Subgroups — #13048178

Order 1296.

251 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	#88907	C₃	1	2	yes	maximal
3	C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic
4	#66364	—	4	5	no	maximal
5	C₂³×3₋¹⁺²	—	3	5	no	—
6	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
7	C₂⁴×C₉	—	3	1	no	—
8	C₂²⋊(C₂²⋊C₉)	—	3	1	no	—
9	C₂²⋊(C₂²⋊C₉)	—	3	2	no	—
10	C₂²×3₋¹⁺²	—	3	10	no	—
11	C₂²×3₋¹⁺²	A₄	1	5	yes	—
12	#1094	—	4	10	no	—
13	C₃×C₂²⋊C₉	—	4	5	no	—
14	#733	—	16	1	no	Sylow
15	C₂³×C₃²	—	3	5	no	—
16	C₂³×C₉	—	3	15	no	—
17	C₂×3₋¹⁺²	—	3	5	no	—
18	C₂⁴×C₃	—	3	1	no	—
19	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
20	C₂²×C₃²	—	3	10	no	—
21	C₂²×C₃²	C₃×A₄	1	5	yes	—
22	C₂²×C₉	—	3	30	no	—
23	C₂²×C₉	—	3	5	no	—
24	C₂²⋊C₉	—	12	5	no	—
25	C₂²⋊C₉	—	12	10	no	—
26	3₋¹⁺²	—	16	2	no	—
27	3₋¹⁺²	C₂²⋊A₄	1	1	yes	characteristic
28	C₃×C₉	—	16	1	no	—
29	C₂³×C₃	—	3	5	no	—
30	C₂³×C₃	—	9	5	no	—
31	C₂×C₃²	—	3	5	no	—
32	C₂×C₉	—	3	15	no	—
33	C₂⁴	#733	1	1	yes	characteristic, complemented, Sylow
34	C₂²×C₃	—	3	10	no	—
35	C₂²×C₃	—	3	5	no	—
36	C₂²×C₃	—	9	10	no	—
37	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
38	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
39	C₉	—	3	1	no	—
40	C₉	—	48	1	no	—
41	C₉	—	48	2	no	—
42	C₂³	—	3	5	no	—
43	C₂×C₃	—	3	5	no	—
44	C₂×C₃	—	9	5	no	—
45	C₂²	—	3	10	no	—
46	C₂²	#66364	1	5	yes	complemented
47	C₃	—	3	1	no	—
48	C₃	C₃²⋊(C₂²⋊A₄)	1	1	yes	characteristic
49	C₂	—	3	5	no	—

	subgroup	G/H
center	C₃	C₃²⋊(C₂²⋊A₄)
commutator, prime radical, cosocle	C₂⁴×C₃²	C₃²
Frattini	C₃²	C₃×C₂²⋊A₄
Fitting	C₂⁴×3₋¹⁺²	C₃
socle	C₂⁴×C₃	H(3,𝔽₃)
hypercenter	3₋¹⁺²	C₂²⋊A₄
hypocenter	C₂⁴	#733
LCS	G ▷ C₂⁴×C₃² ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃ ◁ C₃² ◁ 3₋¹⁺²

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

N	H ≅ G/N	type
33. C₂⁴	#733	semidirect
46. C₂²	#66364	semidirect

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