Group #74357 — subgroups

Subgroups — D₃×#384 #74357

Order 384.

888 conjugacy classes of subgroups; only the 365 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 365 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	C₂×D₃×C₄⋊C₄	C₂	1	6	yes	maximal
2	C₂×C₄²×D₃	C₂	1	1	yes	maximal, characteristic
3	C₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	#4975	C₂	1	6	yes	maximal, complemented
5	#4935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#384	—	3	1	no	maximal, Sylow
7	C₂²×C₄×D₃	C₂²	1	4	yes	—
8	C₂²×C₄×D₃	C₂²	1	3	yes	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	6	yes	—
12	C₂×Dic₃⋊C₄	C₂²	1	12	yes	—
13	C₂×C₄×Dic₃	C₂²	1	3	yes	—
14	D₃×C₄⋊C₄	—	2	24	no	—
15	C₄²×D₃	C₄	1	8	yes	complemented
16	C₂²×C₄⋊C₄	—	3	6	no	—
17	C₂²×C₄²	—	3	1	no	—
18	#384	—	3	6	no	—
19	#384	—	3	1	no	—
20	#384	D₃	1	1	yes	characteristic, direct product factor
21	C₂³×D₃	C₂³	1	1	yes	characteristic
22	C₂²×C₃×C₄	C₂³	1	4	yes	—
23	C₂²×C₃×C₄	C₂³	1	3	yes	—
24	C₂²×Dic₃	C₂³	1	4	yes	—
25	C₂²×Dic₃	C₂³	1	3	yes	—
26	C₂×C₄×D₃	—	2	24	no	—
27	C₂×C₄×D₃	C₂×C₄	1	12	yes	—
28	C₂×C₄×D₃	D₄	1	12	yes	—
29	C₂×C₄×D₃	Q₈	1	12	yes	—
30	C₃×C₄⋊C₄	—	2	12	no	—
31	C₃×C₄²	C₂×C₄	1	4	yes	complemented
32	C₄⋊Dic₃	—	2	12	no	—
33	Dic₃⋊C₄	—	2	24	no	—
34	C₄×Dic₃	C₂×C₄	1	12	yes	complemented
35	C₂³×C₄	—	3	3	no	—
36	C₂³×C₄	—	3	4	no	—
37	C₂×C₄⋊C₄	—	3	12	no	—
38	C₂×C₄⋊C₄	—	3	6	no	—
39	C₂×C₄⋊C₄	—	6	24	no	—
40	C₂×C₄⋊C₄	C₂×D₃	1	6	yes	—
41	C₂×C₄²	—	3	3	no	—
42	C₂×C₄²	—	3	8	no	—
43	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₂³×C₃	C₂⁴	1	1	yes	characteristic
45	C₂²×D₃	C₂²×C₄	1	2	yes	—
46	C₂²×D₃	C₂×D₄	1	6	yes	—
47	C₂²×D₃	C₂×Q₈	1	6	yes	—
48	C₂×C₃×C₄	—	2	12	no	—
49	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
50	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
51	C₂×C₃×C₄	C₂×Q₈	1	6	yes	—
52	C₂×Dic₃	—	2	12	no	—
53	C₂×Dic₃	C₂²×C₄	1	6	yes	—
54	C₂×Dic₃	C₂×D₄	1	6	yes	—
55	C₂×Dic₃	C₂×Q₈	1	6	yes	—
56	C₄×D₃	—	4	16	no	—
57	C₄×D₃	C₄⋊C₄	1	48	yes	complemented
58	C₂⁴	—	3	1	no	—
59	C₂²×C₄	—	3	3	no	—
60	C₂²×C₄	—	3	12	no	—
61	C₂²×C₄	—	3	12	no	—
62	C₂²×C₄	—	3	12	no	—
63	C₂²×C₄	—	3	4	no	—
64	C₂²×C₄	—	6	24	no	—
65	C₂²×C₄	C₂²×D₃	1	4	yes	—
66	C₂²×C₄	C₂²×D₃	1	3	yes	—
67	C₄⋊C₄	—	2	12	no	—
68	C₄⋊C₄	—	6	24	no	—
69	C₄⋊C₄	—	6	12	no	—
70	C₄²	—	3	12	no	—
71	C₄²	C₄×D₃	1	4	yes	complemented
72	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₃	C₂²×D₄	1	3	yes	—
74	C₂²×C₃	C₂²×Q₈	1	3	yes	—
75	C₂×D₃	C₂×C₄⋊C₄	1	12	yes	—
76	C₂×D₃	#127	1	4	yes	—
77	C₂×D₃	C₄⋊Q₈	1	12	yes	—
78	C₃×C₄	—	4	4	no	—
79	C₃×C₄	C₂×C₄⋊C₄	1	12	yes	complemented
80	Dic₃	—	4	4	no	—
81	Dic₃	C₂×C₄⋊C₄	1	12	yes	complemented
82	C₂³	—	3	2	no	—
83	C₂³	—	3	6	no	—
84	C₂³	—	3	6	no	—
85	C₂³	C₂³×D₃	1	1	yes	characteristic
86	C₂×C₄	—	2	12	no	—
87	C₂×C₄	—	3	6	no	—
88	C₂×C₄	—	3	6	no	—
89	C₂×C₄	—	3	6	no	—
90	C₂×C₄	—	3	48	no	—
91	C₂×C₄	—	6	12	no	—
92	C₂×C₄	—	12	16	no	—
93	C₂×C₄	C₂×C₄×D₃	1	6	yes	—
94	C₂×C₄	D₃×D₄	1	6	yes	—
95	C₂×C₄	D₃×Q₈	1	6	yes	—
96	C₂×C₃	C₂²×C₄⋊C₄	1	3	yes	—
97	C₂×C₃	C₂×#127	1	1	yes	characteristic
98	C₂×C₃	C₂×C₄⋊Q₈	1	3	yes	—
99	D₃	#384	1	8	yes	direct product factor
100	C₂²	—	3	4	no	—
101	C₂²	—	3	12	no	—
102	C₂²	—	3	12	no	—
103	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
104	C₂²	C₂×D₃×D₄	1	3	yes	—
105	C₂²	C₂×D₃×Q₈	1	3	yes	—
106	C₄	—	3	12	no	—
107	C₄	—	4	4	no	—
108	C₄	—	12	4	no	—
109	C₄	D₃×C₄⋊C₄	1	12	yes	complemented
110	C₃	C₂×#384	1	1	yes	characteristic, complemented, Sylow
111	C₂	—	3	8	no	—
112	C₂	C₂×D₃×C₄⋊C₄	1	3	yes	—
113	C₂	D₃×#127	1	1	yes	characteristic
114	C₂	D₃×C₄⋊Q₈	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₃×C₄⋊C₄

C₂

yes

maximal

C₂×C₄²×D₃

C₂

yes

maximal, characteristic

C₃×#384

C₂

yes

maximal, characteristic, complemented

#4975

C₂

yes

maximal, complemented

#4935

C₂

yes

maximal, characteristic, complemented

C₂×#384

—

maximal, Sylow

C₂²×C₄×D₃

C₂²

yes

—

C₂²×C₄×D₃

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

D₃×C₄⋊C₄

—

C₄²×D₃

C₄

yes

complemented

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

#384

—

#384

—

#384

D₃

yes

characteristic, direct product factor

C₂³×D₃

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂×C₄×D₃

—

C₂×C₄×D₃

C₂×C₄

yes

—

C₂×C₄×D₃

D₄

yes

—

C₂×C₄×D₃

Q₈

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×C₄

yes

complemented

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

C₂×C₄

yes

complemented

C₂³×C₄

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₂³×C₃

C₂⁴

yes

characteristic

C₂²×D₃

C₂²×C₄

yes

—

C₂²×D₃

C₂×D₄

yes

—

C₂²×D₃

C₂×Q₈

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

C₂²×C₄

yes

—

C₂×Dic₃

C₂×D₄

yes

—

C₂×Dic₃

C₂×Q₈

yes

—

C₄×D₃

—

C₄×D₃

C₄⋊C₄

yes

complemented

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄×D₃

yes

complemented

C₂²×C₃

C₂³×C₄

yes

characteristic

C₂²×C₃

C₂²×D₄

yes

—

C₂²×C₃

C₂²×Q₈

yes

—

C₂×D₃

C₂×C₄⋊C₄

yes

—

C₂×D₃

#127

yes

—

C₂×D₃

C₄⋊Q₈

yes

—

C₃×C₄

—

C₃×C₄

C₂×C₄⋊C₄

yes

complemented

Dic₃

—

Dic₃

C₂×C₄⋊C₄

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₃

yes

—

C₂×C₄

D₃×D₄

yes

—

C₂×C₄

D₃×Q₈

yes

—

C₂×C₃

C₂²×C₄⋊C₄

yes

—

C₂×C₃

C₂×#127

yes

characteristic

C₂×C₃

C₂×C₄⋊Q₈

yes

—

D₃

#384

yes

direct product factor

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂²×C₄×D₃

yes

characteristic

104

C₂²

C₂×D₃×D₄

yes

—

105

C₂²

C₂×D₃×Q₈

yes

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

D₃×C₄⋊C₄

yes

complemented

110

C₃

C₂×#384

yes

characteristic, complemented, Sylow

111

C₂

—

112

C₂

C₂×D₃×C₄⋊C₄

yes

—

113

C₂

D₃×#127

yes

characteristic

114

C₂

D₃×C₄⋊Q₈

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₃×#384	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂³×D₃	C₂³
hypercenter	#384	D₃
hypocenter	C₃	C₂×#384
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator

C₂²×C₃

C₂³×C₄

Fitting

C₃×#384

C₂

socle, cosocle

C₂³×C₃

C₂⁴

prime radical

C₂³×D₃

C₂³

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
3. C₃×#384	C₂	semidirect
4. #4975	C₂	semidirect
5. #4935	C₂	semidirect
15. C₄²×D₃	C₄	semidirect
20. #384	D₃	direct (N × H)
31. C₃×C₄²	C₂×C₄	semidirect
34. C₄×Dic₃	C₂×C₄	semidirect
57. C₄×D₃	C₄⋊C₄	semidirect
71. C₄²	C₄×D₃	semidirect
79. C₃×C₄	C₂×C₄⋊C₄	semidirect
81. Dic₃	C₂×C₄⋊C₄	semidirect
99. D₃	#384	direct (N × H)
109. C₄	D₃×C₄⋊C₄	semidirect
110. C₃	C₂×#384	semidirect

H ≅ G/N

type

3. C₃×#384

C₂

semidirect

4. #4975

C₂

semidirect

5. #4935

C₂

semidirect

15. C₄²×D₃

C₄

semidirect

20. #384

D₃

direct (N × H)

31. C₃×C₄²

C₂×C₄

semidirect

34. C₄×Dic₃

C₂×C₄

semidirect

57. C₄×D₃

C₄⋊C₄

semidirect

71. C₄²

C₄×D₃

semidirect

79. C₃×C₄

C₂×C₄⋊C₄

semidirect

81. Dic₃

C₂×C₄⋊C₄

semidirect

99. D₃

#384

direct (N × H)

109. C₄

D₃×C₄⋊C₄

semidirect

110. C₃

C₂×#384

semidirect

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

subgroup	index
C₂×D₃×C₄⋊C₄	2
C₂×C₄²×D₃	2
C₃×#384	2
#4975	2
#4935	2
C₂×#384	3

subgroup

index

C₂×D₃×C₄⋊C₄

C₂×C₄²×D₃

Subgroups — D₃×#384 #74357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D3×#384 #74357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#384 #74357