Subgroups — C₉×#5465 #13152249

Order 1728.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5465	C₃	1	1	yes	maximal, characteristic
2	C₉×H(3,𝔽₄)	C₃	1	1	yes	maximal, characteristic, complemented
3	#10589567	C₃	1	2	yes	maximal, complemented
4	C₂²×C₉×A₄	—	4	2	no	maximal
5	C₂×C₉×SL(2,𝔽₃)	—	4	3	no	maximal
6	C₉×#124	—	3	3	no	—
7	C₉×#120	—	3	2	no	—
8	C₂×C₉×A₄	—	4	6	no	—
9	C₉×SL(2,𝔽₃)	—	4	3	no	—
10	C₃×H(3,𝔽₄)	C₃²	1	1	yes	characteristic
11	#5465	C₉	1	3	yes	direct product factor
12	C₂²×C₃×A₄	—	4	2	no	—
13	C₂×C₃×SL(2,𝔽₃)	—	4	3	no	—
14	C₂⁴×C₉	A₄	1	2	yes	complemented
15	C₂²×C₂²⋊C₉	—	4	4	no	—
16	C₂×Q₈×C₉	A₄	1	3	yes	complemented
17	C₂×D₄×C₉	—	3	3	no	—
18	C₂×Q₈⋊C₉	—	4	6	no	—
19	C₉×C₂²⋊C₄	—	3	6	no	—
20	C₄²×C₉	—	3	1	no	—
21	C₂²×C₃×C₉	—	16	1	no	—
22	C₉×A₄	—	4	2	no	—
23	C₃×#124	—	3	3	no	—
24	C₃×#120	—	3	2	no	—
25	C₂×C₃×A₄	—	4	6	no	—
26	C₃×SL(2,𝔽₃)	—	4	3	no	—
27	C₂³×C₉	—	3	2	no	—
28	C₂³×C₉	—	4	6	no	—
29	C₂×C₂²⋊C₉	—	4	12	no	—
30	Q₈×C₉	—	4	3	no	—
31	D₄×C₉	—	12	3	no	—
32	C₂×C₄×C₉	—	3	3	no	—
33	Q₈⋊C₉	—	4	6	no	—
34	H(3,𝔽₄)	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₃×C₉	—	16	3	no	—
36	C₂⁴×C₃	C₃×A₄	1	2	yes	—
37	C₂²×A₄	—	4	6	no	—
38	C₂×C₃×Q₈	C₃×A₄	1	3	yes	—
39	C₂×C₃×D₄	—	3	3	no	—
40	C₂×SL(2,𝔽₃)	—	4	9	no	—
41	C₃×C₂²⋊C₄	—	3	6	no	—
42	C₃×C₄²	—	3	1	no	—
43	C₂²×C₃²	—	16	1	no	—
44	C₃×A₄	—	4	2	no	—
45	C₂²×C₉	—	4	2	no	—
46	C₂²×C₉	—	6	6	no	—
47	C₂²×C₉	—	12	2	no	—
48	C₂²×C₉	—	16	2	no	—
49	C₂²×C₉	C₂²⋊A₄	1	1	yes	characteristic
50	C₂²⋊C₉	—	4	4	no	—
51	C₄×C₉	—	6	3	no	—
52	#124	—	3	3	no	—
53	#120	—	3	2	no	—
54	C₃×C₉	—	16	1	no	Sylow
55	C₂³×C₃	—	3	2	no	—
56	C₂³×C₃	—	4	6	no	—
57	C₂×A₄	—	4	18	no	—
58	C₃×Q₈	—	4	3	no	—
59	C₃×D₄	—	12	3	no	—
60	C₂×C₃×C₄	—	3	3	no	—
61	SL(2,𝔽₃)	—	4	9	no	—
62	C₂×C₃²	—	16	3	no	—
63	C₂×C₉	—	12	2	no	—
64	C₂×C₉	—	16	6	no	—
65	C₂×C₉	#997	1	3	yes	—
66	C₂⁴	C₉×A₄	1	2	yes	complemented
67	C₂×Q₈	C₉×A₄	1	3	yes	complemented
68	C₂×D₄	—	3	3	no	—
69	C₂²⋊C₄	—	3	6	no	—
70	C₄²	—	3	1	no	—
71	C₂²×C₃	—	4	2	no	—
72	C₂²×C₃	—	6	6	no	—
73	C₂²×C₃	—	12	2	no	—
74	C₂²×C₃	—	16	3	no	—
75	C₂²×C₃	C₃×C₂²⋊A₄	1	1	yes	characteristic
76	A₄	—	4	6	no	—
77	C₃×C₄	—	6	3	no	—
78	C₃²	—	16	1	no	—
79	C₉	—	16	2	no	—
80	C₉	#5465	1	1	yes	characteristic, direct product factor
81	C₂³	—	3	2	no	—
82	C₂³	—	4	6	no	—
83	Q₈	—	4	3	no	—
84	D₄	—	12	3	no	—
85	C₂×C₄	—	3	3	no	—
86	C₂×C₃	—	12	2	no	—
87	C₂×C₃	—	16	9	no	—
88	C₂×C₃	C₃×#997	1	3	yes	—
89	C₂²	—	4	2	no	—
90	C₂²	—	6	6	no	—
91	C₂²	—	12	2	no	—
92	C₂²	C₉×C₂²⋊A₄	1	1	yes	characteristic
93	C₄	—	6	3	no	—
94	C₃	—	16	3	no	—
95	C₃	C₃×#5465	1	1	yes	characteristic
96	C₂	—	12	2	no	—
97	C₂	C₉×#997	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₉	C₂²⋊A₄
commutator, hypocenter	H(3,𝔽₄)	C₃×C₉
Frattini, socle	C₂²×C₃	C₃×C₂²⋊A₄
Fitting	C₉×H(3,𝔽₄)	C₃
prime radical	C₃×#5465	C₃
cosocle	C₃×H(3,𝔽₄)	C₃²
LCS	G ▷ H(3,𝔽₄)
UCS	C₁ ◁ C₂²×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. #10589567	C₃	semidirect
11. #5465	C₉	direct (N × H)
14. C₂⁴×C₉	A₄	semidirect
16. C₂×Q₈×C₉	A₄	semidirect
34. H(3,𝔽₄)	C₃×C₉	semidirect
66. C₂⁴	C₉×A₄	semidirect
67. C₂×Q₈	C₉×A₄	semidirect
80. C₉	#5465	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×#5465	3
C₉×H(3,𝔽₄)	3
#10589567	3
C₂²×C₉×A₄	4
C₂×C₉×SL(2,𝔽₃)	4

Subgroups — C9×#5465 #13152249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5465 #13152249