Group #11708940 — subgroups

Subgroups — C₂⁴×D₃×C₈ #11708940

Order 768.

15454 conjugacy classes of subgroups; only the 8715 normal subgroups are shown.

normal subgroups only

This group has 15454 conjugacy classes of subgroups — too many to draw. Only its 8715 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	C₂⁴×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃×C₈	C₂	1	60	yes	maximal, direct product factor
5	C₂⁵×C₈	—	3	1	no	maximal, Sylow
6	C₂⁵×D₃	C₄	1	1	yes	characteristic
7	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₃	C₄	1	1	yes	characteristic
9	C₂³×C₄×D₃	C₄	1	30	yes	—
10	C₂³×C₄×D₃	C₂²	1	30	yes	—
11	C₂³×C₃×C₈	C₂²	1	30	yes	complemented
12	C₂³×C₃⋊C₈	C₂²	1	30	yes	complemented
13	C₂²×D₃×C₈	C₂²	1	560	yes	direct product factor
14	C₂⁵×C₄	—	3	1	no	—
15	C₂⁴×C₈	—	3	1	no	—
16	C₂⁴×C₈	—	3	60	no	—
17	C₂⁴×C₈	D₃	1	1	yes	characteristic, direct product factor
18	C₂⁵×C₃	C₂×C₄	1	1	yes	characteristic
19	C₂⁴×D₃	C₈	1	32	yes	direct product factor
20	C₂⁴×D₃	C₂×C₄	1	30	yes	—
21	C₂³×C₃×C₄	C₂×C₄	1	15	yes	—
22	C₂³×C₃×C₄	C₂³	1	15	yes	—
23	C₂³×Dic₃	C₂×C₄	1	30	yes	—
24	C₂²×C₄×D₃	C₂×C₄	1	420	yes	—
25	C₂²×C₄×D₃	C₂³	1	140	yes	—
26	C₂²×C₃×C₈	C₂³	1	140	yes	complemented
27	C₂²×C₃⋊C₈	C₂³	1	140	yes	complemented
28	C₂×D₃×C₈	C₂³	1	960	yes	direct product factor
29	C₂⁶	—	3	1	no	—
30	C₂⁴×C₄	—	3	1	no	—
31	C₂⁴×C₄	—	3	30	no	—
32	C₂⁴×C₄	—	3	30	no	—
33	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
34	C₂³×C₈	—	3	30	no	—
35	C₂³×C₈	—	3	560	no	—
36	C₂³×C₈	C₂×D₃	1	30	yes	direct product factor
37	C₂⁴×C₃	C₂×C₈	1	16	yes	complemented
38	C₂⁴×C₃	C₂²×C₄	1	15	yes	—
39	C₂³×D₃	C₂×C₈	1	480	yes	direct product factor
40	C₂³×D₃	C₂²×C₄	1	140	yes	—
41	C₂²×C₃×C₄	C₂²×C₄	1	105	yes	—
42	C₂²×C₃×C₄	C₂⁴	1	35	yes	—
43	C₂²×Dic₃	C₂²×C₄	1	140	yes	—
44	C₂×C₄×D₃	C₂²×C₄	1	840	yes	—
45	C₂×C₄×D₃	C₂⁴	1	120	yes	—
46	C₂×C₃×C₈	C₂⁴	1	120	yes	complemented
47	C₂×C₃⋊C₈	C₂⁴	1	120	yes	complemented
48	D₃×C₈	C₂⁴	1	256	yes	direct product factor
49	C₂⁵	—	3	32	no	—
50	C₂⁵	—	3	30	no	—
51	C₂⁵	C₄×D₃	1	1	yes	characteristic
52	C₂³×C₄	—	3	30	no	—
53	C₂³×C₄	—	3	420	no	—
54	C₂³×C₄	—	3	140	no	—
55	C₂³×C₄	C₄×D₃	1	15	yes	—
56	C₂³×C₄	C₂²×D₃	1	15	yes	—
57	C₂²×C₈	—	3	140	no	—
58	C₂²×C₈	—	3	960	no	—
59	C₂²×C₈	C₂²×D₃	1	140	yes	direct product factor
60	C₂³×C₃	C₂²×C₈	1	120	yes	complemented
61	C₂³×C₃	C₂³×C₄	1	35	yes	—
62	C₂²×D₃	C₂²×C₈	1	1120	yes	direct product factor
63	C₂²×D₃	C₂³×C₄	1	120	yes	—
64	C₂×C₃×C₄	C₂³×C₄	1	105	yes	—
65	C₂×C₃×C₄	C₂⁵	1	15	yes	—
66	C₂×Dic₃	C₂³×C₄	1	120	yes	—
67	C₄×D₃	C₂³×C₄	1	240	yes	—
68	C₄×D₃	C₂⁵	1	16	yes	—
69	C₃×C₈	C₂⁵	1	16	yes	complemented
70	C₃⋊C₈	C₂⁵	1	16	yes	complemented
71	C₂⁴	—	3	480	no	—
72	C₂⁴	—	3	140	no	—
73	C₂⁴	D₃×C₈	1	16	yes	direct product factor
74	C₂⁴	C₂×C₄×D₃	1	15	yes	—
75	C₂²×C₄	—	3	140	no	—
76	C₂²×C₄	—	3	840	no	—
77	C₂²×C₄	—	3	120	no	—
78	C₂²×C₄	C₂×C₄×D₃	1	105	yes	—
79	C₂²×C₄	C₂³×D₃	1	35	yes	—
80	C₂×C₈	—	3	120	no	—
81	C₂×C₈	—	3	256	no	—
82	C₂×C₈	C₂³×D₃	1	120	yes	direct product factor
83	C₂²×C₃	C₂³×C₈	1	140	yes	complemented
84	C₂²×C₃	C₂⁴×C₄	1	15	yes	—
85	C₂×D₃	C₂³×C₈	1	480	yes	direct product factor
86	C₂×D₃	C₂⁴×C₄	1	16	yes	—
87	C₃×C₄	C₂⁴×C₄	1	15	yes	—
88	C₃×C₄	C₂⁶	1	1	yes	characteristic
89	Dic₃	C₂⁴×C₄	1	16	yes	—
90	C₂³	—	3	120	no	—
91	C₂³	—	3	1120	no	—
92	C₂³	C₂×D₃×C₈	1	120	yes	direct product factor
93	C₂³	C₂²×C₄×D₃	1	35	yes	—
94	C₂×C₄	—	3	120	no	—
95	C₂×C₄	—	3	240	no	—
96	C₂×C₄	—	3	16	no	—
97	C₂×C₄	C₂²×C₄×D₃	1	105	yes	—
98	C₂×C₄	C₂⁴×D₃	1	15	yes	—
99	C₈	—	3	16	no	—
100	C₈	C₂⁴×D₃	1	16	yes	direct product factor
101	C₂×C₃	C₂⁴×C₈	1	30	yes	complemented
102	C₂×C₃	C₂⁵×C₄	1	1	yes	characteristic
103	D₃	C₂⁴×C₈	1	32	yes	direct product factor
104	C₂²	—	3	16	no	—
105	C₂²	—	3	480	no	—
106	C₂²	C₂²×D₃×C₈	1	140	yes	direct product factor
107	C₂²	C₂³×C₄×D₃	1	15	yes	—
108	C₄	—	3	16	no	—
109	C₄	C₂³×C₄×D₃	1	15	yes	—
110	C₄	C₂⁵×D₃	1	1	yes	characteristic
111	C₃	C₂⁵×C₈	1	1	yes	characteristic, complemented, Sylow
112	C₂	—	3	32	no	—
113	C₂	C₂³×D₃×C₈	1	30	yes	direct product factor
114	C₂	C₂⁴×C₄×D₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×C₄×D₃

C₂

yes

maximal, characteristic

C₂⁴×C₃×C₈

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂³×D₃×C₈

C₂

yes

maximal, direct product factor

C₂⁵×C₈

—

maximal, Sylow

C₂⁵×D₃

C₄

yes

characteristic

C₂⁴×C₃×C₄

C₂²

yes

characteristic

C₂⁴×Dic₃

C₄

yes

characteristic

C₂³×C₄×D₃

C₄

yes

—

C₂³×C₄×D₃

C₂²

yes

—

C₂³×C₃×C₈

C₂²

yes

complemented

C₂³×C₃⋊C₈

C₂²

yes

complemented

C₂²×D₃×C₈

C₂²

560

yes

direct product factor

C₂⁵×C₄

—

C₂⁴×C₈

—

C₂⁴×C₈

—

C₂⁴×C₈

D₃

yes

characteristic, direct product factor

C₂⁵×C₃

C₂×C₄

yes

characteristic

C₂⁴×D₃

C₈

yes

direct product factor

C₂⁴×D₃

C₂×C₄

yes

—

C₂³×C₃×C₄

C₂×C₄

yes

—

C₂³×C₃×C₄

C₂³

yes

—

C₂³×Dic₃

C₂×C₄

yes

—

C₂²×C₄×D₃

C₂×C₄

420

yes

—

C₂²×C₄×D₃

C₂³

140

yes

—

C₂²×C₃×C₈

C₂³

140

yes

complemented

C₂²×C₃⋊C₈

C₂³

140

yes

complemented

C₂×D₃×C₈

C₂³

960

yes

direct product factor

C₂⁶

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

C₂×D₃

yes

characteristic

C₂³×C₈

—

C₂³×C₈

—

560

—

C₂³×C₈

C₂×D₃

yes

direct product factor

C₂⁴×C₃

C₂×C₈

yes

complemented

C₂⁴×C₃

C₂²×C₄

yes

—

C₂³×D₃

C₂×C₈

480

yes

direct product factor

C₂³×D₃

C₂²×C₄

140

yes

—

C₂²×C₃×C₄

C₂²×C₄

105

yes

—

C₂²×C₃×C₄

C₂⁴

yes

—

C₂²×Dic₃

C₂²×C₄

140

yes

—

C₂×C₄×D₃

C₂²×C₄

840

yes

—

C₂×C₄×D₃

C₂⁴

120

yes

—

C₂×C₃×C₈

C₂⁴

120

yes

complemented

C₂×C₃⋊C₈

C₂⁴

120

yes

complemented

D₃×C₈

C₂⁴

256

yes

direct product factor

C₂⁵

—

C₂⁵

—

C₂⁵

C₄×D₃

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

420

—

C₂³×C₄

—

140

—

C₂³×C₄

C₄×D₃

yes

—

C₂³×C₄

C₂²×D₃

yes

—

C₂²×C₈

—

140

—

C₂²×C₈

—

960

—

C₂²×C₈

C₂²×D₃

140

yes

direct product factor

C₂³×C₃

C₂²×C₈

120

yes

complemented

C₂³×C₃

C₂³×C₄

yes

—

C₂²×D₃

C₂²×C₈

1120

yes

direct product factor

C₂²×D₃

C₂³×C₄

120

yes

—

C₂×C₃×C₄

C₂³×C₄

105

yes

—

C₂×C₃×C₄

C₂⁵

yes

—

C₂×Dic₃

C₂³×C₄

120

yes

—

C₄×D₃

C₂³×C₄

240

yes

—

C₄×D₃

C₂⁵

yes

—

C₃×C₈

C₂⁵

yes

complemented

C₃⋊C₈

C₂⁵

yes

complemented

C₂⁴

—

480

—

C₂⁴

—

140

—

C₂⁴

D₃×C₈

yes

direct product factor

C₂⁴

C₂×C₄×D₃

yes

—

C₂²×C₄

—

140

—

C₂²×C₄

—

840

—

C₂²×C₄

—

120

—

C₂²×C₄

C₂×C₄×D₃

105

yes

—

C₂²×C₄

C₂³×D₃

yes

—

C₂×C₈

—

120

—

C₂×C₈

—

256

—

C₂×C₈

C₂³×D₃

120

yes

direct product factor

C₂²×C₃

C₂³×C₈

140

yes

complemented

C₂²×C₃

C₂⁴×C₄

yes

—

C₂×D₃

C₂³×C₈

480

yes

direct product factor

C₂×D₃

C₂⁴×C₄

yes

—

C₃×C₄

C₂⁴×C₄

yes

—

C₃×C₄

C₂⁶

yes

characteristic

Dic₃

C₂⁴×C₄

yes

—

C₂³

—

120

—

C₂³

—

1120

—

C₂³

C₂×D₃×C₈

120

yes

direct product factor

C₂³

C₂²×C₄×D₃

yes

—

C₂×C₄

—

120

—

C₂×C₄

—

240

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×D₃

105

yes

—

C₂×C₄

C₂⁴×D₃

yes

—

C₈

—

100

C₈

C₂⁴×D₃

yes

direct product factor

101

C₂×C₃

C₂⁴×C₈

yes

complemented

102

C₂×C₃

C₂⁵×C₄

yes

characteristic

103

D₃

C₂⁴×C₈

yes

direct product factor

104

C₂²

—

105

C₂²

—

480

—

106

C₂²

C₂²×D₃×C₈

140

yes

direct product factor

107

C₂²

C₂³×C₄×D₃

yes

—

108

C₄

—

109

C₄

C₂³×C₄×D₃

yes

—

110

C₄

C₂⁵×D₃

yes

characteristic

111

C₃

C₂⁵×C₈

yes

characteristic, complemented, Sylow

112

C₂

—

113

C₂

C₂³×D₃×C₈

yes

direct product factor

114

C₂

C₂⁴×C₄×D₃

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₈	D₃
commutator, hypocenter	C₃	C₂⁵×C₈
Frattini	C₄	C₂⁵×D₃
Fitting	C₂⁴×C₃×C₈	C₂
socle	C₂⁵×C₃	C₂×C₄
prime radical	C₂⁵×D₃	C₄
cosocle	C₃×C₄	C₂⁶
LCS	G ▷ C₃
UCS	C₁ ◁ C₂⁴×C₈

subgroup

G/H

center, hypercenter

C₂⁴×C₈

D₃

commutator, hypocenter

C₃

C₂⁵×C₈

Frattini

C₄

C₂⁵×D₃

Fitting

C₂⁴×C₃×C₈

C₂

socle

C₂⁵×C₃

C₂×C₄

prime radical

C₂⁵×D₃

C₄

cosocle

C₃×C₄

C₂⁶

LCS

G ▷ C₃

UCS

C₁ ◁ C₂⁴×C₈

N	H ≅ G/N	type
2. C₂⁴×C₃×C₈	C₂	semidirect
3. C₂⁴×C₃⋊C₈	C₂	semidirect
4. C₂³×D₃×C₈	C₂	direct (N × H)
11. C₂³×C₃×C₈	C₂²	semidirect
12. C₂³×C₃⋊C₈	C₂²	semidirect
13. C₂²×D₃×C₈	C₂²	direct (N × H)
17. C₂⁴×C₈	D₃	direct (N × H)
19. C₂⁴×D₃	C₈	direct (N × H)
26. C₂²×C₃×C₈	C₂³	semidirect
27. C₂²×C₃⋊C₈	C₂³	semidirect
28. C₂×D₃×C₈	C₂³	direct (N × H)
36. C₂³×C₈	C₂×D₃	direct (N × H)
37. C₂⁴×C₃	C₂×C₈	semidirect
39. C₂³×D₃	C₂×C₈	direct (N × H)
46. C₂×C₃×C₈	C₂⁴	semidirect
47. C₂×C₃⋊C₈	C₂⁴	semidirect
48. D₃×C₈	C₂⁴	direct (N × H)
59. C₂²×C₈	C₂²×D₃	direct (N × H)
60. C₂³×C₃	C₂²×C₈	semidirect
62. C₂²×D₃	C₂²×C₈	direct (N × H)
69. C₃×C₈	C₂⁵	semidirect
70. C₃⋊C₈	C₂⁵	semidirect
73. C₂⁴	D₃×C₈	direct (N × H)
82. C₂×C₈	C₂³×D₃	direct (N × H)
83. C₂²×C₃	C₂³×C₈	semidirect
85. C₂×D₃	C₂³×C₈	direct (N × H)
92. C₂³	C₂×D₃×C₈	direct (N × H)
100. C₈	C₂⁴×D₃	direct (N × H)
101. C₂×C₃	C₂⁴×C₈	semidirect
103. D₃	C₂⁴×C₈	direct (N × H)
106. C₂²	C₂²×D₃×C₈	direct (N × H)
111. C₃	C₂⁵×C₈	semidirect
113. C₂	C₂³×D₃×C₈	direct (N × H)

H ≅ G/N

type

2. C₂⁴×C₃×C₈

C₂

semidirect

3. C₂⁴×C₃⋊C₈

C₂

semidirect

4. C₂³×D₃×C₈

C₂

direct (N × H)

11. C₂³×C₃×C₈

C₂²

semidirect

12. C₂³×C₃⋊C₈

C₂²

semidirect

13. C₂²×D₃×C₈

C₂²

direct (N × H)

17. C₂⁴×C₈

D₃

direct (N × H)

19. C₂⁴×D₃

C₈

direct (N × H)

26. C₂²×C₃×C₈

C₂³

semidirect

27. C₂²×C₃⋊C₈

C₂³

semidirect

28. C₂×D₃×C₈

C₂³

direct (N × H)

36. C₂³×C₈

C₂×D₃

direct (N × H)

37. C₂⁴×C₃

C₂×C₈

semidirect

39. C₂³×D₃

C₂×C₈

direct (N × H)

46. C₂×C₃×C₈

C₂⁴

semidirect

47. C₂×C₃⋊C₈

C₂⁴

semidirect

48. D₃×C₈

C₂⁴

direct (N × H)

59. C₂²×C₈

C₂²×D₃

direct (N × H)

60. C₂³×C₃

C₂²×C₈

semidirect

62. C₂²×D₃

C₂²×C₈

direct (N × H)

69. C₃×C₈

C₂⁵

semidirect

70. C₃⋊C₈

C₂⁵

semidirect

73. C₂⁴

D₃×C₈

direct (N × H)

82. C₂×C₈

C₂³×D₃

direct (N × H)

83. C₂²×C₃

C₂³×C₈

semidirect

85. C₂×D₃

C₂³×C₈

direct (N × H)

92. C₂³

C₂×D₃×C₈

direct (N × H)

100. C₈

C₂⁴×D₃

direct (N × H)

101. C₂×C₃

C₂⁴×C₈

semidirect

103. D₃

C₂⁴×C₈

direct (N × H)

106. C₂²

C₂²×D₃×C₈

direct (N × H)

111. C₃

C₂⁵×C₈

semidirect

113. C₂

C₂³×D₃×C₈

direct (N × H)

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

subgroup	index
C₂⁴×C₄×D₃	2
C₂⁴×C₃×C₈	2
C₂⁴×C₃⋊C₈	2
C₂³×D₃×C₈	2
C₂⁵×C₈	3

subgroup

index

C₂⁴×C₄×D₃

C₂⁴×C₃×C₈

C₂⁴×C₃⋊C₈

C₂³×D₃×C₈

C₂⁵×C₈

Subgroups — C₂⁴×D₃×C₈ #11708940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C24×D3×C8 #11708940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₃×C₈ #11708940