Subgroups — C₄×(C₃⋊Dic₃)⋊C₃ #88491

Order 432.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₃⋊Dic₃)⋊C₃	C₂	1	2	yes	maximal
3	C₄×C₃⋊Dic₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×Dic₃	—	3	1	no	maximal
5	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₄×H(3,𝔽₃)	C₄	1	2	yes	complemented
7	(C₃⋊Dic₃)⋊C₃	C₄	1	4	yes	direct product factor
8	C₂×C₃²×C₄	—	2	1	no	—
9	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
10	C₂×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
11	C₂×C₃⋊Dic₃	C₂×C₃	1	2	yes	—
12	C₂×C₃×Dic₃	—	3	2	no	—
13	C₂×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
14	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄²	—	9	1	no	—
16	C₄×Dic₃	—	3	1	no	—
17	C₄×Dic₃	—	9	1	no	—
18	C₂²×C₃²	—	2	1	no	—
19	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
20	C₂²×C₃²	C₂²×C₃	1	1	yes	characteristic
21	C₃²×C₄	—	2	2	no	—
22	C₃²×C₄	Dic₃	1	2	yes	complemented
23	C₃²×C₄	C₃×C₄	1	2	yes	complemented
24	C₃⋊Dic₃	C₃×C₄	1	4	yes	complemented
25	C₃×Dic₃	—	3	4	no	—
26	H(3,𝔽₃)	C₄²	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃×C₄	—	3	1	no	—
28	C₂×C₃×C₄	—	3	1	no	—
29	C₂×C₃×C₄	—	6	1	no	—
30	C₂×C₃×C₄	—	9	2	no	—
31	C₂×C₃×C₄	C₃×D₃	1	1	yes	characteristic
32	C₂×Dic₃	—	3	2	no	—
33	C₂×Dic₃	—	9	2	no	—
34	C₂×C₃²	—	2	2	no	—
35	C₂×C₃²	—	2	1	no	—
36	C₂×C₃²	C₄×D₃	1	2	yes	—
37	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
38	C₂×C₃²	C₂×C₃×C₄	1	2	yes	—
39	C₂×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
40	C₄²	—	9	1	no	Sylow
41	C₂²×C₃	—	3	1	no	—
42	C₂²×C₃	—	3	1	no	—
43	C₂²×C₃	—	6	1	no	—
44	C₂²×C₃	C₂×C₃×D₃	1	1	yes	characteristic
45	C₃×C₄	—	3	2	no	—
46	C₃×C₄	—	3	2	no	—
47	C₃×C₄	—	6	2	no	—
48	C₃×C₄	—	9	4	no	—
49	C₃×C₄	C₃×Dic₃	1	2	yes	—
50	Dic₃	—	3	4	no	—
51	Dic₃	—	9	4	no	—
52	C₃²	—	2	1	no	—
53	C₃²	C₄×Dic₃	1	1	yes	characteristic, complemented
54	C₃²	C₃×C₄²	1	1	yes	characteristic, complemented
55	C₂×C₄	—	9	2	no	—
56	C₂×C₄	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
57	C₂×C₃	—	3	2	no	—
58	C₂×C₃	—	3	2	no	—
59	C₂×C₃	—	3	1	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	6	2	no	—
62	C₂×C₃	—	6	1	no	—
63	C₂×C₃	C₃×C₄×D₃	1	2	yes	—
64	C₂×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
65	C₂²	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
66	C₄	—	9	4	no	—
67	C₄	(C₃⋊Dic₃)⋊C₃	1	2	yes	direct product factor
68	C₃	—	3	1	no	—
69	C₃	—	3	1	no	—
70	C₃	—	6	1	no	—
71	C₃	C₃×C₄×Dic₃	1	1	yes	characteristic
72	C₂	C₄×(C₃⋊D₃)⋊C₃	1	2	yes	—
73	C₂	C₂×(C₃⋊Dic₃)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	(C₃⋊D₃)⋊C₃
commutator, hypocenter	C₃²	C₃×C₄²
Frattini, socle	C₂²×C₃	C₂×C₃×D₃
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
prime radical	C₂²×H(3,𝔽₃)	C₂²
cosocle	C₂²×C₃²	C₂²×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. C₄×C₃⋊Dic₃	C₃	semidirect
6. C₄×H(3,𝔽₃)	C₄	semidirect
7. (C₃⋊Dic₃)⋊C₃	C₄	direct (N × H)
22. C₃²×C₄	Dic₃	semidirect
23. C₃²×C₄	C₃×C₄	semidirect
24. C₃⋊Dic₃	C₃×C₄	semidirect
26. H(3,𝔽₃)	C₄²	semidirect
53. C₃²	C₄×Dic₃	semidirect
54. C₃²	C₃×C₄²	semidirect
67. C₄	(C₃⋊Dic₃)⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×H(3,𝔽₃)	2
C₂×(C₃⋊Dic₃)⋊C₃	2
C₄×C₃⋊Dic₃	3
C₃×C₄×Dic₃	3

Subgroups — C4×(C3⋊Dic3)⋊C3 #88491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×(C₃⋊Dic₃)⋊C₃ #88491