Subgroups — C₉×#504⋊C₃ #13141215

Order 1728.

123 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃×#504⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₉×#504	C₃	1	1	yes	maximal, characteristic, complemented
4	#504⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₉×#41⋊C₃	C₄	1	1	yes	characteristic
6	C₃×C₉×C₁₆	—	4	1	no	maximal
7	C₃×#131⋊C₃	C₂×C₃	1	1	yes	characteristic
8	C₉×#131	C₂×C₃	1	1	yes	characteristic
9	#131⋊C₉	C₂×C₃	1	2	yes	—
10	C₉×M₃₂	—	3	1	no	—
11	C₂×C₉×C₁₆	—	3	1	no	—
12	C₉×SL(2,𝔽₃)	C₈	1	1	yes	characteristic
13	C₃×C₈×C₉	—	4	1	no	—
14	C₃×#504	C₃²	1	1	yes	characteristic
15	#504⋊C₃	C₉	1	3	yes	direct product factor
16	C₃×#41⋊C₃	C₃×C₄	1	1	yes	characteristic
17	C₉×#41	C₃×C₄	1	1	yes	characteristic
18	#41⋊C₉	C₃×C₄	1	2	yes	—
19	C₃²×C₁₆	—	4	1	no	—
20	C₉×M₁₆	—	3	1	no	—
21	C₂×C₈×C₉	—	3	1	no	—
22	C₉×C₁₆	—	3	1	no	—
23	C₉×C₁₆	—	4	2	no	—
24	C₉×C₁₆	A₄	1	1	yes	characteristic
25	C₃×C₄×C₉	—	4	1	no	—
26	C₃×#131	C₂×C₃²	1	1	yes	characteristic
27	#131⋊C₃	C₂×C₉	1	3	yes	—
28	C₃×M₃₂	—	3	1	no	—
29	C₂×C₃×C₁₆	—	3	1	no	—
30	C₃×SL(2,𝔽₃)	C₃×C₈	1	1	yes	characteristic
31	C₃²×C₈	—	4	1	no	—
32	Q₈×C₉	C₃×C₈	1	1	yes	characteristic
33	D₄×C₉	—	3	1	no	—
34	C₂×C₄×C₉	—	3	1	no	—
35	Q₈⋊C₉	C₃×C₈	1	2	yes	—
36	C₈×C₉	—	3	1	no	—
37	C₈×C₉	—	4	2	no	—
38	C₈×C₉	C₂×A₄	1	1	yes	characteristic
39	#504	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃×C₉	—	4	1	no	—
41	C₃×#41	C₃²×C₄	1	1	yes	characteristic
42	#41⋊C₃	C₄×C₉	1	3	yes	—
43	C₃×M₁₆	—	3	1	no	—
44	C₂×C₃×C₈	—	3	1	no	—
45	C₃×C₁₆	—	3	1	no	—
46	C₃×C₁₆	—	4	3	no	—
47	C₃×C₁₆	C₃×A₄	1	1	yes	characteristic
48	C₃²×C₄	—	4	1	no	—
49	C₂²×C₉	—	3	1	no	—
50	C₄×C₉	—	3	1	no	—
51	C₄×C₉	—	4	2	no	—
52	C₄×C₉	C₄×A₄	1	1	yes	characteristic
53	#131	C₂×C₃×C₉	1	1	yes	characteristic
54	M₃₂	—	3	1	no	—
55	C₂×C₁₆	—	3	1	no	—
56	C₃×C₉	—	4	1	no	Sylow
57	C₃×Q₈	C₃²×C₈	1	1	yes	characteristic
58	C₃×D₄	—	3	1	no	—
59	C₂×C₃×C₄	—	3	1	no	—
60	SL(2,𝔽₃)	C₈×C₉	1	3	yes	—
61	C₃×C₈	—	3	1	no	—
62	C₃×C₈	—	4	3	no	—
63	C₃×C₈	C₂×C₃×A₄	1	1	yes	characteristic
64	C₂×C₃²	—	4	1	no	—
65	C₂×C₉	—	4	2	no	—
66	C₂×C₉	—	6	1	no	—
67	C₂×C₉	C₈×A₄	1	1	yes	characteristic
68	#41	C₃×C₄×C₉	1	1	yes	characteristic
69	M₁₆	—	3	1	no	—
70	C₂×C₈	—	3	1	no	—
71	C₁₆	—	3	1	no	—
72	C₁₆	C₉×A₄	1	1	yes	characteristic
73	C₂²×C₃	—	3	1	no	—
74	C₃×C₄	—	3	1	no	—
75	C₃×C₄	—	4	3	no	—
76	C₃×C₄	C₃×C₄×A₄	1	1	yes	characteristic
77	C₃²	—	4	1	no	—
78	C₉	—	4	2	no	—
79	C₉	#504⋊C₃	1	1	yes	characteristic, direct product factor
80	Q₈	C₃×C₈×C₉	1	1	yes	characteristic
81	D₄	—	3	1	no	—
82	C₂×C₄	—	3	1	no	—
83	C₈	—	3	1	no	—
84	C₈	C₂×C₉×A₄	1	1	yes	characteristic
85	C₂×C₃	—	4	3	no	—
86	C₂×C₃	—	6	1	no	—
87	C₂×C₃	C₃×C₈×A₄	1	1	yes	characteristic
88	C₂²	—	3	1	no	—
89	C₄	—	3	1	no	—
90	C₄	C₄×C₉×A₄	1	1	yes	characteristic
91	C₃	—	4	3	no	—
92	C₃	C₃×#504⋊C₃	1	1	yes	characteristic
93	C₂	—	6	1	no	—
94	C₂	C₈×C₉×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₉×C₁₆	A₄
commutator, hypocenter	Q₈	C₃×C₈×C₉
Frattini	C₃×C₈	C₂×C₃×A₄
Fitting	C₉×#504	C₃
socle	C₂×C₃	C₃×C₈×A₄
prime radical	C₃×#41⋊C₃	C₃×C₄
cosocle	C₃×#131	C₂×C₃²
LCS	G ▷ Q₈
UCS	C₁ ◁ C₉×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₉×#504	C₃	semidirect
4. #504⋊C₉	C₃	semidirect
15. #504⋊C₃	C₉	direct (N × H)
39. #504	C₃×C₉	semidirect
79. C₉	#504⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#131⋊C₃	2
C₃×#504⋊C₃	3
C₉×#504	3
#504⋊C₉	3
C₃×C₉×C₁₆	4

Subgroups — C9×#504⋊C3 #13141215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#504⋊C₃ #13141215