#10619582	1	1	yes	characteristic
105	C₃	#10619583	1	1	yes	characteristic
106	C₃	#10619584	1	1	yes	characteristic
107	C₂	3₋¹⁺²⋊(C₂²⋊C₉)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×Q₈×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃²×Q₈⋊C₉

C₃

yes

maximal, characteristic

#10619584

C₃

yes

maximal, characteristic

#10619584

C₃

yes

maximal, characteristic

C₂×3₋¹⁺²⋊C₉

—

maximal

C₃×C₄×3₋¹⁺²

—

3₋¹⁺²⋊C₉

—

Sylow

C₃³×Q₈

C₃²

yes

characteristic

Q₈×3₋¹⁺²

—

Q₈×3₋¹⁺²

C₉

yes

complemented

C₃×Q₈×C₉

—

C₃×Q₈⋊C₉

—

C₃×Q₈⋊C₉

—

C₃×Q₈⋊C₉

—

C₃×Q₈⋊C₉

—

C₂×C₃×3₋¹⁺²

A₄

yes

characteristic

C₂×C₃²×C₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

C₃³×C₄

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₃×C₄×C₉

—

C₃×3₋¹⁺²

SL(2,𝔽₃)

yes

characteristic

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

C₃×C₉

yes

characteristic

C₃²×Q₈

H(3,𝔽₃)

yes

characteristic

C₃²×Q₈

3₋¹⁺²

yes

—

Q₈×C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

—

C₂×C₃³

C₃×A₄

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂²⋊C₉

yes

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₄×C₉

—

C₃³

C₃×SL(2,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

Q₈⋊C₉

yes

complemented

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₃²⋊C₉

yes

characteristic

C₃×Q₈

AΣL(1,𝔽₂₇)

yes

C₃×Q₈

yes

C₃×Q₈

yes

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃×C₂²⋊C₉

yes

characteristic

C₂×C₃²

#1094

yes

—

C₂×C₃²

C₂²⋊H(3,𝔽₃)

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×Q₈⋊C₉

yes

characteristic

C₃²

#6204

yes

—

C₃²

Q₈⋊H(3,𝔽₃)

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

Q₈

3₋¹⁺²⋊C₉

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

C₃²⋊(C₂²⋊C₉)

yes

characteristic

100

C₄

—

101

C₃

—

102

C₃

—

103

C₃

#10619581

yes

characteristic

104

C₃

#10619582

yes

characteristic

105

C₃

#10619583

yes

characteristic

106

C₃

#10619584

yes

characteristic

107

C₂

3₋¹⁺²⋊(C₂²⋊C₉)

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₂²⋊H(3,𝔽₃)
commutator	C₃²×Q₈	C₃×C₉
Frattini, prime radical	C₂×C₃³	C₃×A₄
Fitting	C₃×Q₈×3₋¹⁺²	C₃
cosocle	C₃³×Q₈	C₃²
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	3₋¹⁺²⋊C₉
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃³ ◁ C₂×C₃×3₋¹⁺²

subgroup

G/H

center, socle

C₂×C₃²

C₂²⋊H(3,𝔽₃)

commutator

C₃²×Q₈

C₃×C₉

Frattini, prime radical

C₂×C₃³

C₃×A₄

Fitting

C₃×Q₈×3₋¹⁺²

C₃

cosocle

C₃³×Q₈

C₃²

hypercenter

C₂×C₃×3₋¹⁺²

A₄

hypocenter

Q₈

3₋¹⁺²⋊C₉

LCS

G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂×C₃² ◁ C₂×C₃³ ◁ C₂×C₃×3₋¹⁺²

N	H ≅ G/N	type
10. Q₈×3₋¹⁺²	C₉	semidirect
55. 3₋¹⁺²	Q₈⋊C₉	semidirect
93. Q₈	3₋¹⁺²⋊C₉	semidirect

H ≅ G/N

type

10. Q₈×3₋¹⁺²

C₉

semidirect

55. 3₋¹⁺²

Q₈⋊C₉

semidirect

93. Q₈

3₋¹⁺²⋊C₉

semidirect

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

subgroup	index
C₃×Q₈×3₋¹⁺²	3
C₃²×Q₈⋊C₉	3
#10619584	3
C₂×3₋¹⁺²⋊C₉	4

subgroup

index

C₃×Q₈×3₋¹⁺²

C₃²×Q₈⋊C₉

#10619584

C₂×3₋¹⁺²⋊C₉

Subgroups — 3₋¹⁺²⋊(Q₈⋊C₉) #14515892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2⋊(Q8⋊C9) #14515892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²⋊(Q₈⋊C₉) #14515892