Subgroups — C₂²×C₄²×D₃ #85672

Order 384.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₃	C₂	1	3	yes	maximal
2	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×Dic₃	C₂	1	3	yes	maximal, complemented
4	C₂×C₄²×D₃	C₂	1	24	yes	maximal, direct product factor
5	C₂³×C₄²	—	3	1	no	maximal, Sylow
6	C₂⁴×D₃	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄	C₂²	1	3	yes	—
8	C₂³×Dic₃	C₂²	1	3	yes	—
9	C₂²×C₄×D₃	C₄	1	48	yes	direct product factor
10	C₂²×C₄×D₃	C₂²	1	36	yes	—
11	C₂×C₃×C₄²	C₂²	1	12	yes	complemented
12	C₂×C₄×Dic₃	C₂²	1	36	yes	complemented
13	C₄²×D₃	C₂²	1	64	yes	direct product factor
14	C₂⁴×C₄	—	3	3	no	—
15	C₂²×C₄²	—	3	3	no	—
16	C₂²×C₄²	—	3	24	no	—
17	C₂²×C₄²	D₃	1	1	yes	characteristic, direct product factor
18	C₂⁴×C₃	C₂³	1	1	yes	characteristic
19	C₂³×D₃	C₂×C₄	1	24	yes	—
20	C₂³×D₃	C₂³	1	6	yes	—
21	C₂²×C₃×C₄	C₂×C₄	1	24	yes	complemented
22	C₂²×C₃×C₄	C₂³	1	18	yes	—
23	C₂²×Dic₃	C₂×C₄	1	24	yes	complemented
24	C₂²×Dic₃	C₂³	1	18	yes	—
25	C₂×C₄×D₃	C₂×C₄	1	288	yes	direct product factor
26	C₂×C₄×D₃	C₂³	1	48	yes	—
27	C₃×C₄²	C₂³	1	16	yes	complemented
28	C₄×Dic₃	C₂³	1	48	yes	complemented
29	C₂⁵	—	3	1	no	—
30	C₂³×C₄	—	3	3	no	—
31	C₂³×C₄	—	3	48	no	—
32	C₂³×C₄	—	3	36	no	—
33	C₂³×C₄	C₂×D₃	1	3	yes	—
34	C₂×C₄²	—	3	36	no	—
35	C₂×C₄²	—	3	64	no	—
36	C₂×C₄²	C₂×D₃	1	12	yes	direct product factor
37	C₂³×C₃	C₂²×C₄	1	12	yes	—
38	C₂³×C₃	C₂⁴	1	3	yes	—
39	C₂²×D₃	C₄²	1	64	yes	direct product factor
40	C₂²×D₃	C₂²×C₄	1	72	yes	—
41	C₂²×D₃	C₂⁴	1	4	yes	—
42	C₂×C₃×C₄	C₂²×C₄	1	72	yes	complemented
43	C₂×C₃×C₄	C₂⁴	1	12	yes	—
44	C₂×Dic₃	C₂²×C₄	1	72	yes	complemented
45	C₂×Dic₃	C₂⁴	1	12	yes	—
46	C₄×D₃	C₂²×C₄	1	192	yes	direct product factor
47	C₂⁴	—	3	24	no	—
48	C₂⁴	—	3	6	no	—
49	C₂⁴	C₂²×D₃	1	1	yes	characteristic
50	C₂²×C₄	—	3	24	no	—
51	C₂²×C₄	—	3	18	no	—
52	C₂²×C₄	—	3	288	no	—
53	C₂²×C₄	—	3	48	no	—
54	C₂²×C₄	C₄×D₃	1	24	yes	direct product factor
55	C₂²×C₄	C₂²×D₃	1	18	yes	—
56	C₄²	—	3	48	no	—
57	C₄²	C₂²×D₃	1	16	yes	direct product factor
58	C₂²×C₃	C₂×C₄²	1	16	yes	complemented
59	C₂²×C₃	C₂³×C₄	1	18	yes	—
60	C₂²×C₃	C₂⁵	1	1	yes	characteristic
61	C₂×D₃	C₂×C₄²	1	96	yes	direct product factor
62	C₂×D₃	C₂³×C₄	1	24	yes	—
63	C₃×C₄	C₂³×C₄	1	24	yes	complemented
64	Dic₃	C₂³×C₄	1	24	yes	complemented
65	C₂³	—	3	64	no	—
66	C₂³	—	3	72	no	—
67	C₂³	—	3	4	no	—
68	C₂³	C₂×C₄×D₃	1	12	yes	—
69	C₂³	C₂³×D₃	1	3	yes	—
70	C₂×C₄	—	3	72	no	—
71	C₂×C₄	—	3	12	no	—
72	C₂×C₄	—	3	192	no	—
73	C₂×C₄	C₂×C₄×D₃	1	72	yes	direct product factor
74	C₂×C₄	C₂³×D₃	1	12	yes	—
75	C₂×C₃	C₂²×C₄²	1	12	yes	complemented
76	C₂×C₃	C₂⁴×C₄	1	3	yes	—
77	D₃	C₂²×C₄²	1	16	yes	direct product factor
78	C₂²	—	3	24	no	—
79	C₂²	—	3	96	no	—
80	C₂²	C₄²×D₃	1	16	yes	direct product factor
81	C₂²	C₂²×C₄×D₃	1	18	yes	—
82	C₂²	C₂⁴×D₃	1	1	yes	characteristic
83	C₄	—	3	24	no	—
84	C₄	C₂²×C₄×D₃	1	24	yes	direct product factor
85	C₃	C₂³×C₄²	1	1	yes	characteristic, complemented, Sylow
86	C₂	—	3	16	no	—
87	C₂	C₂×C₄²×D₃	1	12	yes	direct product factor
88	C₂	C₂³×C₄×D₃	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₃
commutator, hypocenter	C₃	C₂³×C₄²
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle	C₂⁴×C₃	C₂³
prime radical	C₂⁴×D₃	C₂²
cosocle	C₂²×C₃	C₂⁵
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×C₃×C₄²	C₂	semidirect
3. C₂²×C₄×Dic₃	C₂	semidirect
4. C₂×C₄²×D₃	C₂	direct (N × H)
9. C₂²×C₄×D₃	C₄	direct (N × H)
11. C₂×C₃×C₄²	C₂²	semidirect
12. C₂×C₄×Dic₃	C₂²	semidirect
13. C₄²×D₃	C₂²	direct (N × H)
17. C₂²×C₄²	D₃	direct (N × H)
21. C₂²×C₃×C₄	C₂×C₄	semidirect
23. C₂²×Dic₃	C₂×C₄	semidirect
25. C₂×C₄×D₃	C₂×C₄	direct (N × H)
27. C₃×C₄²	C₂³	semidirect
28. C₄×Dic₃	C₂³	semidirect
36. C₂×C₄²	C₂×D₃	direct (N × H)
39. C₂²×D₃	C₄²	direct (N × H)
42. C₂×C₃×C₄	C₂²×C₄	semidirect
44. C₂×Dic₃	C₂²×C₄	semidirect
46. C₄×D₃	C₂²×C₄	direct (N × H)
54. C₂²×C₄	C₄×D₃	direct (N × H)
57. C₄²	C₂²×D₃	direct (N × H)
58. C₂²×C₃	C₂×C₄²	semidirect
61. C₂×D₃	C₂×C₄²	direct (N × H)
63. C₃×C₄	C₂³×C₄	semidirect
64. Dic₃	C₂³×C₄	semidirect
73. C₂×C₄	C₂×C₄×D₃	direct (N × H)
75. C₂×C₃	C₂²×C₄²	semidirect
77. D₃	C₂²×C₄²	direct (N × H)
80. C₂²	C₄²×D₃	direct (N × H)
84. C₄	C₂²×C₄×D₃	direct (N × H)
85. C₃	C₂³×C₄²	semidirect
87. C₂	C₂×C₄²×D₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₃	2
C₂²×C₃×C₄²	2
C₂²×C₄×Dic₃	2
C₂×C₄²×D₃	2
C₂³×C₄²	3

Subgroups — C22×C42×D3 #85672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄²×D₃ #85672