Subgroups — #11721865

Order 864.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	#88532	C₂	1	3	yes	maximal, complemented
3	C₂×C₃⋊(C₉⋊C₈)	C₂	1	3	yes	maximal, complemented
4	#64157	—	3	1	no	maximal
5	#63510	—	3	3	no	maximal
6	C₃×Q₈×C₉	C₄	1	1	yes	characteristic
7	C₃×D₄×C₉	C₄	1	3	yes	—
8	C₂×C₃×C₄×C₉	C₂²	1	3	yes	—
9	C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₃⋊(C₉⋊C₈)	C₂²	1	3	yes	—
11	C₃²×#41	D₃	1	1	yes	characteristic
12	#3748	—	3	3	no	—
13	C₂×C₃⋊(C₃⋊C₈)	—	3	3	no	—
14	C₉×#41	D₃	1	3	yes	—
15	#3667	—	3	9	no	—
16	C₂×C₉⋊C₈	—	3	9	no	—
17	C₂²×C₃×C₉	C₂×C₄	1	3	yes	—
18	C₃×C₄×C₉	C₂×C₄	1	3	yes	—
19	C₃×C₄×C₉	C₂³	1	1	yes	characteristic
20	#948	—	9	3	no	—
21	#948	—	9	1	no	—
22	C₃²×Q₈	Dic₃	1	1	yes	characteristic
23	C₃²×D₄	Dic₃	1	3	yes	—
24	C₂×C₃²×C₄	C₂×D₃	1	3	yes	—
25	C₃⋊(C₃⋊C₈)	—	3	1	no	—
26	C₃⋊(C₃⋊C₈)	—	3	3	no	—
27	Q₈×C₉	Dic₃	1	3	yes	—
28	D₄×C₉	Dic₃	1	9	yes	—
29	C₂×C₄×C₉	C₂×D₃	1	9	yes	—
30	C₉⋊C₈	—	3	3	no	—
31	C₉⋊C₈	—	3	9	no	—
32	C₂×C₃×C₉	—	2	3	no	—
33	C₂×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
34	C₃×#41	D₉	1	3	yes	—
35	C₃×#41	C₃⋊D₃	1	1	yes	characteristic
36	#208	—	9	9	no	—
37	#208	—	9	3	no	—
38	C₂×C₃⋊C₈	—	9	9	no	—
39	C₂×C₃⋊C₈	—	9	3	no	—
40	C₂²×C₃²	C₂×Dic₃	1	3	yes	—
41	C₃²×C₄	C₂×Dic₃	1	3	yes	—
42	C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
43	C₂²×C₉	C₂×Dic₃	1	9	yes	—
44	C₄×C₉	C₂×Dic₃	1	9	yes	—
45	C₄×C₉	C₂²×D₃	1	3	yes	—
46	#131	—	27	1	no	Sylow
47	C₃×C₉	#131	1	1	yes	characteristic, complemented, Sylow
48	C₃×Q₈	Dic₉	1	3	yes	—
49	C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
50	C₃×D₄	Dic₉	1	9	yes	—
51	C₃×D₄	C₃⋊Dic₃	1	3	yes	—
52	C₂×C₃×C₄	C₂×D₉	1	9	yes	—
53	C₂×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
54	C₃⋊C₈	—	9	3	no	—
55	C₃⋊C₈	—	9	9	no	—
56	C₃⋊C₈	—	9	1	no	—
57	C₃⋊C₈	—	9	3	no	—
58	C₂×C₃²	—	2	3	no	—
59	C₂×C₃²	C₂²×Dic₃	1	1	yes	characteristic
60	C₂×C₉	—	2	9	no	—
61	C₂×C₉	C₂²×Dic₃	1	3	yes	—
62	#41	C₃⋊D₉	1	1	yes	characteristic
63	M₁₆	—	27	3	no	—
64	C₂×C₈	—	27	3	no	—
65	C₂²×C₃	C₂×Dic₉	1	9	yes	—
66	C₂²×C₃	C₂×C₃⋊Dic₃	1	3	yes	—
67	C₃×C₄	C₂×Dic₉	1	9	yes	—
68	C₃×C₄	C₂²×D₉	1	3	yes	—
69	C₃×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
70	C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
71	C₃²	#948	1	1	yes	characteristic
72	C₉	#948	1	3	yes	complemented
73	Q₈	C₃⋊Dic₉	1	1	yes	characteristic
74	D₄	C₃⋊Dic₉	1	3	yes	—
75	C₂×C₄	C₂×C₃⋊D₉	1	3	yes	—
76	C₈	—	27	1	no	—
77	C₈	—	27	3	no	—
78	C₂×C₃	—	2	9	no	—
79	C₂×C₃	—	2	3	no	—
80	C₂×C₃	C₂²×Dic₉	1	3	yes	—
81	C₂×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
82	C₂²	C₂×C₃⋊Dic₉	1	3	yes	—
83	C₄	C₂×C₃⋊Dic₉	1	3	yes	—
84	C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
85	C₃	#63510	1	3	yes	complemented
86	C₃	#64157	1	1	yes	characteristic
87	C₂	—	2	3	no	—
88	C₂	C₂²×C₃⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂²×C₄
Frattini	C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#41	C₂
socle	C₂×C₃²	C₂²×Dic₃
prime radical	C₃²×#41	D₃
cosocle	C₃×C₄×C₉	C₂³
hypercenter	#41	C₃⋊D₉
hypocenter	C₃×C₉	#131
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #88532	C₂	semidirect
3. C₂×C₃⋊(C₉⋊C₈)	C₂	semidirect
47. C₃×C₉	#131	semidirect
72. C₉	#948	semidirect
85. C₃	#63510	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#41	2
#88532	2
C₂×C₃⋊(C₉⋊C₈)	2
#64157	3
#63510	3