Subgroups — #13048174

Order 1296.

361 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂²⋊A₄	C₃	1	1	yes	maximal, characteristic
2	C₂⁴×C₃×C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
4	(C₃×A₄)⋊C₉	—	4	5	no	maximal
5	C₂³×C₃×C₉	—	3	5	no	—
6	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
7	C₃×C₂²⋊A₄	—	3	1	no	—
8	C₃×C₂²⋊A₄	—	3	2	no	—
9	C₃×C₂²⋊A₄	C₉	1	3	yes	complemented
10	C₂⁴×C₉	—	3	1	no	—
11	C₂²⋊(C₂²⋊C₉)	—	3	2	no	—
12	C₃²×A₄	—	4	5	no	—
13	C₂²×C₃×C₉	—	3	10	no	—
14	C₂²×C₃×C₉	A₄	1	5	yes	complemented
15	C₃×C₂²⋊C₉	—	4	10	no	—
16	C₃²⋊C₉	—	16	1	no	Sylow
17	C₂³×C₃²	—	3	5	no	—
18	C₂³×C₉	—	3	15	no	—
19	C₂×C₃×C₉	—	3	5	no	—
20	C₂⁴×C₃	C₃×C₉	1	1	yes	characteristic
21	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₂⁴×C₃	3₋¹⁺²	1	2	yes	—
23	C₂²⋊A₄	—	3	3	no	—
24	C₂²×C₃²	—	3	10	no	—
25	C₂²×C₃²	C₃×A₄	1	5	yes	—
26	C₃×A₄	—	4	15	no	—
27	C₃×A₄	—	12	10	no	—
28	C₃×A₄	—	12	5	no	—
29	C₂²×C₉	—	3	30	no	—
30	C₂²×C₉	—	3	5	no	—
31	C₂²⋊C₉	—	12	10	no	—
32	C₃³	—	16	1	no	—
33	C₃×C₉	—	16	2	no	—
34	C₃×C₉	C₂²⋊A₄	1	1	yes	characteristic, complemented
35	C₂³×C₃	—	3	5	no	—
36	C₂³×C₃	—	3	10	no	—
37	C₂³×C₃	—	3	5	no	—
38	C₂×C₃²	—	3	5	no	—
39	C₂×C₉	—	3	15	no	—
40	C₂⁴	C₃²⋊C₉	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₃	—	3	10	no	—
42	C₂²×C₃	—	3	10	no	—
43	C₂²×C₃	—	3	20	no	—
44	C₂²×C₃	C₉×A₄	1	5	yes	—
45	C₂²×C₃	C₉⋊A₄	1	10	yes	—
46	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
47	A₄	—	12	15	no	—
48	C₃²	—	16	3	no	—
49	C₃²	—	48	2	no	—
50	C₃²	—	48	1	no	—
51	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
52	C₉	—	3	1	no	—
53	C₉	—	48	2	no	—
54	C₂³	—	3	5	no	—
55	C₂×C₃	—	3	5	no	—
56	C₂×C₃	—	3	10	no	—
57	C₂×C₃	—	3	5	no	—
58	C₂²	—	3	10	no	—
59	C₂²	(C₃×A₄)⋊C₉	1	5	yes	complemented
60	C₃	—	48	3	no	—
61	C₃	C₉×C₂²⋊A₄	1	1	yes	characteristic
62	C₃	C₉⋊(C₂²⋊A₄)	1	2	yes	—
63	C₃	C₃²⋊(C₂²⋊A₄)	1	1	yes	characteristic
64	C₂	—	3	5	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃×C₂²⋊A₄
commutator	C₂⁴×C₃	C₃×C₉
Fitting	C₂⁴×C₃×C₉	C₃
socle, cosocle	C₂⁴×C₃²	C₃²
prime radical	C₃²×C₂²⋊A₄	C₃
hypercenter	C₃×C₉	C₂²⋊A₄
hypocenter	C₂⁴	C₃²⋊C₉
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃² ◁ C₃×C₉

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₃×C₉	C₃	semidirect
3. C₃×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
9. C₃×C₂²⋊A₄	C₉	semidirect
14. C₂²×C₃×C₉	A₄	semidirect
34. C₃×C₉	C₂²⋊A₄	semidirect
40. C₂⁴	C₃²⋊C₉	semidirect
59. C₂²	(C₃×A₄)⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂²⋊A₄	3
C₂⁴×C₃×C₉	3
C₃×C₂²⋊(C₂²⋊C₉)	3
(C₃×A₄)⋊C₉	4