Subgroups — #13047785

Order 1296.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊(C₄²⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#88455	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₉⋊(C₄²⋊C₃)	C₃	1	6	yes	maximal, complemented
6	C₉×C₄²⋊C₃	C₃	1	3	yes	maximal, complemented
7	#66440	—	4	1	no	maximal
8	C₂×C₄×3₋¹⁺²	—	3	1	no	—
9	C₃²×C₄²	C₃²	1	1	yes	characteristic
10	C₃×C₄²⋊C₃	C₃²	1	3	yes	—
11	C₄²×C₉	C₃²	1	3	yes	—
12	C₄²⋊C₉	C₃²	1	1	yes	characteristic
13	C₄²⋊C₉	C₃²	1	3	yes	—
14	C₄²⋊C₉	C₃²	1	2	yes	—
15	C₂²×3₋¹⁺²	A₄	1	1	yes	characteristic
16	C₂²⋊H(3,𝔽₃)	—	4	1	no	—
17	#1094	—	4	1	no	—
18	C₃×C₂²⋊C₉	—	4	1	no	—
19	C₉⋊A₄	—	4	6	no	—
20	C₉×A₄	—	4	3	no	—
21	C₄×3₋¹⁺²	—	3	2	no	—
22	#737	—	16	1	no	Sylow
23	C₂×C₃²×C₄	—	3	1	no	—
24	C₂×C₄×C₉	—	3	3	no	—
25	C₂×3₋¹⁺²	—	3	1	no	—
26	C₃×C₄²	—	3	1	no	—
27	C₃×C₄²	C₃³	1	1	yes	characteristic
28	C₄²⋊C₃	—	3	3	no	—
29	C₂²×C₃²	C₃×A₄	1	1	yes	characteristic
30	C₃×A₄	—	4	3	no	—
31	C₃²×C₄	—	3	2	no	—
32	C₂²×C₉	C₃×A₄	1	3	yes	—
33	C₂²⋊C₉	—	4	1	no	—
34	C₂²⋊C₉	—	4	3	no	—
35	C₂²⋊C₉	—	4	2	no	—
36	C₄×C₉	—	3	6	no	—
37	3₋¹⁺²	—	16	1	no	—
38	3₋¹⁺²	—	16	6	no	—
39	3₋¹⁺²	C₄²⋊C₃	1	1	yes	characteristic, complemented
40	H(3,𝔽₃)	—	16	1	no	—
41	C₃×C₉	—	16	1	no	—
42	C₃×C₉	—	16	3	no	—
43	C₂×C₃×C₄	—	3	1	no	—
44	C₂×C₃×C₄	—	9	1	no	—
45	C₂×C₃²	—	3	1	no	—
46	C₂×C₉	—	3	3	no	—
47	C₄²	#737	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₃	—	3	1	no	—
49	C₂²×C₃	C₃²×A₄	1	1	yes	characteristic
50	A₄	—	12	3	no	—
51	C₃×C₄	—	3	2	no	—
52	C₃×C₄	—	9	2	no	—
53	C₃²	—	16	3	no	—
54	C₃²	C₃×C₄²⋊C₃	1	1	yes	characteristic
55	C₉	—	16	1	no	—
56	C₉	—	16	2	no	—
57	C₉	—	16	3	no	—
58	C₉	C₃×C₄²⋊C₃	1	3	yes	—
59	C₂×C₄	—	3	1	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	9	1	no	—
62	C₂²	#66440	1	1	yes	characteristic
63	C₄	—	3	2	no	—
64	C₃	—	3	1	no	—
65	C₃	—	48	3	no	—
66	C₃	C₃²×C₄²⋊C₃	1	1	yes	characteristic
67	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	C₃²×C₄²⋊C₃
commutator, cosocle	C₃×C₄²	C₃³
Frattini, socle	C₂²×C₃	C₃²×A₄
Fitting	C₄²×3₋¹⁺²	C₃
prime radical	C₃²⋊(C₄²⋊C₃)	C₃
hypercenter	3₋¹⁺²	C₄²⋊C₃
hypocenter	C₄²	#737
LCS	G ▷ C₃×C₄² ▷ C₄²
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₄²×3₋¹⁺²	C₃	semidirect
3. #88455	C₃	semidirect
4. C₃×C₄²⋊C₉	C₃	semidirect
5. C₉⋊(C₄²⋊C₃)	C₃	semidirect
6. C₉×C₄²⋊C₃	C₃	semidirect
39. 3₋¹⁺²	C₄²⋊C₃	semidirect
47. C₄²	#737	semidirect

Maximal subgroups

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

subgroup	index
C₄²×3₋¹⁺²	3
C₃²⋊(C₄²⋊C₃)	3
#88455	3
C₃×C₄²⋊C₉	3
C₉⋊(C₄²⋊C₃)	3
C₉×C₄²⋊C₃	3
#66440	4