Subgroups — #3748⋊C₃ #88490

Order 432.

86 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	(C₃⋊(C₃⋊C₈))⋊C₃	C₂	1	2	yes	maximal, complemented
3	#3748	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#208	—	3	1	no	maximal
5	C₂²×H(3,𝔽₃)	C₄	1	1	yes	characteristic
6	C₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
7	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
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	C₃⋊(C₃⋊C₈)	C₂×C₃	1	2	yes	complemented
12	C₃×C₃⋊C₈	—	3	2	no	—
13	C₂×H(3,𝔽₃)	—	2	1	no	—
14	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₃×M₁₆	—	9	1	no	—
16	#208	—	3	1	no	—
17	#208	—	9	1	no	—
18	C₂²×C₃²	—	2	1	no	—
19	C₂²×C₃²	Dic₃	1	1	yes	characteristic
20	C₂²×C₃²	C₃×C₄	1	1	yes	characteristic
21	C₃²×C₄	—	2	1	no	—
22	C₃²×C₄	—	2	1	no	—
23	C₃²×C₄	Dic₃	1	1	yes	characteristic
24	C₃²×C₄	C₃×C₄	1	1	yes	characteristic
25	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
27	H(3,𝔽₃)	M₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃×C₄	—	3	1	no	—
29	C₂×C₃×C₄	—	3	1	no	—
30	C₂×C₃×C₄	—	6	1	no	—
31	C₂×C₃×C₄	C₃×D₃	1	1	yes	characteristic
32	C₃×C₈	—	9	2	no	—
33	C₃⋊C₈	—	3	2	no	—
34	C₃⋊C₈	—	9	2	no	—
35	C₂×C₃²	—	2	2	no	—
36	C₂×C₃²	—	2	1	no	—
37	C₂×C₃²	—	2	1	no	—
38	C₂×C₃²	—	2	1	no	—
39	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
40	C₂×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
41	M₁₆	—	9	1	no	Sylow
42	C₂²×C₃	—	3	1	no	—
43	C₂²×C₃	—	3	1	no	—
44	C₂²×C₃	—	6	1	no	—
45	C₂²×C₃	C₃×Dic₃	1	1	yes	characteristic
46	C₃×C₄	—	3	1	no	—
47	C₃×C₄	—	3	1	no	—
48	C₃×C₄	—	3	1	no	—
49	C₃×C₄	—	3	1	no	—
50	C₃×C₄	—	6	1	no	—
51	C₃×C₄	—	6	1	no	—
52	C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
53	C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
54	C₃²	—	2	1	no	—
55	C₃²	#208	1	1	yes	characteristic, complemented
56	C₃²	C₃×M₁₆	1	1	yes	characteristic, complemented
57	C₂×C₄	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
58	C₈	—	9	2	no	—
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	3	1	no	—
62	C₂×C₃	—	6	1	no	—
63	C₂×C₃	—	6	1	no	—
64	C₂×C₃	—	6	2	no	—
65	C₂×C₃	—	6	1	no	—
66	C₂×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
67	C₂²	(C₃⋊Dic₃)⋊C₃	1	1	yes	characteristic
68	C₄	(C₃⋊Dic₃)⋊C₃	1	1	yes	characteristic
69	C₄	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
70	C₃	—	3	1	no	—
71	C₃	—	3	1	no	—
72	C₃	—	6	1	no	—
73	C₃	C₃×#208	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	C₂×(C₃⋊Dic₃)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×(C₃⋊D₃)⋊C₃
commutator	C₂×C₃²	C₂×C₃×C₄
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
socle	C₂×C₃	C₂×C₃×Dic₃
prime radical	C₂²×H(3,𝔽₃)	C₄
cosocle	C₃²×C₄	C₂²×C₃
hypercenter	C₂×C₄	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₃×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. (C₃⋊(C₃⋊C₈))⋊C₃	C₂	semidirect
3. #3748	C₃	semidirect
11. C₃⋊(C₃⋊C₈)	C₂×C₃	semidirect
27. H(3,𝔽₃)	M₁₆	semidirect
55. C₃²	#208	semidirect
56. C₃²	C₃×M₁₆	semidirect

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
(C₃⋊(C₃⋊C₈))⋊C₃	2
#3748	3
C₃×#208	3

Subgroups — #3748⋊C3 #88490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3748⋊C₃ #88490