Group #11783009 — subgroups

Subgroups — C₃×C₄²×C₃⋊C₈ #11783009

Order 1152.

945 conjugacy classes of subgroups; only the 666 normal subgroups are shown.

normal subgroups only

This group has 945 conjugacy classes of subgroups — too many to draw. Only its 666 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	C₃²×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₃⋊C₈	C₂	1	6	yes	maximal
3	C₃×C₄²×C₈	—	3	1	no	maximal
4	C₄²×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃²×C₄²	C₄	1	4	yes	—
6	C₂×C₃²×C₄²	C₂²	1	3	yes	—
7	C₂²×C₃×C₃⋊C₈	C₂²	1	4	yes	—
8	C₃×C₄×C₃⋊C₈	C₄	1	24	yes	direct product factor
9	C₂×C₃×C₄×C₈	—	3	6	no	—
10	C₃×C₄³	—	2	1	no	—
11	C₃×C₄³	D₃	1	1	yes	characteristic
12	C₃×C₄³	C₂×C₃	1	1	yes	characteristic
13	C₂×C₄×C₃⋊C₈	C₂×C₃	1	6	yes	—
14	C₂²×C₃²×C₄	C₂×C₄	1	6	yes	—
15	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
16	C₃²×C₄²	C₈	1	16	yes	complemented
17	C₃²×C₄²	C₂×C₄	1	12	yes	—
18	C₂×C₃×C₃⋊C₈	C₂×C₄	1	24	yes	—
19	C₄²×C₈	—	3	1	no	Sylow
20	C₂²×C₃×C₈	—	3	4	no	—
21	C₂×C₃×C₄²	—	2	4	no	—
22	C₂×C₃×C₄²	—	2	3	no	—
23	C₂×C₃×C₄²	Dic₃	1	4	yes	—
24	C₂×C₃×C₄²	C₃×C₄	1	4	yes	—
25	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
26	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
27	C₂²×C₃⋊C₈	C₂²×C₃	1	4	yes	—
28	C₃×C₄×C₈	—	3	24	no	—
29	C₄×C₃⋊C₈	C₃×C₄	1	24	yes	direct product factor
30	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₄	C₄²	1	12	yes	—
32	C₂×C₃²×C₄	C₂×C₈	1	24	yes	—
33	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
34	C₃×C₃⋊C₈	C₄²	1	16	yes	direct product factor
35	C₂×C₄×C₈	—	3	6	no	—
36	C₄³	C₃×D₃	1	1	yes	characteristic
37	C₂²×C₃×C₄	—	2	6	no	—
38	C₂²×C₃×C₄	—	2	1	no	—
39	C₂²×C₃×C₄	C₂×Dic₃	1	6	yes	—
40	C₂²×C₃×C₄	C₂×C₃×C₄	1	6	yes	—
41	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
43	C₂×C₃×C₈	—	3	24	no	—
44	C₃×C₄²	—	2	16	no	—
45	C₃×C₄²	—	2	12	no	—
46	C₃×C₄²	C₃⋊C₈	1	16	yes	direct product factor
47	C₃×C₄²	C₃×C₈	1	16	yes	complemented
48	C₃×C₄²	C₄×D₃	1	12	yes	—
49	C₃×C₄²	C₂×C₃×C₄	1	12	yes	—
50	C₂×C₃⋊C₈	C₂×C₃×C₄	1	24	yes	—
51	C₂²×C₃²	C₂×C₄²	1	3	yes	—
52	C₂²×C₃²	C₂²×C₈	1	4	yes	—
53	C₃²×C₄	C₄×C₈	1	24	yes	complemented
54	C₃²×C₄	C₂×C₄²	1	4	yes	—
55	C₂²×C₈	—	3	4	no	—
56	C₂×C₄²	C₃×Dic₃	1	4	yes	—
57	C₂×C₄²	C₂×C₃×D₃	1	3	yes	—
58	C₄×C₈	—	3	24	no	—
59	C₂³×C₃	—	2	1	no	—
60	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
61	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄	—	2	24	no	—
63	C₂×C₃×C₄	—	2	12	no	—
64	C₂×C₃×C₄	—	2	6	no	—
65	C₂×C₃×C₄	C₂×C₃⋊C₈	1	24	yes	—
66	C₂×C₃×C₄	C₄×Dic₃	1	12	yes	—
67	C₂×C₃×C₄	C₃×C₄²	1	12	yes	—
68	C₂×C₃×C₄	C₂×C₃×C₈	1	24	yes	—
69	C₂×C₃×C₄	C₂×C₄×D₃	1	6	yes	—
70	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
71	C₃×C₈	—	3	16	no	—
72	C₃⋊C₈	C₃×C₄²	1	16	yes	direct product factor
73	C₂×C₃²	C₄³	1	1	yes	characteristic
74	C₂×C₃²	C₂×C₄×C₈	1	6	yes	—
75	C₂²×C₄	C₂×C₃×Dic₃	1	6	yes	—
76	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
77	C₂×C₈	—	3	24	no	—
78	C₄²	C₃×C₃⋊C₈	1	16	yes	direct product factor
79	C₄²	C₃×C₄×D₃	1	12	yes	—
80	C₂²×C₃	—	2	4	no	—
81	C₂²×C₃	—	2	3	no	—
82	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
83	C₂²×C₃	C₂×C₄×Dic₃	1	3	yes	—
84	C₂²×C₃	C₂×C₃×C₄²	1	3	yes	—
85	C₂²×C₃	C₂²×C₃×C₈	1	4	yes	—
86	C₃×C₄	—	2	24	no	—
87	C₃×C₄	—	2	4	no	—
88	C₃×C₄	C₄×C₃⋊C₈	1	24	yes	direct product factor
89	C₃×C₄	C₃×C₄×C₈	1	24	yes	complemented
90	C₃×C₄	C₄²×D₃	1	4	yes	—
91	C₃×C₄	C₂×C₃×C₄²	1	4	yes	—
92	C₃²	C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
93	C₂³	C₂²×C₃×Dic₃	1	1	yes	characteristic
94	C₂×C₄	C₂×C₃×C₃⋊C₈	1	24	yes	—
95	C₂×C₄	C₃×C₄×Dic₃	1	12	yes	—
96	C₂×C₄	C₂×C₃×C₄×D₃	1	6	yes	—
97	C₈	—	3	16	no	—
98	C₂×C₃	—	2	6	no	—
99	C₂×C₃	—	2	1	no	—
100	C₂×C₃	C₂×C₄×C₃⋊C₈	1	6	yes	—
101	C₂×C₃	C₄²×Dic₃	1	1	yes	characteristic
102	C₂×C₃	C₃×C₄³	1	1	yes	characteristic
103	C₂×C₃	C₂×C₃×C₄×C₈	1	6	yes	—
104	C₂²	C₂²×C₃×C₃⋊C₈	1	4	yes	—
105	C₂²	C₂×C₃×C₄×Dic₃	1	3	yes	—
106	C₄	C₃×C₄×C₃⋊C₈	1	24	yes	direct product factor
107	C₄	C₃×C₄²×D₃	1	4	yes	—
108	C₃	—	2	1	no	—
109	C₃	C₄²×C₃⋊C₈	1	1	yes	characteristic, direct product factor
110	C₃	C₃×C₄²×C₈	1	1	yes	characteristic, complemented
111	C₂	C₂×C₃×C₄×C₃⋊C₈	1	6	yes	—
112	C₂	C₃×C₄²×Dic₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₄³

C₂

yes

maximal, characteristic

C₂×C₃×C₄×C₃⋊C₈

C₂

yes

maximal

C₃×C₄²×C₈

—

maximal

C₄²×C₃⋊C₈

C₃

yes

maximal, characteristic, direct product factor

C₂×C₃²×C₄²

C₄

yes

—

C₂×C₃²×C₄²

C₂²

yes

—

C₂²×C₃×C₃⋊C₈

C₂²

yes

—

C₃×C₄×C₃⋊C₈

C₄

yes

direct product factor

C₂×C₃×C₄×C₈

—

C₃×C₄³

—

C₃×C₄³

D₃

yes

characteristic

C₃×C₄³

C₂×C₃

yes

characteristic

C₂×C₄×C₃⋊C₈

C₂×C₃

yes

—

C₂²×C₃²×C₄

C₂×C₄

yes

—

C₂²×C₃²×C₄

C₂³

yes

characteristic

C₃²×C₄²

C₈

yes

complemented

C₃²×C₄²

C₂×C₄

yes

—

C₂×C₃×C₃⋊C₈

C₂×C₄

yes

—

C₄²×C₈

—

Sylow

C₂²×C₃×C₈

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

Dic₃

yes

—

C₂×C₃×C₄²

C₃×C₄

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂²×C₃

yes

—

C₂²×C₃⋊C₈

C₂²×C₃

yes

—

C₃×C₄×C₈

—

C₄×C₃⋊C₈

C₃×C₄

yes

direct product factor

C₂³×C₃²

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

C₄²

yes

—

C₂×C₃²×C₄

C₂×C₈

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

—

C₃×C₃⋊C₈

C₄²

yes

direct product factor

C₂×C₄×C₈

—

C₄³

C₃×D₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₂²×C₃×C₄

C₂×C₃×C₄

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×C₃×C₄

C₂³×C₃

yes

characteristic

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₃⋊C₈

yes

direct product factor

C₃×C₄²

C₃×C₈

yes

complemented

C₃×C₄²

C₄×D₃

yes

—

C₃×C₄²

C₂×C₃×C₄

yes

—

C₂×C₃⋊C₈

C₂×C₃×C₄

yes

—

C₂²×C₃²

C₂×C₄²

yes

—

C₂²×C₃²

C₂²×C₈

yes

—

C₃²×C₄

C₄×C₈

yes

complemented

C₃²×C₄

C₂×C₄²

yes

—

C₂²×C₈

—

C₂×C₄²

C₃×Dic₃

yes

—

C₂×C₄²

C₂×C₃×D₃

yes

—

C₄×C₈

—

C₂³×C₃

—

C₂³×C₃

C₂²×Dic₃

yes

characteristic

C₂³×C₃

C₂²×C₃×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊C₈

yes

—

C₂×C₃×C₄

C₄×Dic₃

yes

—

C₂×C₃×C₄

C₃×C₄²

yes

—

C₂×C₃×C₄

C₂×C₃×C₈

yes

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

C₂²×C₃×C₄

yes

—

C₃×C₈

—

C₃⋊C₈

C₃×C₄²

yes

direct product factor

C₂×C₃²

C₄³

yes

characteristic

C₂×C₃²

C₂×C₄×C₈

yes

—

C₂²×C₄

C₂×C₃×Dic₃

yes

—

C₂²×C₄

C₂²×C₃×D₃

yes

characteristic

C₂×C₈

—

C₄²

C₃×C₃⋊C₈

yes

direct product factor

C₄²

C₃×C₄×D₃

yes

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²×C₃⋊C₈

yes

—

C₂²×C₃

C₂×C₄×Dic₃

yes

—

C₂²×C₃

C₂×C₃×C₄²

yes

—

C₂²×C₃

C₂²×C₃×C₈

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×C₃⋊C₈

yes

direct product factor

C₃×C₄

C₃×C₄×C₈

yes

complemented

C₃×C₄

C₄²×D₃

yes

—

C₃×C₄

C₂×C₃×C₄²

yes

—

C₃²

C₄²×C₈

yes

characteristic, complemented, Sylow

C₂³

C₂²×C₃×Dic₃

yes

characteristic

C₂×C₄

C₂×C₃×C₃⋊C₈

yes

—

C₂×C₄

C₃×C₄×Dic₃

yes

—

C₂×C₄

C₂×C₃×C₄×D₃

yes

—

C₈

—

C₂×C₃

—

C₂×C₃

—

100

C₂×C₃

C₂×C₄×C₃⋊C₈

yes

—

101

C₂×C₃

C₄²×Dic₃

yes

characteristic

102

C₂×C₃

C₃×C₄³

yes

characteristic

103

C₂×C₃

C₂×C₃×C₄×C₈

yes

—

104

C₂²

C₂²×C₃×C₃⋊C₈

yes

—

105

C₂²

C₂×C₃×C₄×Dic₃

yes

—

106

C₄

C₃×C₄×C₃⋊C₈

yes

direct product factor

107

C₄

C₃×C₄²×D₃

yes

—

108

C₃

—

109

C₃

C₄²×C₃⋊C₈

yes

characteristic, direct product factor

110

C₃

C₃×C₄²×C₈

yes

characteristic, complemented

111

C₂

C₂×C₃×C₄×C₃⋊C₈

yes

—

112

C₂

C₃×C₄²×Dic₃

yes

characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄³	D₃
commutator, hypocenter	C₃	C₃×C₄²×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×C₄³	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₃×C₄³

subgroup

G/H

center, hypercenter

C₃×C₄³

D₃

commutator, hypocenter

C₃

C₃×C₄²×C₈

Frattini

C₂²×C₄

C₂²×C₃×D₃

Fitting

C₃²×C₄³

C₂

socle, prime radical

C₂³×C₃²

C₂²×C₄

cosocle

C₂²×C₃×C₄

C₂³×C₃

LCS

G ▷ C₃

UCS

C₁ ◁ C₃×C₄³

N	H ≅ G/N	type
4. C₄²×C₃⋊C₈	C₃	direct (N × H)
8. C₃×C₄×C₃⋊C₈	C₄	direct (N × H)
16. C₃²×C₄²	C₈	semidirect
29. C₄×C₃⋊C₈	C₃×C₄	direct (N × H)
34. C₃×C₃⋊C₈	C₄²	direct (N × H)
46. C₃×C₄²	C₃⋊C₈	direct (N × H)
47. C₃×C₄²	C₃×C₈	semidirect
53. C₃²×C₄	C₄×C₈	semidirect
72. C₃⋊C₈	C₃×C₄²	direct (N × H)
78. C₄²	C₃×C₃⋊C₈	direct (N × H)
88. C₃×C₄	C₄×C₃⋊C₈	direct (N × H)
89. C₃×C₄	C₃×C₄×C₈	semidirect
92. C₃²	C₄²×C₈	semidirect
106. C₄	C₃×C₄×C₃⋊C₈	direct (N × H)
109. C₃	C₄²×C₃⋊C₈	direct (N × H)
110. C₃	C₃×C₄²×C₈	semidirect

H ≅ G/N

type

4. C₄²×C₃⋊C₈

C₃

direct (N × H)

8. C₃×C₄×C₃⋊C₈

C₄

direct (N × H)

16. C₃²×C₄²

C₈

semidirect

29. C₄×C₃⋊C₈

C₃×C₄

direct (N × H)

34. C₃×C₃⋊C₈

C₄²

direct (N × H)

46. C₃×C₄²

C₃⋊C₈

direct (N × H)

47. C₃×C₄²

C₃×C₈

semidirect

53. C₃²×C₄

C₄×C₈

semidirect

72. C₃⋊C₈

C₃×C₄²

direct (N × H)

78. C₄²

C₃×C₃⋊C₈

direct (N × H)

88. C₃×C₄

C₄×C₃⋊C₈

direct (N × H)

89. C₃×C₄

C₃×C₄×C₈

semidirect

92. C₃²

C₄²×C₈

semidirect

106. C₄

C₃×C₄×C₃⋊C₈

direct (N × H)

109. C₃

C₄²×C₃⋊C₈

direct (N × H)

110. C₃

C₃×C₄²×C₈

semidirect

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

subgroup	index
C₃²×C₄³	2
C₂×C₃×C₄×C₃⋊C₈	2
C₃×C₄²×C₈	3
C₄²×C₃⋊C₈	3

subgroup

index

C₃²×C₄³

C₂×C₃×C₄×C₃⋊C₈

C₃×C₄²×C₈

C₄²×C₃⋊C₈

Subgroups — C₃×C₄²×C₃⋊C₈ #11783009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C42×C3⋊C8 #11783009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×C₃⋊C₈ #11783009