Subgroups — #13135867

Order 1728.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×D₁₆	C₂	1	1	yes	maximal, characteristic
2	#11720509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
4	#10589386	—	3	1	no	maximal
5	#10588197	—	3	3	no	maximal
6	C₃×C₉×D₈	—	2	1	no	—
7	#88521	—	2	1	no	—
8	C₃×C₉×C₁₆	C₂²	1	1	yes	characteristic
9	C₃²×D₁₆	D₃	1	1	yes	characteristic
10	#63628	—	3	1	no	—
11	C₃⋊(C₃⋊C₃₂)	—	3	1	no	—
12	C₉×D₁₆	D₃	1	3	yes	—
13	Dic₇₂	—	3	3	no	—
14	C₉⋊C₃₂	—	3	3	no	—
15	C₃×D₄×C₉	—	4	1	no	—
16	#6226	—	4	1	no	—
17	C₃×C₈×C₉	D₄	1	1	yes	characteristic
18	#4520	—	9	3	no	—
19	#4520	—	9	1	no	—
20	C₃²×D₈	—	2	1	no	—
21	#3746	—	6	1	no	—
22	C₃²×C₁₆	C₂×D₃	1	1	yes	characteristic
23	C₉×D₈	—	2	3	no	—
24	Dic₃₆	—	6	3	no	—
25	C₉×C₁₆	C₂×D₃	1	3	yes	—
26	C₂²×C₃×C₉	—	8	1	no	—
27	C₃×C₄×C₉	D₈	1	1	yes	characteristic
28	C₃⋊Dic₉	—	8	1	no	—
29	C₃×D₁₆	D₉	1	3	yes	—
30	C₃×D₁₆	C₃⋊D₃	1	1	yes	characteristic
31	Dic₂₄	—	9	3	no	—
32	Dic₂₄	—	9	1	no	—
33	C₃⋊C₃₂	—	9	3	no	—
34	C₃⋊C₃₂	—	9	1	no	—
35	C₃²×D₄	—	4	1	no	—
36	#634	—	12	1	no	—
37	C₃²×C₈	#67	1	1	yes	characteristic
38	D₄×C₉	—	4	3	no	—
39	Dic₁₈	—	12	3	no	—
40	C₈×C₉	#67	1	3	yes	—
41	SD₆₄	—	27	1	no	Sylow
42	C₂×C₃×C₉	—	16	1	no	—
43	C₂×C₃×C₉	D₁₆	1	1	yes	characteristic
44	C₃×D₈	—	2	3	no	—
45	C₃×D₈	—	2	1	no	—
46	Dic₁₂	—	18	3	no	—
47	Dic₁₂	—	18	1	no	—
48	C₃×C₁₆	C₂×D₉	1	3	yes	—
49	C₃×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
50	C₂²×C₃²	—	8	1	no	—
51	C₃²×C₄	#213	1	1	yes	characteristic
52	C₃⋊Dic₃	—	24	1	no	—
53	C₂²×C₉	—	8	3	no	—
54	C₄×C₉	#213	1	3	yes	—
55	Dic₉	—	24	3	no	—
56	Q₃₂	—	27	1	no	—
57	D₁₆	C₃⋊D₉	1	1	yes	characteristic
58	C₃₂	—	27	1	no	—
59	C₃×C₉	SD₆₄	1	1	yes	characteristic, complemented, Sylow
60	C₃×D₄	—	4	3	no	—
61	C₃×D₄	—	4	1	no	—
62	Dic₆	—	36	3	no	—
63	Dic₆	—	36	1	no	—
64	C₃×C₈	#611	1	3	yes	—
65	C₃×C₈	#638	1	1	yes	characteristic
66	C₂×C₃²	—	16	1	no	—
67	C₂×C₃²	#826	1	1	yes	characteristic
68	C₂×C₉	—	16	3	no	—
69	C₂×C₉	#826	1	3	yes	—
70	Q₁₆	—	54	1	no	—
71	D₈	—	2	1	no	—
72	C₁₆	C₂×C₃⋊D₉	1	1	yes	characteristic
73	C₂²×C₃	—	8	3	no	—
74	C₂²×C₃	—	8	1	no	—
75	C₃×C₄	#3673	1	3	yes	—
76	C₃×C₄	#3753	1	1	yes	characteristic
77	Dic₃	—	72	3	no	—
78	Dic₃	—	72	1	no	—
79	C₃²	#4520	1	1	yes	characteristic
80	C₉	#4520	1	3	yes	complemented
81	Q₈	—	108	1	no	—
82	D₄	—	4	1	no	—
83	C₈	#6233	1	1	yes	characteristic
84	C₂×C₃	—	16	3	no	—
85	C₂×C₃	—	16	1	no	—
86	C₂×C₃	#63385	1	3	yes	—
87	C₂×C₃	#63653	1	1	yes	characteristic
88	C₂²	—	8	1	no	—
89	C₄	—	216	1	no	—
90	C₄	#88544	1	1	yes	characteristic
91	C₃	#10588197	1	3	yes	complemented
92	C₃	#10589386	1	1	yes	characteristic
93	C₂	—	16	1	no	—
94	C₂	#11720563	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11720563
commutator, cosocle	C₃×C₉×C₁₆	C₂²
Frattini	C₃×C₁₆	C₂×C₃⋊D₃
Fitting	C₃×C₉×D₁₆	C₂
socle	C₂×C₃²	#826
prime radical	C₃²×D₁₆	D₃
hypercenter	D₁₆	C₃⋊D₉
hypocenter	C₃×C₉	SD₆₄
LCS	G ▷ C₃×C₉×C₁₆ ▷ C₃×C₈×C₉ ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
2. #11720509	C₂	semidirect
3. C₃⋊(C₉⋊C₃₂)	C₂	semidirect
59. C₃×C₉	SD₆₄	semidirect
80. C₉	#4520	semidirect
91. C₃	#10588197	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×D₁₆	2
#11720509	2
C₃⋊(C₉⋊C₃₂)	2
#10589386	3
#10588197	3