Subgroups — #13049241

Order 1296.

215 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	C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×#1094	C₃	1	2	yes	maximal, complemented
4	#88881	C₃	1	6	yes	maximal, complemented
5	(C₂²⋊A₄)⋊C₉	C₃	1	3	yes	maximal, complemented
6	#66440	—	4	2	no	maximal
7	C₂×#1094	—	3	2	no	—
8	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
9	C₃×C₂²⋊A₄	C₃²	1	3	yes	—
10	C₂²⋊(C₂²⋊C₉)	C₃²	1	1	yes	characteristic
11	C₂²⋊(C₂²⋊C₉)	C₃²	1	2	yes	—
12	C₂²×C₂²⋊C₉	C₃²	1	6	yes	—
13	C₂²×3₋¹⁺²	—	4	2	no	—
14	C₂²⋊H(3,𝔽₃)	—	4	2	no	—
15	C₂²⋊H(3,𝔽₃)	—	12	1	no	—
16	#1094	A₄	1	2	yes	complemented
17	C₃×C₂²⋊C₉	—	4	2	no	—
18	C₃×C₂²⋊C₉	—	12	1	no	—
19	C₉⋊A₄	—	4	12	no	—
20	C₉×A₄	—	4	6	no	—
21	#737	—	16	1	no	Sylow
22	C₂³×C₃²	—	3	2	no	—
23	C₂³×C₃²	—	9	1	no	—
24	C₂×C₂²⋊C₉	—	3	6	no	—
25	C₂×3₋¹⁺²	—	12	2	no	—
26	C₂⁴×C₃	—	3	1	no	—
27	C₂⁴×C₃	C₃³	1	1	yes	characteristic
28	C₂²⋊A₄	—	3	3	no	—
29	C₂²×C₃²	—	3	1	no	—
30	C₂²×C₃²	—	3	1	no	—
31	C₂²×C₃²	—	9	1	no	—
32	C₂²×C₃²	—	9	2	no	—
33	C₂²×C₃²	C₃×A₄	1	2	yes	—
34	C₃×A₄	—	4	6	no	—
35	C₃×A₄	—	12	3	no	—
36	C₂²×C₉	—	4	6	no	—
37	C₂²⋊C₉	—	4	2	no	—
38	C₂²⋊C₉	—	4	4	no	—
39	C₂²⋊C₉	—	12	1	no	—
40	C₂²⋊C₉	—	12	2	no	—
41	C₂²⋊C₉	C₃×A₄	1	6	yes	—
42	3₋¹⁺²	—	4	2	no	—
43	3₋¹⁺²	—	16	6	no	—
44	H(3,𝔽₃)	—	16	1	no	—
45	C₃×C₉	—	16	3	no	—
46	C₃×C₉	—	16	1	no	—
47	C₂³×C₃	—	3	2	no	—
48	C₂³×C₃	—	9	2	no	—
49	C₂³×C₃	—	9	1	no	—
50	C₂³×C₃	—	9	3	no	—
51	C₂×C₃²	—	3	2	no	—
52	C₂×C₃²	—	9	1	no	—
53	C₂×C₉	—	12	6	no	—
54	C₂⁴	#737	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₃	—	3	1	no	—
56	C₂²×C₃	—	3	1	no	—
57	C₂²×C₃	—	3	3	no	—
58	C₂²×C₃	—	3	2	no	—
59	C₂²×C₃	—	9	1	no	—
60	C₂²×C₃	—	9	1	no	—
61	C₂²×C₃	—	9	2	no	—
62	C₂²×C₃	—	9	3	no	—
63	C₂²×C₃	—	9	6	no	—
64	C₂²×C₃	C₃²×A₄	1	2	yes	—
65	A₄	—	12	6	no	—
66	A₄	—	36	3	no	—
67	C₃²	—	16	3	no	—
68	C₃²	A₄²	1	1	yes	characteristic
69	C₉	—	4	6	no	—
70	C₉	—	16	1	no	—
71	C₉	—	16	2	no	—
72	C₂³	—	3	2	no	—
73	C₂³	—	9	1	no	—
74	C₂×C₃	—	3	2	no	—
75	C₂×C₃	—	9	2	no	—
76	C₂×C₃	—	9	3	no	—
77	C₂×C₃	—	9	1	no	—
78	C₂²	—	3	1	no	—
79	C₂²	—	3	1	no	—
80	C₂²	—	9	1	no	—
81	C₂²	—	9	2	no	—
82	C₂²	#66440	1	2	yes	complemented
83	C₃	—	3	1	no	—
84	C₃	—	48	3	no	—
85	C₃	C₃×A₄²	1	1	yes	characteristic
86	C₂	—	3	2	no	—
87	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini	C₃	C₃×A₄²
commutator, socle, cosocle	C₂⁴×C₃	C₃³
Fitting	C₂⁴×C₃²	C₃²
prime radical	C₃²⋊(C₂²⋊A₄)	C₃
hypercenter	C₃²	A₄²
hypocenter	C₂⁴	#737
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃ ◁ C₃²

Complements

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

N	H ≅ G/N	type
2. C₃×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
3. C₂²×#1094	C₃	semidirect
4. #88881	C₃	semidirect
5. (C₂²⋊A₄)⋊C₉	C₃	semidirect
16. #1094	A₄	semidirect
54. C₂⁴	#737	semidirect
82. C₂²	#66440	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
C₃×C₂²⋊(C₂²⋊C₉)	3
C₂²×#1094	3
#88881	3
(C₂²⋊A₄)⋊C₉	3
#66440	4