Group #13145877 — subgroups

Subgroups — C₃×C₈×C₃⋊(C₃⋊C₈) #13145877

Order 1728.

620 conjugacy classes of subgroups; only the 294 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 294 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	C₃³×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
3	C₈×C₃⋊(C₃⋊C₈)	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×C₈×C₃⋊C₈	—	3	4	no	maximal
5	C₂×C₃³×C₈	C₄	1	2	yes	—
6	C₃³×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	—
8	C₃²×C₄×C₈	—	2	4	no	—
9	C₃²×C₄×C₈	D₃	1	4	yes	—
10	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
11	C₄×C₃⋊(C₃⋊C₈)	C₂×C₃	1	2	yes	—
12	C₃×C₄×C₃⋊C₈	—	3	8	no	—
13	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
14	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
15	C₃³×C₈	C₈	1	4	yes	complemented
16	C₃×C₃⋊(C₃⋊C₈)	C₈	1	8	yes	direct product factor
17	C₃×C₈²	—	9	1	no	—
18	C₈×C₃⋊C₈	—	3	4	no	—
19	C₂×C₃²×C₈	—	2	8	no	—
20	C₂×C₃²×C₈	Dic₃	1	8	yes	—
21	C₂×C₃²×C₈	C₃×C₄	1	2	yes	—
22	C₃²×C₄²	—	2	4	no	—
23	C₃²×C₄²	C₂×D₃	1	4	yes	—
24	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
25	C₂×C₃⋊(C₃⋊C₈)	C₃×C₄	1	4	yes	—
26	C₂×C₃×C₃⋊C₈	—	3	16	no	—
27	C₂²×C₃³	C₄²	1	1	yes	characteristic
28	C₃³×C₄	C₂×C₈	1	4	yes	—
29	C₃³×C₄	C₂×C₈	1	2	yes	—
30	C₃×C₄×C₈	—	2	4	no	—
31	C₃×C₄×C₈	—	9	2	no	—
32	C₃×C₄×C₈	C₃×D₃	1	4	yes	—
33	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
34	C₄×C₃⋊C₈	—	3	8	no	—
35	C₂×C₃²×C₄	—	2	8	no	—
36	C₂×C₃²×C₄	—	2	4	no	—
37	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
38	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
39	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
40	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
41	C₃²×C₈	—	2	16	no	—
42	C₃²×C₈	C₃⋊C₈	1	16	yes	complemented
43	C₃²×C₈	C₃×C₈	1	4	yes	complemented
44	C₃⋊(C₃⋊C₈)	C₃×C₈	1	8	yes	direct product factor
45	C₃×C₃⋊C₈	—	3	32	no	—
46	C₈²	—	9	1	no	Sylow
47	C₂×C₃³	C₄×C₈	1	1	yes	characteristic
48	C₂×C₃³	C₄×C₈	1	2	yes	—
49	C₂×C₃×C₈	—	2	8	no	—
50	C₂×C₃×C₈	—	9	4	no	—
51	C₂×C₃×C₈	C₃×Dic₃	1	8	yes	—
52	C₂×C₃×C₈	C₃⋊Dic₃	1	2	yes	—
53	C₃×C₄²	—	2	4	no	—
54	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
55	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
56	C₂×C₃⋊C₈	—	3	16	no	—
57	C₂²×C₃²	—	2	4	no	—
58	C₂²×C₃²	C₄×Dic₃	1	4	yes	—
59	C₂²×C₃²	C₃×C₄²	1	1	yes	characteristic
60	C₃²×C₄	—	2	16	no	—
61	C₃²×C₄	—	2	8	no	—
62	C₃²×C₄	D₃×C₈	1	16	yes	—
63	C₃²×C₄	C₂×C₃⋊C₈	1	8	yes	—
64	C₃²×C₄	C₂×C₃×C₈	1	4	yes	—
65	C₃²×C₄	C₂×C₃×C₈	1	2	yes	—
66	C₄×C₈	—	9	2	no	—
67	C₄×C₈	C₃×C₃⋊D₃	1	1	yes	characteristic
68	C₃³	C₈²	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₃×C₄	—	2	8	no	—
70	C₂×C₃×C₄	—	2	4	no	—
71	C₂×C₃×C₄	C₃×C₄×D₃	1	8	yes	—
72	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
73	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
74	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
75	C₃×C₈	—	2	16	no	—
76	C₃×C₈	—	9	8	no	—
77	C₃×C₈	C₃×C₃⋊C₈	1	16	yes	complemented
78	C₃×C₈	C₃⋊(C₃⋊C₈)	1	4	yes	direct product factor
79	C₃⋊C₈	—	3	32	no	—
80	C₂×C₃²	—	2	4	no	—
81	C₂×C₃²	—	2	8	no	—
82	C₂×C₃²	C₄×C₃⋊C₈	1	4	yes	—
83	C₂×C₃²	C₈×Dic₃	1	8	yes	—
84	C₂×C₃²	C₃×C₄×C₈	1	1	yes	characteristic
85	C₂×C₃²	C₃×C₄×C₈	1	2	yes	—
86	C₂×C₈	—	9	4	no	—
87	C₂×C₈	C₃×C₃⋊Dic₃	1	2	yes	—
88	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
89	C₂²×C₃	—	2	4	no	—
90	C₂²×C₃	C₃×C₄×Dic₃	1	4	yes	—
91	C₂²×C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic
92	C₃×C₄	—	2	16	no	—
93	C₃×C₄	—	2	8	no	—
94	C₃×C₄	C₃×D₃×C₈	1	16	yes	—
95	C₃×C₄	C₂×C₃×C₃⋊C₈	1	8	yes	—
96	C₃×C₄	C₈×C₃⋊D₃	1	4	yes	—
97	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
98	C₃²	—	2	4	no	—
99	C₃²	C₈×C₃⋊C₈	1	4	yes	complemented
100	C₃²	C₃×C₈²	1	1	yes	characteristic, complemented
101	C₂×C₄	C₃×C₄×C₃⋊D₃	1	2	yes	—
102	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
103	C₈	—	9	8	no	—
104	C₈	C₃×C₃⋊(C₃⋊C₈)	1	4	yes	direct product factor
105	C₂×C₃	—	2	4	no	—
106	C₂×C₃	—	2	8	no	—
107	C₂×C₃	C₃×C₄×C₃⋊C₈	1	4	yes	—
108	C₂×C₃	C₃×C₈×Dic₃	1	8	yes	—
109	C₂×C₃	C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
110	C₂×C₃	C₈×C₃⋊Dic₃	1	2	yes	—
111	C₂²	C₃×C₄×C₃⋊Dic₃	1	1	yes	characteristic
112	C₄	C₃×C₈×C₃⋊D₃	1	4	yes	—
113	C₄	C₂×C₃×C₃⋊(C₃⋊C₈)	1	2	yes	—
114	C₃	—	2	4	no	—
115	C₃	C₃×C₈×C₃⋊C₈	1	4	yes	complemented
116	C₃	C₈×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic, direct product factor
117	C₂	C₃×C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
118	C₂	C₃×C₈×C₃⋊Dic₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×C₄×C₈

C₂

yes

maximal, characteristic

C₃×C₄×C₃⋊(C₃⋊C₈)

C₂

yes

maximal

C₈×C₃⋊(C₃⋊C₈)

C₃

yes

maximal, characteristic, direct product factor

C₃×C₈×C₃⋊C₈

—

maximal

C₂×C₃³×C₈

C₄

yes

—

C₃³×C₄²

C₂²

yes

characteristic

C₂×C₃×C₃⋊(C₃⋊C₈)

C₄

yes

—

C₃²×C₄×C₈

—

C₃²×C₄×C₈

D₃

yes

—

C₃²×C₄×C₈

C₂×C₃

yes

characteristic

C₄×C₃⋊(C₃⋊C₈)

C₂×C₃

yes

—

C₃×C₄×C₃⋊C₈

—

C₂×C₃³×C₄

C₂×C₄

yes

—

C₂×C₃³×C₄

C₂×C₄

yes

characteristic

C₃³×C₈

C₈

yes

complemented

C₃×C₃⋊(C₃⋊C₈)

C₈

yes

direct product factor

C₃×C₈²

—

C₈×C₃⋊C₈

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

Dic₃

yes

—

C₂×C₃²×C₈

C₃×C₄

yes

—

C₃²×C₄²

—

C₃²×C₄²

C₂×D₃

yes

—

C₃²×C₄²

C₂²×C₃

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₃×C₄

yes

—

C₂×C₃×C₃⋊C₈

—

C₂²×C₃³

C₄²

yes

characteristic

C₃³×C₄

C₂×C₈

yes

—

C₃³×C₄

C₂×C₈

yes

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₃×D₃

yes

—

C₃×C₄×C₈

C₃⋊D₃

yes

characteristic

C₄×C₃⋊C₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂×C₃×C₄

yes

—

C₂×C₃²×C₄

C₂×C₃×C₄

yes

characteristic

C₃²×C₈

—

C₃²×C₈

C₃⋊C₈

yes

complemented

C₃²×C₈

C₃×C₈

yes

complemented

C₃⋊(C₃⋊C₈)

C₃×C₈

yes

direct product factor

C₃×C₃⋊C₈

—

C₈²

—

Sylow

C₂×C₃³

C₄×C₈

yes

characteristic

C₂×C₃³

C₄×C₈

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×Dic₃

yes

—

C₂×C₃×C₈

C₃⋊Dic₃

yes

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃×D₃

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₄×Dic₃

yes

—

C₂²×C₃²

C₃×C₄²

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

D₃×C₈

yes

—

C₃²×C₄

C₂×C₃⋊C₈

yes

—

C₃²×C₄

C₂×C₃×C₈

yes

—

C₃²×C₄

C₂×C₃×C₈

yes

—

C₄×C₈

—

C₄×C₈

C₃×C₃⋊D₃

yes

characteristic

C₃³

C₈²

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×C₄×D₃

yes

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₃×C₃⋊C₈

yes

complemented

C₃×C₈

C₃⋊(C₃⋊C₈)

yes

direct product factor

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₄×C₃⋊C₈

yes

—

C₂×C₃²

C₈×Dic₃

yes

—

C₂×C₃²

C₃×C₄×C₈

yes

characteristic

C₂×C₃²

C₃×C₄×C₈

yes

—

C₂×C₈

—

C₂×C₈

C₃×C₃⋊Dic₃

yes

—

C₄²

C₂×C₃×C₃⋊D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

C₃×C₄×Dic₃

yes

—

C₂²×C₃

C₄×C₃⋊Dic₃

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₃×D₃×C₈

yes

—

C₃×C₄

C₂×C₃×C₃⋊C₈

yes

—

C₃×C₄

C₈×C₃⋊D₃

yes

—

C₃×C₄

C₂×C₃⋊(C₃⋊C₈)

yes

—

C₃²

—

C₃²

C₈×C₃⋊C₈

yes

complemented

100

C₃²

C₃×C₈²

yes

characteristic, complemented

101

C₂×C₄

C₃×C₄×C₃⋊D₃

yes

—

102

C₂×C₄

C₂×C₃×C₃⋊Dic₃

yes

characteristic

103

C₈

—

104

C₈

C₃×C₃⋊(C₃⋊C₈)

yes

direct product factor

105

C₂×C₃

—

106

C₂×C₃

—

107

C₂×C₃

C₃×C₄×C₃⋊C₈

yes

—

108

C₂×C₃

C₃×C₈×Dic₃

yes

—

109

C₂×C₃

C₄×C₃⋊(C₃⋊C₈)

yes

characteristic

110

C₂×C₃

C₈×C₃⋊Dic₃

yes

—

111

C₂²

C₃×C₄×C₃⋊Dic₃

yes

characteristic

112

C₄

C₃×C₈×C₃⋊D₃

yes

—

113

C₄

C₂×C₃×C₃⋊(C₃⋊C₈)

yes

—

114

C₃

—

115

C₃

C₃×C₈×C₃⋊C₈

yes

complemented

116

C₃

C₈×C₃⋊(C₃⋊C₈)

yes

characteristic, direct product factor

117

C₂

C₃×C₄×C₃⋊(C₃⋊C₈)

yes

characteristic

118

C₂

C₃×C₈×C₃⋊Dic₃

yes

—

	subgroup	G/H
center, hypercenter	C₃×C₄×C₈	C₃⋊D₃
commutator, hypocenter	C₃²	C₃×C₈²
Frattini	C₄²	C₂×C₃×C₃⋊D₃
Fitting	C₃³×C₄×C₈	C₂
socle, prime radical	C₂²×C₃³	C₄²
cosocle	C₃²×C₄²	C₂²×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃×C₄×C₈

subgroup

G/H

center, hypercenter

C₃×C₄×C₈

C₃⋊D₃

commutator, hypocenter

C₃²

C₃×C₈²

Frattini

C₄²

C₂×C₃×C₃⋊D₃

Fitting

C₃³×C₄×C₈

C₂

socle, prime radical

C₂²×C₃³

C₄²

cosocle

C₃²×C₄²

C₂²×C₃

LCS

G ▷ C₃²

UCS

C₁ ◁ C₃×C₄×C₈

N	H ≅ G/N	type
3. C₈×C₃⋊(C₃⋊C₈)	C₃	direct (N × H)
15. C₃³×C₈	C₈	semidirect
16. C₃×C₃⋊(C₃⋊C₈)	C₈	direct (N × H)
42. C₃²×C₈	C₃⋊C₈	semidirect
43. C₃²×C₈	C₃×C₈	semidirect
44. C₃⋊(C₃⋊C₈)	C₃×C₈	direct (N × H)
68. C₃³	C₈²	semidirect
77. C₃×C₈	C₃×C₃⋊C₈	semidirect
78. C₃×C₈	C₃⋊(C₃⋊C₈)	direct (N × H)
99. C₃²	C₈×C₃⋊C₈	semidirect
100. C₃²	C₃×C₈²	semidirect
104. C₈	C₃×C₃⋊(C₃⋊C₈)	direct (N × H)
115. C₃	C₃×C₈×C₃⋊C₈	semidirect
116. C₃	C₈×C₃⋊(C₃⋊C₈)	direct (N × H)

H ≅ G/N

type

3. C₈×C₃⋊(C₃⋊C₈)

C₃

direct (N × H)

15. C₃³×C₈

C₈

semidirect

16. C₃×C₃⋊(C₃⋊C₈)

C₈

direct (N × H)

42. C₃²×C₈

C₃⋊C₈

semidirect

43. C₃²×C₈

C₃×C₈

semidirect

44. C₃⋊(C₃⋊C₈)

C₃×C₈

direct (N × H)

68. C₃³

C₈²

semidirect

77. C₃×C₈

C₃×C₃⋊C₈

semidirect

78. C₃×C₈

C₃⋊(C₃⋊C₈)

direct (N × H)

99. C₃²

C₈×C₃⋊C₈

semidirect

100. C₃²

C₃×C₈²

semidirect

104. C₈

C₃×C₃⋊(C₃⋊C₈)

direct (N × H)

115. C₃

C₃×C₈×C₃⋊C₈

semidirect

116. C₃

C₈×C₃⋊(C₃⋊C₈)

direct (N × H)

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

subgroup	index
C₃³×C₄×C₈	2
C₃×C₄×C₃⋊(C₃⋊C₈)	2
C₈×C₃⋊(C₃⋊C₈)	3
C₃×C₈×C₃⋊C₈	3

subgroup

index

C₃³×C₄×C₈

C₃×C₄×C₃⋊(C₃⋊C₈)

C₈×C₃⋊(C₃⋊C₈)

C₃×C₈×C₃⋊C₈

Subgroups — C₃×C₈×C₃⋊(C₃⋊C₈) #13145877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C8×C3⋊(C3⋊C8) #13145877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×C₃⋊(C₃⋊C₈) #13145877