Subgroups — #88431

Order 432.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊((C₃⋊D₃)⋊C₃)	C₂	1	1	yes	maximal, characteristic
2	#634⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#6180	C₂	1	1	yes	maximal, characteristic, complemented
4	#3717	—	3	1	no	maximal
5	#3716	—	3	1	no	maximal
6	C₂×(C₃⋊D₃)⋊C₃	—	2	1	no	—
7	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	(C₃⋊Dic₃)⋊C₃	—	2	1	no	—
9	C₄⋊(C₃⋊D₃)	D₃	1	1	yes	characteristic
10	#634	D₃	1	1	yes	characteristic, complemented
11	C₃×D₁₂	—	3	1	no	—
12	C₃×Dic₆	—	3	1	no	—
13	C₃×C₃⋊C₈	—	3	1	no	—
14	C₃×C₃⋊C₈	—	3	1	no	—
15	C₃×C₃⋊C₈	—	6	1	no	—
16	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	(C₃⋊D₃)⋊C₃	—	4	1	no	—
18	#215	—	9	1	no	—
19	#214	—	9	1	no	—
20	#204	—	9	1	no	—
21	C₂×C₃⋊D₃	—	2	1	no	—
22	C₂×C₃×D₃	—	6	1	no	—
23	C₃²×C₄	—	2	1	no	—
24	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
25	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₃⋊Dic₃	—	2	1	no	—
27	C₃×Dic₃	—	6	1	no	—
28	H(3,𝔽₃)	SD₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃×Q₈	—	9	1	no	—
30	C₃×D₄	—	9	1	no	—
31	D₁₂	—	3	1	no	—
32	D₁₂	—	18	1	no	—
33	Dic₆	—	3	1	no	—
34	Dic₆	—	9	1	no	—
35	C₃×C₈	—	9	1	no	—
36	C₃⋊C₈	—	3	1	no	—
37	C₃⋊C₈	—	3	1	no	—
38	C₃⋊C₈	—	6	1	no	—
39	C₂×C₃²	—	2	1	no	—
40	C₂×C₃²	#67	1	1	yes	characteristic
41	C₂×C₃²	#67	1	1	yes	characteristic
42	C₃⋊D₃	—	4	1	no	—
43	C₃×D₃	—	12	1	no	—
44	SD₁₆	—	27	1	no	Sylow
45	C₂²×C₃	—	18	1	no	—
46	C₂×D₃	—	6	1	no	—
47	C₂×D₃	—	18	1	no	—
48	C₃×C₄	—	3	1	no	—
49	C₃×C₄	—	6	1	no	—
50	C₃×C₄	—	6	1	no	—
51	C₃×C₄	—	18	1	no	—
52	C₃×C₄	D₃²	1	1	yes	characteristic
53	Dic₃	—	6	1	no	—
54	Dic₃	—	18	1	no	—
55	C₃²	—	2	1	no	—
56	C₃²	#214	1	1	yes	characteristic, complemented
57	C₃²	#215	1	1	yes	characteristic, complemented
58	Q₈	—	9	1	no	—
59	D₄	—	9	1	no	—
60	C₈	—	9	1	no	—
61	C₂×C₃	—	3	1	no	—
62	C₂×C₃	—	6	1	no	—
63	C₂×C₃	—	6	1	no	—
64	C₂×C₃	—	36	1	no	—
65	C₂×C₃	#625	1	1	yes	characteristic
66	D₃	—	12	1	no	—
67	D₃	—	36	1	no	—
68	C₂²	—	18	1	no	—
69	C₄	—	18	1	no	—
70	C₄	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
71	C₃	—	3	1	no	—
72	C₃	—	6	1	no	—
73	C₃	—	6	1	no	—
74	C₃	#3715	1	1	yes	characteristic
75	C₂	—	36	1	no	—
76	C₂	#6198	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6198
commutator, Fitting, cosocle	C₄×H(3,𝔽₃)	C₂²
Frattini	C₃×C₄	D₃²
socle	C₂×C₃	#625
prime radical	C₄⋊((C₃⋊D₃)⋊C₃)	C₂
hypercenter	C₄	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	SD₁₆
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
2. #634⋊C₃	C₂	semidirect
3. #6180	C₂	semidirect
10. #634	D₃	semidirect
28. H(3,𝔽₃)	SD₁₆	semidirect
56. C₃²	#214	semidirect
57. C₃²	#215	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊((C₃⋊D₃)⋊C₃)	2
#634⋊C₃	2
#6180	2
#3717	3
#3716	3