Subgroups — C₉×#5464 #13152247

Order 1728.

274 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5464	C₃	1	1	yes	maximal, characteristic
2	C₉×H(3,𝔽₄)	C₃	1	1	yes	maximal, characteristic, complemented
3	#10589566	C₃	1	2	yes	maximal, complemented
4	C₉×C₂²⋊A₄	—	4	2	no	maximal
5	C₉×C₄²⋊C₃	—	4	3	no	maximal
6	C₉×#124	—	3	3	no	—
7	C₉×#120	—	3	2	no	—
8	C₃×H(3,𝔽₄)	C₃²	1	1	yes	characteristic
9	#5464	C₉	1	3	yes	direct product factor
10	C₃×C₂²⋊A₄	—	4	2	no	—
11	C₂⁴×C₉	A₄	1	2	yes	complemented
12	C₂²⋊(C₂²⋊C₉)	—	4	4	no	—
13	C₃×C₄²⋊C₃	—	4	3	no	—
14	C₂×Q₈×C₉	—	3	1	no	—
15	C₂×D₄×C₉	—	3	3	no	—
16	C₉×C₂²⋊C₄	—	3	6	no	—
17	C₄²×C₉	A₄	1	3	yes	complemented
18	C₄²⋊C₉	—	4	6	no	—
19	C₉×A₄	—	16	8	no	—
20	C₉×A₄	—	16	1	no	—
21	C₃×#124	—	3	3	no	—
22	C₃×#120	—	3	2	no	—
23	C₂³×C₉	—	3	2	no	—
24	C₂³×C₉	—	12	2	no	—
25	Q₈×C₉	—	12	1	no	—
26	D₄×C₉	—	12	3	no	—
27	C₂×C₄×C₉	—	3	3	no	—
28	H(3,𝔽₄)	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
29	C₂⁴×C₃	C₃×A₄	1	2	yes	—
30	C₂²⋊A₄	—	4	6	no	—
31	C₂×C₃×Q₈	—	3	1	no	—
32	C₂×C₃×D₄	—	3	3	no	—
33	C₃×C₂²⋊C₄	—	3	6	no	—
34	C₃×C₄²	C₃×A₄	1	3	yes	—
35	C₄²⋊C₃	—	4	9	no	—
36	C₃×A₄	—	16	8	no	—
37	C₃×A₄	—	16	1	no	—
38	C₂²×C₉	—	4	8	no	—
39	C₂²×C₉	—	6	6	no	—
40	C₂²×C₉	C₂²⋊A₄	1	1	yes	characteristic
41	C₂²⋊C₉	—	16	16	no	—
42	C₂²⋊C₉	—	16	2	no	—
43	C₄×C₉	—	6	3	no	—
44	#124	—	3	3	no	—
45	#120	—	3	2	no	—
46	C₃×C₉	—	64	1	no	Sylow
47	C₂³×C₃	—	3	2	no	—
48	C₂³×C₃	—	12	2	no	—
49	C₃×Q₈	—	12	1	no	—
50	C₃×D₄	—	12	3	no	—
51	C₂×C₃×C₄	—	3	3	no	—
52	C₂×C₉	—	3	1	no	—
53	C₂×C₉	—	12	2	no	—
54	C₂⁴	C₉×A₄	1	2	yes	complemented
55	C₂×Q₈	—	3	1	no	—
56	C₂×D₄	—	3	3	no	—
57	C₂²⋊C₄	—	3	6	no	—
58	C₄²	C₉×A₄	1	3	yes	complemented
59	C₂²×C₃	—	4	8	no	—
60	C₂²×C₃	—	6	6	no	—
61	C₂²×C₃	C₃×C₂²⋊A₄	1	1	yes	characteristic
62	A₄	—	16	24	no	—
63	A₄	—	16	3	no	—
64	C₃×C₄	—	6	3	no	—
65	C₃²	—	64	1	no	—
66	C₉	—	64	2	no	—
67	C₉	#5464	1	1	yes	characteristic, direct product factor
68	C₂³	—	3	2	no	—
69	C₂³	—	12	2	no	—
70	Q₈	—	12	1	no	—
71	D₄	—	12	3	no	—
72	C₂×C₄	—	3	3	no	—
73	C₂×C₃	—	3	1	no	—
74	C₂×C₃	—	12	2	no	—
75	C₂²	—	4	8	no	—
76	C₂²	—	6	6	no	—
77	C₂²	C₉×C₂²⋊A₄	1	1	yes	characteristic
78	C₄	—	6	3	no	—
79	C₃	—	64	3	no	—
80	C₃	C₃×#5464	1	1	yes	characteristic
81	C₂	—	3	1	no	—
82	C₂	—	12	2	no	—

	subgroup	G/H
center, hypercenter	C₉	#5464
commutator, hypocenter	H(3,𝔽₄)	C₃×C₉
Frattini, socle	C₂²×C₃	C₃×C₂²⋊A₄
Fitting	C₉×H(3,𝔽₄)	C₃
prime radical	C₃×#5464	C₃
cosocle	C₃×H(3,𝔽₄)	C₃²
LCS	G ▷ H(3,𝔽₄)
UCS	C₁ ◁ C₉

Complements

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

N	H ≅ G/N	type
2. C₉×H(3,𝔽₄)	C₃	semidirect
3. #10589566	C₃	semidirect
9. #5464	C₉	direct (N × H)
11. C₂⁴×C₉	A₄	semidirect
17. C₄²×C₉	A₄	semidirect
28. H(3,𝔽₄)	C₃×C₉	semidirect
54. C₂⁴	C₉×A₄	semidirect
58. C₄²	C₉×A₄	semidirect
67. C₉	#5464	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#5464	3
C₉×H(3,𝔽₄)	3
#10589566	3
C₉×C₂²⋊A₄	4
C₉×C₄²⋊C₃	4

Subgroups — C9×#5464 #13152247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5464 #13152247