Subgroups — #11720555

Order 864.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#88532	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#63649	—	3	1	no	maximal
5	#63381	—	3	3	no	maximal
6	C₂×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₈×C₉	—	2	1	no	—
8	C₃⋊(C₉⋊C₈)	—	2	1	no	—
9	C₃⋊(C₉⋊C₈)	C₄	1	2	yes	—
10	C₃²×M₁₆	D₃	1	1	yes	characteristic
11	#3748	—	3	1	no	—
12	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
13	C₉×M₁₆	D₃	1	3	yes	—
14	#3667	—	3	3	no	—
15	C₂×C₉⋊C₈	—	3	3	no	—
16	C₂²×C₃×C₉	Q₈	1	1	yes	characteristic
17	C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
18	C₃×C₄×C₉	D₄	1	1	yes	characteristic
19	#822	—	9	3	no	—
20	#822	—	9	1	no	—
21	C₂×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
22	C₃²×C₈	—	2	1	no	—
23	C₃⋊(C₃⋊C₈)	—	3	2	no	—
24	C₃⋊(C₃⋊C₈)	—	6	1	no	—
25	C₂×C₄×C₉	C₂×D₃	1	3	yes	—
26	C₈×C₉	—	2	3	no	—
27	C₉⋊C₈	—	3	6	no	—
28	C₉⋊C₈	—	6	3	no	—
29	C₂×C₃×C₉	—	2	1	no	—
30	C₂×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
31	C₃×M₁₆	D₉	1	3	yes	—
32	C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
33	#208	—	9	3	no	—
34	#208	—	9	1	no	—
35	C₂×C₃⋊C₈	—	9	3	no	—
36	C₂×C₃⋊C₈	—	9	1	no	—
37	C₂²×C₃²	Dic₆	1	1	yes	characteristic
38	C₃²×C₄	C₄×D₃	1	1	yes	characteristic
39	C₃²×C₄	#67	1	1	yes	characteristic
40	C₂²×C₉	Dic₆	1	3	yes	—
41	C₄×C₉	C₄×D₃	1	3	yes	—
42	C₄×C₉	#67	1	3	yes	—
43	#108	—	27	1	no	Sylow
44	C₃×C₉	#108	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃×C₄	C₂×D₉	1	3	yes	—
46	C₂×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
47	C₃×C₈	—	2	3	no	—
48	C₃×C₈	—	2	1	no	—
49	C₃⋊C₈	—	9	6	no	—
50	C₃⋊C₈	—	9	2	no	—
51	C₃⋊C₈	—	18	3	no	—
52	C₃⋊C₈	—	18	1	no	—
53	C₂×C₃²	—	2	1	no	—
54	C₂×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
55	C₂×C₉	—	2	3	no	—
56	C₂×C₉	Dic₃⋊C₄	1	3	yes	—
57	M₁₆	—	27	1	no	—
58	M₁₆	C₃⋊D₉	1	1	yes	characteristic
59	C₂×C₈	—	27	1	no	—
60	C₂²×C₃	Dic₁₈	1	3	yes	—
61	C₂²×C₃	#634	1	1	yes	characteristic
62	C₃×C₄	C₄×D₉	1	3	yes	—
63	C₃×C₄	#611	1	3	yes	—
64	C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
65	C₃×C₄	#638	1	1	yes	characteristic
66	C₃²	#822	1	1	yes	characteristic
67	C₉	#822	1	3	yes	complemented
68	C₂×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	27	2	no	—
71	C₈	—	54	1	no	—
72	C₂×C₃	—	2	3	no	—
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	Dic₉⋊C₄	1	3	yes	—
75	C₂×C₃	#3750	1	1	yes	characteristic
76	C₂²	#6226	1	1	yes	characteristic
77	C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
78	C₄	#6233	1	1	yes	characteristic
79	C₃	#63381	1	3	yes	complemented
80	C₃	#63649	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	#88534	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#6233
commutator	C₃×C₄×C₉	C₂×C₄
Frattini	C₂×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₃×C₉×M₁₆	C₂
socle	C₂×C₃²	Dic₃⋊C₄
prime radical	C₂²×C₃²	Dic₆
cosocle	C₂×C₃×C₄×C₉	C₂²
hypercenter	M₁₆	C₃⋊D₉
hypocenter	C₃×C₉	#108
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
44. C₃×C₉	#108	semidirect
67. C₉	#822	semidirect
79. C₃	#63381	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×M₁₆	2
#88532	2
C₂×C₃⋊(C₉⋊C₈)	2
#63649	3
#63381	3