Subgroups — #88880

Order 432.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×A₄	C₃	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	#1094	—	4	1	no	maximal
5	C₉⋊A₄	—	4	1	no	maximal
6	C₂×C₃×A₄	—	3	1	no	—
7	C₂×C₂²⋊C₉	—	3	1	no	—
8	C₂⁴×C₃	C₃²	1	1	yes	characteristic
9	C₂²×A₄	—	3	1	no	—
10	C₂²×C₃²	—	4	1	no	—
11	C₃×A₄	A₄	1	1	yes	characteristic
12	C₂²×C₉	—	4	1	no	—
13	C₂²⋊C₉	—	4	2	no	—
14	C₂²⋊C₉	—	4	2	no	—
15	C₂²⋊C₉	—	12	2	no	—
16	C₂²⋊C₉	A₄	1	1	yes	characteristic, complemented
17	3₋¹⁺²	—	16	1	no	Sylow
18	C₂³×C₃	—	3	1	no	—
19	C₂³×C₃	—	3	1	no	—
20	C₂³×C₃	—	9	1	no	—
21	C₂×A₄	—	3	3	no	—
22	C₂×C₃²	—	12	1	no	—
23	C₂×C₉	—	12	1	no	—
24	C₂⁴	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
25	C₂²×C₃	—	3	2	no	—
26	C₂²×C₃	—	9	1	no	—
27	C₂²×C₃	—	9	1	no	—
28	C₂²×C₃	—	9	1	no	—
29	C₂²×C₃	—	12	1	no	—
30	C₂²×C₃	C₃×A₄	1	1	yes	characteristic
31	C₂²×C₃	C₃×A₄	1	1	yes	characteristic
32	A₄	—	3	1	no	—
33	C₃²	—	4	1	no	—
34	C₉	—	4	1	no	—
35	C₉	—	16	2	no	—
36	C₂³	—	3	1	no	—
37	C₂³	—	3	1	no	—
38	C₂³	—	9	1	no	—
39	C₂×C₃	—	3	1	no	—
40	C₂×C₃	—	3	1	no	—
41	C₂×C₃	—	9	1	no	—
42	C₂×C₃	—	12	3	no	—
43	C₂²	—	3	2	no	—
44	C₂²	—	9	1	no	—
45	C₂²	—	9	1	no	—
46	C₂²	—	9	1	no	—
47	C₂²	C₉⋊A₄	1	1	yes	characteristic, complemented
48	C₂²	#1094	1	1	yes	characteristic, complemented
49	C₃	—	12	1	no	—
50	C₃	A₄²	1	1	yes	characteristic
51	C₂	—	3	1	no	—
52	C₂	—	3	1	no	—
53	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	A₄²
commutator, Fitting, socle, cosocle	C₂⁴×C₃	C₃²
prime radical	C₂²×C₃×A₄	C₃
hypocenter	C₂⁴	3₋¹⁺²
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₂²⋊(C₂²⋊C₉)	C₃	semidirect
3. C₂²×C₂²⋊C₉	C₃	semidirect
16. C₂²⋊C₉	A₄	semidirect
24. C₂⁴	3₋¹⁺²	semidirect
47. C₂²	C₉⋊A₄	semidirect
48. C₂²	#1094	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×A₄	3
C₂²⋊(C₂²⋊C₉)	3
C₂²×C₂²⋊C₉	3
#1094	4
C₉⋊A₄	4