Subgroups — #88425

Order 432.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃₆	C₂	1	1	yes	maximal, characteristic
2	C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3715	—	3	1	no	maximal
5	#3675	—	3	1	no	maximal
6	C₂×C₃×D₉	—	2	1	no	—
7	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₉×Dic₃	—	2	1	no	—
9	C₃×D₁₂	—	3	1	no	—
10	C₃×Dic₆	D₃	1	1	yes	characteristic
11	C₃⋊(C₃⋊C₈)	—	3	1	no	—
12	Q₈×C₉	—	3	1	no	—
13	D₃₆	D₃	1	1	yes	characteristic
14	C₉⋊C₈	—	3	1	no	—
15	C₉⋊C₈	—	6	1	no	—
16	C₂×C₃×C₉	D₄	1	1	yes	characteristic
17	C₃×D₉	—	4	1	no	—
18	#215	—	9	1	no	—
19	#214	—	9	1	no	—
20	C₂×C₃×D₃	—	6	1	no	—
21	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
22	C₃×Dic₃	—	2	1	no	—
23	C₂×D₉	—	2	1	no	—
24	C₄×C₉	—	2	1	no	—
25	C₄×C₉	—	6	1	no	—
26	C₄×C₉	C₂×D₃	1	1	yes	characteristic
27	C₃×C₉	SD₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	3	1	no	—
29	C₃×D₄	—	9	1	no	—
30	D₁₂	—	3	1	no	—
31	Dic₆	D₉	1	1	yes	characteristic, complemented
32	C₃⋊C₈	—	9	1	no	—
33	C₃⋊C₈	—	9	1	no	—
34	C₃⋊C₈	—	18	1	no	—
35	C₂×C₃²	#67	1	1	yes	characteristic
36	C₃×D₃	—	12	1	no	—
37	C₂×C₉	—	2	1	no	—
38	C₂×C₉	#67	1	1	yes	characteristic
39	D₉	—	4	1	no	—
40	SD₁₆	—	27	1	no	Sylow
41	C₂²×C₃	—	18	1	no	—
42	C₂×D₃	—	6	1	no	—
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	6	1	no	—
45	C₃×C₄	C₂×D₉	1	1	yes	characteristic
46	C₃×C₄	D₃²	1	1	yes	characteristic
47	Dic₃	—	2	1	no	—
48	C₃²	#215	1	1	yes	characteristic
49	C₉	—	2	1	no	—
50	C₉	#214	1	1	yes	characteristic, complemented
51	Q₈	—	3	1	no	—
52	D₄	—	9	1	no	—
53	C₈	—	27	1	no	—
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	—	36	1	no	—
56	C₂×C₃	#611	1	1	yes	characteristic
57	C₂×C₃	#625	1	1	yes	characteristic
58	D₃	—	12	1	no	—
59	C₂²	—	18	1	no	—
60	C₄	—	6	1	no	—
61	C₄	D₃×D₉	1	1	yes	characteristic
62	C₃	—	2	1	no	—
63	C₃	#3675	1	1	yes	characteristic, complemented
64	C₃	#3715	1	1	yes	characteristic
65	C₂	—	36	1	no	—
66	C₂	#6194	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6194
commutator, Fitting, cosocle	C₃×C₄×C₉	C₂²
Frattini	C₃×C₄	D₃²
socle	C₂×C₃²	#67
prime radical	C₃×D₃₆	C₂
hypercenter	C₄	D₃×D₉
hypocenter	C₃×C₉	SD₁₆
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₉×Dic₆	C₂	semidirect
3. C₃⋊(C₉⋊C₈)	C₂	semidirect
27. C₃×C₉	SD₁₆	semidirect
31. Dic₆	D₉	semidirect
50. C₉	#214	semidirect
63. C₃	#3675	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃₆	2
C₉×Dic₆	2
C₃⋊(C₉⋊C₈)	2
#3715	3
#3675	3