Group #11800644 — subgroups

Subgroups — C₂×C₄²×C₃⋊Dic₃ #11800644

Order 1152.

3816 conjugacy classes of subgroups; only the 1881 normal subgroups are shown.

normal subgroups only

This group has 3816 conjugacy classes of subgroups — too many to draw. Only its 1881 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	C₂²×C₃²×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₃⋊Dic₃	C₂	1	6	yes	maximal
3	C₄²×C₃⋊Dic₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄²×Dic₃	—	3	4	no	maximal
5	C₂³×C₃²×C₄	C₂²	1	3	yes	—
6	C₂³×C₃⋊Dic₃	C₂²	1	4	yes	—
7	C₂×C₃²×C₄²	C₄	1	8	yes	complemented
8	C₂×C₃²×C₄²	C₂²	1	4	yes	—
9	C₂×C₄×C₃⋊Dic₃	C₄	1	48	yes	direct product factor
10	C₂×C₄×C₃⋊Dic₃	C₂²	1	24	yes	—
11	C₂²×C₃×C₄²	D₃	1	4	yes	—
12	C₂²×C₄×Dic₃	—	3	24	no	—
13	C₄²×Dic₃	—	3	32	no	—
14	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂×C₄	1	24	yes	—
16	C₂²×C₃²×C₄	C₂×C₄	1	12	yes	—
17	C₂²×C₃²×C₄	C₂³	1	6	yes	—
18	C₂²×C₃⋊Dic₃	C₂×C₄	1	48	yes	—
19	C₂²×C₃⋊Dic₃	C₂³	1	8	yes	—
20	C₃²×C₄²	C₂×C₄	1	16	yes	complemented
21	C₄×C₃⋊Dic₃	C₂×C₄	1	96	yes	direct product factor
22	C₂×C₄³	—	9	1	no	Sylow
23	C₂³×C₃×C₄	C₂×D₃	1	12	yes	—
24	C₂³×Dic₃	—	3	16	no	—
25	C₂×C₃×C₄²	Dic₃	1	32	yes	complemented
26	C₂×C₃×C₄²	C₂×D₃	1	16	yes	—
27	C₂×C₄×Dic₃	—	3	192	no	—
28	C₂×C₄×Dic₃	—	3	96	no	—
29	C₂³×C₃²	C₂²×C₄	1	12	yes	—
30	C₂³×C₃²	C₂²×C₄	1	2	yes	—
31	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
32	C₂×C₃²×C₄	C₄²	1	48	yes	complemented
33	C₂×C₃²×C₄	C₂²×C₄	1	24	yes	—
34	C₂×C₃²×C₄	C₂²×C₄	1	12	yes	—
35	C₂×C₃⋊Dic₃	C₄²	1	64	yes	direct product factor
36	C₂×C₃⋊Dic₃	C₂²×C₄	1	48	yes	—
37	C₂²×C₄²	—	9	6	no	—
38	C₂²×C₄²	C₃⋊D₃	1	1	yes	characteristic
39	C₄³	—	9	8	no	—
40	C₂⁴×C₃	C₂²×D₃	1	4	yes	—
41	C₂²×C₃×C₄	C₄×D₃	1	96	yes	—
42	C₂²×C₃×C₄	C₂×Dic₃	1	48	yes	—
43	C₂²×C₃×C₄	C₂²×D₃	1	24	yes	—
44	C₂²×Dic₃	—	3	192	no	—
45	C₂²×Dic₃	—	3	32	no	—
46	C₃×C₄²	C₂×Dic₃	1	64	yes	complemented
47	C₄×Dic₃	—	3	384	no	—
48	C₂²×C₃²	C₂×C₄²	1	16	yes	—
49	C₂²×C₃²	C₂×C₄²	1	12	yes	—
50	C₂²×C₃²	C₂³×C₄	1	6	yes	—
51	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
52	C₃²×C₄	C₂×C₄²	1	24	yes	complemented
53	C₃⋊Dic₃	C₂×C₄²	1	32	yes	direct product factor
54	C₂³×C₄	—	9	4	no	—
55	C₂³×C₄	C₂×C₃⋊D₃	1	3	yes	—
56	C₂×C₄²	—	9	48	no	—
57	C₂×C₄²	—	9	24	no	—
58	C₂×C₄²	C₃⋊Dic₃	1	8	yes	direct product factor
59	C₂×C₄²	C₂×C₃⋊D₃	1	4	yes	—
60	C₂³×C₃	C₂×C₄×D₃	1	48	yes	—
61	C₂³×C₃	C₂²×Dic₃	1	8	yes	—
62	C₂³×C₃	C₂³×D₃	1	4	yes	—
63	C₂×C₃×C₄	C₄×Dic₃	1	192	yes	complemented
64	C₂×C₃×C₄	C₂×C₄×D₃	1	96	yes	—
65	C₂×C₃×C₄	C₂²×Dic₃	1	48	yes	—
66	C₂×Dic₃	—	3	256	no	—
67	C₂×Dic₃	—	3	192	no	—
68	C₂×C₃²	C₄³	1	8	yes	complemented
69	C₂×C₃²	C₂²×C₄²	1	4	yes	—
70	C₂×C₃²	C₂²×C₄²	1	3	yes	—
71	C₂⁴	C₂²×C₃⋊D₃	1	1	yes	characteristic
72	C₂²×C₄	—	9	8	no	—
73	C₂²×C₄	—	9	48	no	—
74	C₂²×C₄	C₄×C₃⋊D₃	1	24	yes	—
75	C₂²×C₄	C₂×C₃⋊Dic₃	1	12	yes	—
76	C₂²×C₄	C₂²×C₃⋊D₃	1	6	yes	—
77	C₄²	—	9	96	no	—
78	C₄²	C₂×C₃⋊Dic₃	1	16	yes	direct product factor
79	C₂²×C₃	C₄²×D₃	1	64	yes	—
80	C₂²×C₃	C₂×C₄×Dic₃	1	48	yes	—
81	C₂²×C₃	C₂²×C₄×D₃	1	24	yes	—
82	C₂²×C₃	C₂³×Dic₃	1	4	yes	—
83	C₃×C₄	C₂×C₄×Dic₃	1	96	yes	complemented
84	Dic₃	—	3	128	no	—
85	C₃²	C₂×C₄³	1	1	yes	characteristic, complemented, Sylow
86	C₂³	C₂×C₄×C₃⋊D₃	1	12	yes	—
87	C₂³	C₂²×C₃⋊Dic₃	1	2	yes	—
88	C₂³	C₂³×C₃⋊D₃	1	1	yes	characteristic
89	C₂×C₄	—	9	64	no	—
90	C₂×C₄	—	9	48	no	—
91	C₂×C₄	C₄×C₃⋊Dic₃	1	48	yes	direct product factor
92	C₂×C₄	C₂×C₄×C₃⋊D₃	1	24	yes	—
93	C₂×C₄	C₂²×C₃⋊Dic₃	1	12	yes	—
94	C₂×C₃	C₄²×Dic₃	1	32	yes	complemented
95	C₂×C₃	C₂×C₄²×D₃	1	16	yes	—
96	C₂×C₃	C₂²×C₄×Dic₃	1	12	yes	—
97	C₂²	C₄²×C₃⋊D₃	1	16	yes	—
98	C₂²	C₂×C₄×C₃⋊Dic₃	1	12	yes	—
99	C₂²	C₂²×C₄×C₃⋊D₃	1	6	yes	—
100	C₂²	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
101	C₄	—	9	32	no	—
102	C₄	C₂×C₄×C₃⋊Dic₃	1	24	yes	direct product factor
103	C₃	C₂×C₄²×Dic₃	1	4	yes	complemented
104	C₂	C₄²×C₃⋊Dic₃	1	8	yes	direct product factor
105	C₂	C₂×C₄²×C₃⋊D₃	1	4	yes	—
106	C₂	C₂²×C₄×C₃⋊Dic₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃²×C₄²

C₂

yes

maximal, characteristic

C₂²×C₄×C₃⋊Dic₃

C₂

yes

maximal

C₄²×C₃⋊Dic₃

C₂

yes

maximal, direct product factor

C₂×C₄²×Dic₃

—

maximal

C₂³×C₃²×C₄

C₂²

yes

—

C₂³×C₃⋊Dic₃

C₂²

yes

—

C₂×C₃²×C₄²

C₄

yes

complemented

C₂×C₃²×C₄²

C₂²

yes

—

C₂×C₄×C₃⋊Dic₃

C₄

yes

direct product factor

C₂×C₄×C₃⋊Dic₃

C₂²

yes

—

C₂²×C₃×C₄²

D₃

yes

—

C₂²×C₄×Dic₃

—

C₄²×Dic₃

—

C₂⁴×C₃²

C₂³

yes

characteristic

C₂²×C₃²×C₄

C₂×C₄

yes

—

C₂²×C₃²×C₄

C₂×C₄

yes

—

C₂²×C₃²×C₄

C₂³

yes

—

C₂²×C₃⋊Dic₃

C₂×C₄

yes

—

C₂²×C₃⋊Dic₃

C₂³

yes

—

C₃²×C₄²

C₂×C₄

yes

complemented

C₄×C₃⋊Dic₃

C₂×C₄

yes

direct product factor

C₂×C₄³

—

Sylow

C₂³×C₃×C₄

C₂×D₃

yes

—

C₂³×Dic₃

—

C₂×C₃×C₄²

Dic₃

yes

complemented

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×C₄×Dic₃

—

192

—

C₂×C₄×Dic₃

—

C₂³×C₃²

C₂²×C₄

yes

—

C₂³×C₃²

C₂²×C₄

yes

—

C₂³×C₃²

C₂⁴

yes

characteristic

C₂×C₃²×C₄

C₄²

yes

complemented

C₂×C₃²×C₄

C₂²×C₄

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

—

C₂×C₃⋊Dic₃

C₄²

yes

direct product factor

C₂×C₃⋊Dic₃

C₂²×C₄

yes

—

C₂²×C₄²

—

C₂²×C₄²

C₃⋊D₃

yes

characteristic

C₄³

—

C₂⁴×C₃

C₂²×D₃

yes

—

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂²×Dic₃

—

192

—

C₂²×Dic₃

—

C₃×C₄²

C₂×Dic₃

yes

complemented

C₄×Dic₃

—

384

—

C₂²×C₃²

C₂×C₄²

yes

—

C₂²×C₃²

C₂×C₄²

yes

—

C₂²×C₃²

C₂³×C₄

yes

—

C₂²×C₃²

C₂³×C₄

yes

characteristic

C₃²×C₄

C₂×C₄²

yes

complemented

C₃⋊Dic₃

C₂×C₄²

yes

direct product factor

C₂³×C₄

—

C₂³×C₄

C₂×C₃⋊D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₃⋊Dic₃

yes

direct product factor

C₂×C₄²

C₂×C₃⋊D₃

yes

—

C₂³×C₃

C₂×C₄×D₃

yes

—

C₂³×C₃

C₂²×Dic₃

yes

—

C₂³×C₃

C₂³×D₃

yes

—

C₂×C₃×C₄

C₄×Dic₃

192

yes

complemented

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

C₂²×Dic₃

yes

—

C₂×Dic₃

—

256

—

C₂×Dic₃

—

192

—

C₂×C₃²

C₄³

yes

complemented

C₂×C₃²

C₂²×C₄²

yes

—

C₂×C₃²

C₂²×C₄²

yes

—

C₂⁴

C₂²×C₃⋊D₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×C₃⋊D₃

yes

—

C₂²×C₄

C₂×C₃⋊Dic₃

yes

—

C₂²×C₄

C₂²×C₃⋊D₃

yes

—

C₄²

—

C₄²

C₂×C₃⋊Dic₃

yes

direct product factor

C₂²×C₃

C₄²×D₃

yes

—

C₂²×C₃

C₂×C₄×Dic₃

yes

—

C₂²×C₃

C₂²×C₄×D₃

yes

—

C₂²×C₃

C₂³×Dic₃

yes

—

C₃×C₄

C₂×C₄×Dic₃

yes

complemented

Dic₃

—

128

—

C₃²

C₂×C₄³

yes

characteristic, complemented, Sylow

C₂³

C₂×C₄×C₃⋊D₃

yes

—

C₂³

C₂²×C₃⋊Dic₃

yes

—

C₂³

C₂³×C₃⋊D₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₃⋊Dic₃

yes

direct product factor

C₂×C₄

C₂×C₄×C₃⋊D₃

yes

—

C₂×C₄

C₂²×C₃⋊Dic₃

yes

—

C₂×C₃

C₄²×Dic₃

yes

complemented

C₂×C₃

C₂×C₄²×D₃

yes

—

C₂×C₃

C₂²×C₄×Dic₃

yes

—

C₂²

C₄²×C₃⋊D₃

yes

—

C₂²

C₂×C₄×C₃⋊Dic₃

yes

—

C₂²

C₂²×C₄×C₃⋊D₃

yes

—

100

C₂²

C₂³×C₃⋊Dic₃

yes

characteristic

101

C₄

—

102

C₄

C₂×C₄×C₃⋊Dic₃

yes

direct product factor

103

C₃

C₂×C₄²×Dic₃

yes

complemented

104

C₂

C₄²×C₃⋊Dic₃

yes

direct product factor

105

C₂

C₂×C₄²×C₃⋊D₃

yes

—

106

C₂

C₂²×C₄×C₃⋊Dic₃

yes

—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	C₃⋊D₃
commutator, hypocenter	C₃²	C₂×C₄³
Frattini	C₂³	C₂³×C₃⋊D₃
Fitting	C₂²×C₃²×C₄²	C₂
socle, prime radical	C₂⁴×C₃²	C₂³
cosocle	C₂³×C₃²	C₂⁴
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₄²

subgroup

G/H

center, hypercenter

C₂²×C₄²

C₃⋊D₃

commutator, hypocenter

C₃²

C₂×C₄³

Frattini

C₂³

C₂³×C₃⋊D₃

Fitting

C₂²×C₃²×C₄²

C₂

socle, prime radical

C₂⁴×C₃²

C₂³

cosocle

C₂³×C₃²

C₂⁴

LCS

G ▷ C₃²

UCS

C₁ ◁ C₂²×C₄²

N	H ≅ G/N	type
3. C₄²×C₃⋊Dic₃	C₂	direct (N × H)
7. C₂×C₃²×C₄²	C₄	semidirect
9. C₂×C₄×C₃⋊Dic₃	C₄	direct (N × H)
20. C₃²×C₄²	C₂×C₄	semidirect
21. C₄×C₃⋊Dic₃	C₂×C₄	direct (N × H)
25. C₂×C₃×C₄²	Dic₃	semidirect
32. C₂×C₃²×C₄	C₄²	semidirect
35. C₂×C₃⋊Dic₃	C₄²	direct (N × H)
46. C₃×C₄²	C₂×Dic₃	semidirect
52. C₃²×C₄	C₂×C₄²	semidirect
53. C₃⋊Dic₃	C₂×C₄²	direct (N × H)
58. C₂×C₄²	C₃⋊Dic₃	direct (N × H)
63. C₂×C₃×C₄	C₄×Dic₃	semidirect
68. C₂×C₃²	C₄³	semidirect
78. C₄²	C₂×C₃⋊Dic₃	direct (N × H)
83. C₃×C₄	C₂×C₄×Dic₃	semidirect
85. C₃²	C₂×C₄³	semidirect
91. C₂×C₄	C₄×C₃⋊Dic₃	direct (N × H)
94. C₂×C₃	C₄²×Dic₃	semidirect
102. C₄	C₂×C₄×C₃⋊Dic₃	direct (N × H)
103. C₃	C₂×C₄²×Dic₃	semidirect
104. C₂	C₄²×C₃⋊Dic₃	direct (N × H)

H ≅ G/N

type

3. C₄²×C₃⋊Dic₃

C₂

direct (N × H)

7. C₂×C₃²×C₄²

C₄

semidirect

9. C₂×C₄×C₃⋊Dic₃

C₄

direct (N × H)

20. C₃²×C₄²

C₂×C₄

semidirect

21. C₄×C₃⋊Dic₃

C₂×C₄

direct (N × H)

25. C₂×C₃×C₄²

Dic₃

semidirect

32. C₂×C₃²×C₄

C₄²

semidirect

35. C₂×C₃⋊Dic₃

C₄²

direct (N × H)

46. C₃×C₄²

C₂×Dic₃

semidirect

52. C₃²×C₄

C₂×C₄²

semidirect

53. C₃⋊Dic₃

C₂×C₄²

direct (N × H)

58. C₂×C₄²

C₃⋊Dic₃

direct (N × H)

63. C₂×C₃×C₄

C₄×Dic₃

semidirect

68. C₂×C₃²

C₄³

semidirect

78. C₄²

C₂×C₃⋊Dic₃

direct (N × H)

83. C₃×C₄

C₂×C₄×Dic₃

semidirect

85. C₃²

C₂×C₄³

semidirect

91. C₂×C₄

C₄×C₃⋊Dic₃

direct (N × H)

94. C₂×C₃

C₄²×Dic₃

semidirect

102. C₄

C₂×C₄×C₃⋊Dic₃

direct (N × H)

103. C₃

C₂×C₄²×Dic₃

semidirect

104. C₂

C₄²×C₃⋊Dic₃

direct (N × H)

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

subgroup	index
C₂²×C₃²×C₄²	2
C₂²×C₄×C₃⋊Dic₃	2
C₄²×C₃⋊Dic₃	2
C₂×C₄²×Dic₃	3

subgroup

index

C₂²×C₃²×C₄²

C₂²×C₄×C₃⋊Dic₃

C₄²×C₃⋊Dic₃

C₂×C₄²×Dic₃

Subgroups — C₂×C₄²×C₃⋊Dic₃ #11800644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C42×C3⋊Dic3 #11800644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₃⋊Dic₃ #11800644