Subgroups — Dic₃×C₃⋊Dic₃ #88801

Order 432.

244 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃⋊Dic₃	—	3	1	no	maximal
5	Dic₃²	—	3	4	no	maximal
6	C₂²×C₃³	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊Dic₃)	C₄	1	2	yes	complemented
8	C₃×C₃⋊Dic₃	C₄	1	2	yes	complemented
9	C₃²×Dic₃	C₄	1	2	yes	complemented
10	C₂×C₃²×C₄	—	3	1	no	—
11	C₂×C₃⋊Dic₃	—	3	1	no	—
12	C₂×C₃⋊Dic₃	—	3	4	no	—
13	C₂×C₃⋊Dic₃	—	6	4	no	—
14	C₂×C₃⋊Dic₃	D₃	1	1	yes	characteristic
15	C₂×C₃×Dic₃	—	3	4	no	—
16	C₂×C₃×Dic₃	D₃	1	4	yes	—
17	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₃	—	9	4	no	—
21	C₄×Dic₃	—	9	1	no	—
22	C₂²×C₃²	—	2	4	no	—
23	C₂²×C₃²	C₂×D₃	1	4	yes	—
24	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
25	C₃²×C₄	—	3	2	no	—
26	C₃⋊Dic₃	—	3	2	no	—
27	C₃⋊Dic₃	—	3	8	no	—
28	C₃⋊Dic₃	—	6	8	no	—
29	C₃⋊Dic₃	Dic₃	1	2	yes	direct product factor
30	C₃×Dic₃	—	3	8	no	—
31	C₃×Dic₃	Dic₃	1	8	yes	complemented
32	C₃³	C₄²	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃×C₄	—	3	4	no	—
34	C₂×C₃×C₄	—	9	1	no	—
35	C₂×Dic₃	—	3	4	no	—
36	C₂×Dic₃	—	9	4	no	—
37	C₂×Dic₃	—	9	1	no	—
38	C₂×Dic₃	—	18	4	no	—
39	C₂×Dic₃	C₃⋊D₃	1	1	yes	characteristic
40	C₂×C₃²	—	2	4	no	—
41	C₂×C₃²	—	2	4	no	—
42	C₂×C₃²	—	2	4	no	—
43	C₂×C₃²	C₄×D₃	1	4	yes	—
44	C₂×C₃²	C₄×D₃	1	4	yes	—
45	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
46	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
47	C₂×C₃²	C₂×Dic₃	1	4	yes	—
48	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
49	C₄²	—	27	1	no	Sylow
50	C₂²×C₃	—	2	4	no	—
51	C₂²×C₃	D₃²	1	4	yes	—
52	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₃×C₄	—	3	8	no	—
54	C₃×C₄	—	9	2	no	—
55	Dic₃	—	3	8	no	—
56	Dic₃	—	9	8	no	—
57	Dic₃	—	9	2	no	—
58	Dic₃	—	18	8	no	—
59	Dic₃	C₃⋊Dic₃	1	2	yes	direct product factor
60	C₃²	—	2	4	no	—
61	C₃²	C₄×Dic₃	1	4	yes	complemented
62	C₃²	C₄×Dic₃	1	1	yes	characteristic, complemented
63	C₂×C₄	—	3	1	no	—
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	—	27	1	no	—
66	C₂×C₃	—	2	4	no	—
67	C₂×C₃	—	2	4	no	—
68	C₂×C₃	—	2	4	no	—
69	C₂×C₃	D₃×Dic₃	1	4	yes	—
70	C₂×C₃	D₃×Dic₃	1	4	yes	—
71	C₂×C₃	#624	1	4	yes	—
72	C₂×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
73	C₂×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
74	C₂×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
75	C₂²	D₃×C₃⋊D₃	1	1	yes	characteristic
76	C₄	—	3	2	no	—
77	C₄	—	9	2	no	—
78	C₄	—	27	2	no	—
79	C₃	—	2	4	no	—
80	C₃	Dic₃²	1	4	yes	complemented
81	C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic, complemented
82	C₂	D₃×C₃⋊Dic₃	1	1	yes	characteristic
83	C₂	Dic₃×C₃⋊D₃	1	1	yes	characteristic
84	C₂	#6289	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃×C₃⋊D₃
commutator, hypocenter	C₃³	C₄²
Fitting, socle, prime radical, cosocle	C₂²×C₃³	C₂²
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
7. C₃⋊(C₃⋊Dic₃)	C₄	semidirect
8. C₃×C₃⋊Dic₃	C₄	semidirect
9. C₃²×Dic₃	C₄	semidirect
29. C₃⋊Dic₃	Dic₃	direct (N × H)
31. C₃×Dic₃	Dic₃	semidirect
32. C₃³	C₄²	semidirect
59. Dic₃	C₃⋊Dic₃	direct (N × H)
61. C₃²	C₄×Dic₃	semidirect
62. C₃²	C₄×Dic₃	semidirect
80. C₃	Dic₃²	semidirect
81. C₃	C₄×C₃⋊Dic₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊(C₃⋊Dic₃)	2
C₂×C₃×C₃⋊Dic₃	2
C₂×C₃²×Dic₃	2
C₄×C₃⋊Dic₃	3
Dic₃²	3

Subgroups — Dic3×C3⋊Dic3 #88801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×C₃⋊Dic₃ #88801