Subgroups — #88551

Order 432.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊Dic₉	C₂	1	2	yes	maximal, complemented
3	#3757	—	3	1	no	maximal
4	#3676	—	3	3	no	maximal
5	C₂²×C₃×C₉	—	2	2	no	—
6	C₂²×C₃×C₉	C₄	1	2	yes	complemented
7	C₂²×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₃⋊Dic₉	—	2	2	no	—
9	C₂³×C₃²	D₃	1	1	yes	characteristic
10	C₂×C₃⋊Dic₃	—	3	2	no	—
11	C₂³×C₉	D₃	1	3	yes	—
12	C₂×Dic₉	—	3	6	no	—
13	C₂×C₃×C₉	—	2	2	no	—
14	C₂×C₃×C₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₉	D₄	1	2	yes	—
16	#217	—	9	3	no	—
17	#217	—	9	1	no	—
18	C₂²×C₃²	—	2	2	no	—
19	C₂²×C₃²	Dic₃	1	2	yes	—
20	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
21	C₃⋊Dic₃	—	6	2	no	—
22	C₂²×C₉	—	2	6	no	—
23	C₂²×C₉	Dic₃	1	6	yes	complemented
24	C₂²×C₉	C₂×D₃	1	3	yes	—
25	Dic₉	—	6	6	no	—
26	C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
27	C₂³×C₃	D₉	1	3	yes	—
28	C₂³×C₃	C₃⋊D₃	1	1	yes	characteristic
29	C₂×Dic₃	—	9	6	no	—
30	C₂×Dic₃	—	9	2	no	—
31	C₂×C₃²	—	2	2	no	—
32	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
33	C₂×C₃²	#67	1	2	yes	—
34	C₂×C₉	—	2	6	no	—
35	C₂×C₉	C₂×Dic₃	1	3	yes	—
36	C₂×C₉	#67	1	6	yes	—
37	C₂²⋊C₄	—	27	1	no	Sylow
38	C₂²×C₃	—	2	6	no	—
39	C₂²×C₃	—	2	2	no	—
40	C₂²×C₃	Dic₉	1	6	yes	complemented
41	C₂²×C₃	C₂×D₉	1	3	yes	—
42	C₂²×C₃	C₃⋊Dic₃	1	2	yes	—
43	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
44	Dic₃	—	18	6	no	—
45	Dic₃	—	18	2	no	—
46	C₃²	#217	1	1	yes	characteristic
47	C₉	#217	1	3	yes	complemented
48	C₂³	C₃⋊D₉	1	1	yes	characteristic
49	C₂×C₄	—	27	2	no	—
50	C₂×C₃	—	2	6	no	—
51	C₂×C₃	—	2	2	no	—
52	C₂×C₃	C₂×Dic₉	1	3	yes	—
53	C₂×C₃	#611	1	6	yes	—
54	C₂×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
55	C₂×C₃	#638	1	2	yes	—
56	C₂²	—	2	2	no	—
57	C₂²	C₃⋊Dic₉	1	2	yes	complemented
58	C₂²	C₂×C₃⋊D₉	1	1	yes	characteristic
59	C₄	—	54	2	no	—
60	C₃	#3676	1	3	yes	complemented
61	C₃	#3757	1	1	yes	characteristic
62	C₂	—	2	2	no	—
63	C₂	C₂×C₃⋊Dic₉	1	1	yes	characteristic
64	C₂	#6233	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₄
Frattini	C₂²×C₃	C₂×C₃⋊D₃
Fitting	C₂³×C₃×C₉	C₂
socle	C₂²×C₃²	C₂×D₃
prime radical	C₂³×C₃²	D₃
cosocle	C₂²×C₃×C₉	C₂²
hypercenter	C₂³	C₃⋊D₉
hypocenter	C₃×C₉	C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊Dic₉	C₂	semidirect
6. C₂²×C₃×C₉	C₄	semidirect
23. C₂²×C₉	Dic₃	semidirect
26. C₃×C₉	C₂²⋊C₄	semidirect
40. C₂²×C₃	Dic₉	semidirect
47. C₉	#217	semidirect
57. C₂²	C₃⋊Dic₉	semidirect
60. C₃	#3676	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₉	2
C₂×C₃⋊Dic₉	2
#3757	3
#3676	3