Subgroups — #3756⋊C₃ #88513

Order 432.

75 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#634⋊C₃	C₂	1	1	yes	maximal, characteristic
3	(C₃⋊(C₃⋊C₈))⋊C₃	C₂	1	1	yes	maximal, characteristic
4	#3756	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#216	—	3	1	no	maximal
6	C₄×H(3,𝔽₃)	—	2	1	no	—
7	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	(C₃⋊Dic₃)⋊C₃	—	2	1	no	—
9	C₃²×Q₈	—	2	1	no	—
10	C₃²×Q₈	D₃	1	1	yes	characteristic
11	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
12	#634	C₂×C₃	1	1	yes	characteristic
13	C₃×Dic₆	—	3	1	no	—
14	C₃⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
15	C₃×C₃⋊C₈	—	3	1	no	—
16	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	C₃×Q₁₆	—	9	1	no	—
18	#216	—	3	1	no	—
19	#216	—	9	1	no	—
20	C₃²×C₄	—	2	2	no	—
21	C₃²×C₄	—	2	1	no	—
22	C₃²×C₄	—	2	1	no	—
23	C₃²×C₄	—	2	1	no	—
24	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
25	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
26	C₃⋊Dic₃	—	2	1	no	—
27	C₃×Dic₃	—	6	1	no	—
28	H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃×Q₈	—	3	1	no	—
30	C₃×Q₈	—	3	1	no	—
31	C₃×Q₈	—	6	1	no	—
32	C₃×Q₈	—	9	1	no	—
33	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
34	Dic₆	—	3	1	no	—
35	Dic₆	—	9	1	no	—
36	C₃×C₈	—	9	1	no	—
37	C₃⋊C₈	—	3	1	no	—
38	C₃⋊C₈	—	9	1	no	—
39	C₂×C₃²	—	2	1	no	—
40	C₂×C₃²	#67	1	1	yes	characteristic
41	C₂×C₃²	C₃×D₄	1	1	yes	characteristic
42	Q₁₆	—	9	1	no	Sylow
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	3	1	no	—
45	C₃×C₄	—	3	1	no	—
46	C₃×C₄	—	6	1	no	—
47	C₃×C₄	—	6	1	no	—
48	C₃×C₄	—	6	2	no	—
49	C₃×C₄	—	6	1	no	—
50	C₃×C₄	—	18	1	no	—
51	C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
52	Dic₃	—	6	1	no	—
53	Dic₃	—	18	1	no	—
54	C₃²	—	2	1	no	—
55	C₃²	#216	1	1	yes	characteristic, complemented
56	C₃²	C₃×Q₁₆	1	1	yes	characteristic, complemented
57	Q₈	—	9	1	no	—
58	Q₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
59	C₈	—	9	1	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	3	1	no	—
62	C₂×C₃	—	6	1	no	—
63	C₂×C₃	C₃×#67	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	18	1	no	—
66	C₄	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
67	C₃	—	3	1	no	—
68	C₃	—	3	1	no	—
69	C₃	—	6	1	no	—
70	C₃	C₃×#216	1	1	yes	characteristic
71	C₂	#638⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#638⋊C₃
commutator, cosocle	C₃²×C₄	C₂²×C₃
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	Q₈×H(3,𝔽₃)	C₂
socle	C₂×C₃	C₃×#67
prime radical	C₂×H(3,𝔽₃)	D₄
hypercenter	Q₈	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₃×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
4. #3756	C₃	semidirect
28. H(3,𝔽₃)	Q₁₆	semidirect
55. C₃²	#216	semidirect
56. C₃²	C₃×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×H(3,𝔽₃)	2
#634⋊C₃	2
(C₃⋊(C₃⋊C₈))⋊C₃	2
#3756	3
C₃×#216	3

Subgroups — #3756⋊C3 #88513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3756⋊C₃ #88513