Subgroups — #88439

Order 432.

75 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6229	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	#3719	—	3	1	no	maximal
5	#3718	—	3	1	no	maximal
6	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	#1084	—	2	1	no	—
8	(C₃⋊Dic₃)⋊C₃	—	2	1	no	—
9	#634	D₃	1	1	yes	characteristic
10	C₃×Dic₆	—	3	1	no	—
11	C₃×Dic₆	—	3	1	no	—
12	C₃×Dic₆	—	3	1	no	—
13	C₃×Dic₆	—	6	1	no	—
14	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
15	C₃×C₃⋊C₈	—	3	1	no	—
16	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	#216	—	9	1	no	—
18	#216	—	9	1	no	—
19	Dic₁₂	—	9	1	no	—
20	C₃²×C₄	—	2	1	no	—
21	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
22	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
23	C₃⋊Dic₃	—	2	1	no	—
24	C₃×Dic₃	—	6	2	no	—
25	C₃×Dic₃	—	6	1	no	—
26	C₃×Dic₃	—	6	1	no	—
27	C₃×Dic₃	—	6	1	no	—
28	H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃×Q₈	—	9	1	no	—
30	C₃×Q₈	—	9	1	no	—
31	Dic₆	—	3	1	no	—
32	Dic₆	—	3	1	no	—
33	Dic₆	—	3	1	no	—
34	Dic₆	—	6	1	no	—
35	Dic₆	—	18	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	D₁₂	1	1	yes	characteristic
41	C₂×C₃²	#67	1	1	yes	characteristic
42	Q₁₆	—	27	1	no	Sylow
43	C₃×C₄	—	3	1	no	—
44	C₃×C₄	—	6	1	no	—
45	C₃×C₄	—	12	1	no	—
46	C₃×C₄	—	18	1	no	—
47	C₃×C₄	—	18	1	no	—
48	C₃×C₄	D₃²	1	1	yes	characteristic
49	Dic₃	—	6	1	no	—
50	Dic₃	—	6	1	no	—
51	Dic₃	—	6	2	no	—
52	Dic₃	—	6	1	no	—
53	Dic₃	—	18	1	no	—
54	C₃²	—	2	1	no	—
55	C₃²	Dic₁₂	1	1	yes	characteristic, complemented
56	C₃²	#216	1	1	yes	characteristic, complemented
57	Q₈	—	9	1	no	—
58	Q₈	—	9	1	no	—
59	C₈	—	9	1	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	6	1	no	—
62	C₂×C₃	—	12	1	no	—
63	C₂×C₃	#626	1	1	yes	characteristic
64	C₄	—	18	1	no	—
65	C₄	—	18	1	no	—
66	C₄	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
67	C₃	—	3	1	no	—
68	C₃	—	6	1	no	—
69	C₃	—	12	1	no	—
70	C₃	#3719	1	1	yes	characteristic
71	C₂	#6200	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6200
commutator, Fitting, cosocle	C₄×H(3,𝔽₃)	C₂²
Frattini	C₃×C₄	D₃²
socle	C₂×C₃	#626
prime radical	C₂×H(3,𝔽₃)	D₄
hypercenter	C₄	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	Q₁₆
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
28. H(3,𝔽₃)	Q₁₆	semidirect
55. C₃²	Dic₁₂	semidirect
56. C₃²	#216	semidirect

Maximal subgroups

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

subgroup	index
#6229	2
#634⋊C₃	2
(C₃⋊(C₃⋊C₈))⋊C₃	2
#3719	3
#3718	3