Subgroups — #11720245

Order 864.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#88521	C₂	1	1	yes	maximal, characteristic
2	C₉×Dic₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#63551	—	3	1	no	maximal
5	#63388	—	3	1	no	maximal
6	#6226	—	2	1	no	—
7	C₉×Dic₆	—	2	1	no	—
8	C₃×C₈×C₉	C₂²	1	1	yes	characteristic
9	#3746	—	3	1	no	—
10	C₃×Dic₁₂	D₃	1	1	yes	characteristic
11	C₃×C₃⋊C₁₆	—	3	1	no	—
12	C₉×Q₁₆	—	3	1	no	—
13	Dic₃₆	—	3	1	no	—
14	Dic₃₆	—	6	1	no	—
15	C₉⋊C₁₆	D₃	1	1	yes	characteristic
16	C₃×C₄×C₉	D₄	1	1	yes	characteristic
17	C₃⋊Dic₉	—	4	1	no	—
18	C₉×Dic₃	—	4	1	no	—
19	#829	—	9	1	no	—
20	Dic₂₄	—	9	1	no	—
21	#634	—	6	1	no	—
22	C₃×Dic₆	—	2	1	no	—
23	C₃²×C₈	C₂×D₃	1	1	yes	characteristic
24	Q₈×C₉	—	6	1	no	—
25	Dic₁₈	—	6	2	no	—
26	Dic₁₈	—	6	1	no	—
27	C₈×C₉	—	2	1	no	—
28	C₈×C₉	C₂×D₃	1	1	yes	characteristic
29	C₂×C₃×C₉	D₈	1	1	yes	characteristic
30	C₃×Q₁₆	—	3	1	no	—
31	Dic₁₂	—	9	1	no	—
32	Dic₁₂	—	9	1	no	—
33	Dic₁₂	—	18	1	no	—
34	Dic₁₂	D₉	1	1	yes	characteristic
35	C₃×C₁₆	—	9	1	no	—
36	C₃⋊C₁₆	—	3	1	no	—
37	C₃²×C₄	#67	1	1	yes	characteristic
38	C₃⋊Dic₃	—	12	1	no	—
39	C₃×Dic₃	—	4	1	no	—
40	C₄×C₉	—	2	1	no	—
41	C₄×C₉	—	12	1	no	—
42	C₄×C₉	D₁₂	1	1	yes	characteristic
43	Dic₉	—	12	2	no	—
44	Dic₉	—	12	1	no	—
45	Q₃₂	—	27	1	no	Sylow
46	C₃×C₉	Q₃₂	1	1	yes	characteristic, complemented, Sylow
47	C₃×Q₈	—	6	1	no	—
48	Dic₆	—	2	1	no	—
49	Dic₆	—	18	2	no	—
50	Dic₆	—	18	1	no	—
51	Dic₆	—	18	1	no	—
52	C₃×C₈	—	2	1	no	—
53	C₃×C₈	C₂×D₉	1	1	yes	characteristic
54	C₃×C₈	D₃²	1	1	yes	characteristic
55	C₂×C₃²	#213	1	1	yes	characteristic
56	C₂×C₉	—	2	1	no	—
57	C₂×C₉	D₂₄	1	1	yes	characteristic
58	Q₁₆	—	3	1	no	—
59	Q₁₆	—	27	1	no	—
60	C₁₆	—	9	1	no	—
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	—	12	1	no	—
63	C₃×C₄	#611	1	1	yes	characteristic
64	C₃×C₄	#626	1	1	yes	characteristic
65	Dic₃	—	4	1	no	—
66	Dic₃	—	36	2	no	—
67	Dic₃	—	36	1	no	—
68	Dic₃	—	36	1	no	—
69	C₃²	#829	1	1	yes	characteristic
70	C₉	—	2	1	no	—
71	C₉	Dic₂₄	1	1	yes	characteristic, complemented
72	Q₈	—	6	1	no	—
73	Q₈	—	54	1	no	—
74	C₈	D₃×D₉	1	1	yes	characteristic
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	#3673	1	1	yes	characteristic
77	C₂×C₃	#3714	1	1	yes	characteristic
78	C₄	—	12	1	no	—
79	C₄	—	108	1	no	—
80	C₄	#6195	1	1	yes	characteristic
81	C₃	—	2	1	no	—
82	C₃	#63388	1	1	yes	characteristic, complemented
83	C₃	#63551	1	1	yes	characteristic
84	C₂	#88422	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#88422
commutator, Fitting, cosocle	C₃×C₈×C₉	C₂²
Frattini	C₃×C₈	D₃²
socle, prime radical	C₂×C₃²	#213
hypercenter	C₈	D₃×D₉
hypocenter	C₃×C₉	Q₃₂
LCS	G ▷ C₃×C₈×C₉ ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
46. C₃×C₉	Q₃₂	semidirect
71. C₉	Dic₂₄	semidirect
82. C₃	#63388	semidirect

Maximal subgroups

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

subgroup	index
#88521	2
C₉×Dic₁₂	2
C₃×C₉⋊C₁₆	2
#63551	3
#63388	3