Subgroups — #88532

Order 432.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
3	#3748	—	3	1	no	maximal
4	#3667	—	3	3	no	maximal
5	C₂²×C₃×C₉	C₄	1	1	yes	characteristic
6	C₃×C₄×C₉	C₄	1	1	yes	characteristic
7	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
9	C₃⋊(C₃⋊C₈)	—	3	2	no	—
10	C₂×C₄×C₉	D₃	1	3	yes	—
11	C₉⋊C₈	—	3	6	no	—
12	C₂×C₃×C₉	—	2	1	no	—
13	C₂×C₃×C₉	C₂×C₄	1	1	yes	characteristic
14	#208	—	9	3	no	—
15	#208	—	9	1	no	—
16	C₂²×C₃²	Dic₃	1	1	yes	characteristic
17	C₃²×C₄	Dic₃	1	1	yes	characteristic
18	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
19	C₂²×C₉	Dic₃	1	3	yes	—
20	C₄×C₉	Dic₃	1	3	yes	—
21	C₄×C₉	C₂×D₃	1	3	yes	—
22	C₃×C₉	M₁₆	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₃×C₄	D₉	1	3	yes	—
24	C₂×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
25	C₃⋊C₈	—	9	6	no	—
26	C₃⋊C₈	—	9	2	no	—
27	C₂×C₃²	—	2	1	no	—
28	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
29	C₂×C₉	—	2	3	no	—
30	C₂×C₉	C₂×Dic₃	1	3	yes	—
31	M₁₆	—	27	1	no	Sylow
32	C₂²×C₃	Dic₉	1	3	yes	—
33	C₂²×C₃	C₃⋊Dic₃	1	1	yes	characteristic
34	C₃×C₄	Dic₉	1	3	yes	—
35	C₃×C₄	C₂×D₉	1	3	yes	—
36	C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
37	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
38	C₃²	#208	1	1	yes	characteristic
39	C₉	#208	1	3	yes	complemented
40	C₂×C₄	C₃⋊D₉	1	1	yes	characteristic
41	C₈	—	27	2	no	—
42	C₂×C₃	—	2	3	no	—
43	C₂×C₃	—	2	1	no	—
44	C₂×C₃	C₂×Dic₉	1	3	yes	—
45	C₂×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
46	C₂²	C₃⋊Dic₉	1	1	yes	characteristic
47	C₄	C₃⋊Dic₉	1	1	yes	characteristic
48	C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
49	C₃	#3667	1	3	yes	complemented
50	C₃	#3748	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	C₂×C₃⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₄
Frattini	C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₃×C₄×C₉	C₂
socle	C₂×C₃²	C₂×Dic₃
prime radical	C₂²×C₃²	Dic₃
cosocle	C₃×C₄×C₉	C₂²
hypercenter	C₂×C₄	C₃⋊D₉
hypocenter	C₃×C₉	M₁₆
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₉⋊C₈)	C₂	semidirect
22. C₃×C₉	M₁₆	semidirect
39. C₉	#208	semidirect
49. C₃	#3667	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₉	2
C₃⋊(C₉⋊C₈)	2
#3748	3
#3667	3