Subgroups — #13141266

Order 1728.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11721359	C₂	1	1	yes	maximal, characteristic
2	C₃²×#504	C₃	1	1	yes	maximal, characteristic
3	#504⋊C₉	C₃	1	3	yes	maximal, complemented
4	#88691	C₄	1	1	yes	characteristic
5	C₁₆×3₋¹⁺²	—	4	1	no	maximal
6	C₃²×#131	C₂×C₃	1	1	yes	characteristic
7	C₃²×M₃₂	—	3	1	no	—
8	C₂×C₃²×C₁₆	—	3	1	no	—
9	#131⋊C₉	C₂×C₃	1	3	yes	—
10	#6204	C₈	1	1	yes	characteristic
11	C₈×3₋¹⁺²	—	4	1	no	—
12	C₃×#504	—	3	1	no	—
13	C₃×#504	C₃²	1	1	yes	characteristic
14	C₃²×#41	C₃×C₄	1	1	yes	characteristic
15	C₃²×M₁₆	—	3	1	no	—
16	C₂×C₃²×C₈	—	3	1	no	—
17	#41⋊C₉	C₃×C₄	1	3	yes	—
18	C₃²×C₁₆	—	3	1	no	—
19	C₃²×C₁₆	A₄	1	1	yes	characteristic
20	C₉×C₁₆	—	4	3	no	—
21	C₄×3₋¹⁺²	—	4	1	no	—
22	C₃×#131	—	3	1	no	—
23	C₃×#131	C₂×C₃²	1	1	yes	characteristic
24	C₃×M₃₂	—	3	3	no	—
25	C₃×M₃₂	—	3	1	no	—
26	C₂×C₃×C₁₆	—	3	3	no	—
27	C₂×C₃×C₁₆	—	3	1	no	—
28	C₃²×Q₈	C₃×C₈	1	1	yes	characteristic
29	C₃²×D₄	—	3	1	no	—
30	C₂×C₃²×C₄	—	3	1	no	—
31	C₃²×C₈	—	3	1	no	—
32	C₃²×C₈	C₂×A₄	1	1	yes	characteristic
33	Q₈⋊C₉	C₃×C₈	1	3	yes	—
34	C₈×C₉	—	4	3	no	—
35	#504	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
36	C₂×3₋¹⁺²	—	4	1	no	—
37	C₃×#41	—	3	1	no	—
38	C₃×#41	C₃²×C₄	1	1	yes	characteristic
39	C₃×M₁₆	—	3	3	no	—
40	C₃×M₁₆	—	3	1	no	—
41	C₂×C₃×C₈	—	3	3	no	—
42	C₂×C₃×C₈	—	3	1	no	—
43	C₃×C₁₆	—	3	3	no	—
44	C₃×C₁₆	—	3	1	no	—
45	C₃×C₁₆	—	3	1	no	—
46	C₃×C₁₆	C₃×A₄	1	1	yes	characteristic
47	C₂²×C₃²	—	3	1	no	—
48	C₃²×C₄	—	3	1	no	—
49	C₃²×C₄	C₄×A₄	1	1	yes	characteristic
50	C₄×C₉	—	4	3	no	—
51	#131	C₂×3₋¹⁺²	1	1	yes	characteristic
52	M₃₂	—	3	1	no	—
53	C₂×C₁₆	—	3	1	no	—
54	3₋¹⁺²	—	4	1	no	Sylow
55	C₃×Q₈	—	3	1	no	—
56	C₃×Q₈	C₃²×C₈	1	1	yes	characteristic
57	C₃×D₄	—	3	1	no	—
58	C₃×D₄	—	3	3	no	—
59	C₂×C₃×C₄	—	3	3	no	—
60	C₂×C₃×C₄	—	3	1	no	—
61	C₃×C₈	—	3	3	no	—
62	C₃×C₈	—	3	1	no	—
63	C₃×C₈	—	3	1	no	—
64	C₃×C₈	C₂×C₃×A₄	1	1	yes	characteristic
65	C₂×C₃²	—	6	1	no	—
66	C₂×C₃²	C₈×A₄	1	1	yes	characteristic
67	C₂×C₉	—	4	3	no	—
68	#41	C₄×3₋¹⁺²	1	1	yes	characteristic
69	M₁₆	—	3	1	no	—
70	C₂×C₈	—	3	1	no	—
71	C₁₆	—	3	1	no	—
72	C₁₆	#1094	1	1	yes	characteristic
73	C₂²×C₃	—	3	3	no	—
74	C₂²×C₃	—	3	1	no	—
75	C₃×C₄	—	3	3	no	—
76	C₃×C₄	—	3	1	no	—
77	C₃×C₄	—	3	1	no	—
78	C₃×C₄	C₃×C₄×A₄	1	1	yes	characteristic
79	C₃²	#504⋊C₃	1	1	yes	characteristic
80	C₉	—	4	3	no	—
81	Q₈	C₈×3₋¹⁺²	1	1	yes	characteristic
82	D₄	—	3	1	no	—
83	C₂×C₄	—	3	1	no	—
84	C₈	—	3	1	no	—
85	C₈	C₂×#1094	1	1	yes	characteristic
86	C₂×C₃	—	3	1	no	—
87	C₂×C₃	—	6	3	no	—
88	C₂×C₃	—	6	1	no	—
89	C₂×C₃	C₃×C₈×A₄	1	1	yes	characteristic
90	C₂²	—	3	1	no	—
91	C₄	—	3	1	no	—
92	C₄	C₄×#1094	1	1	yes	characteristic
93	C₃	—	3	1	no	—
94	C₃	C₃×#504⋊C₃	1	1	yes	characteristic
95	C₂	—	6	1	no	—
96	C₂	C₈×#1094	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₁₆	C₃×A₄
commutator	C₃×Q₈	C₃²×C₈
Frattini	C₃×C₈	C₂×C₃×A₄
Fitting	C₃²×#504	C₃
socle	C₂×C₃	C₃×C₈×A₄
prime radical	C₃²×#41	C₃×C₄
cosocle	C₃×#131	C₂×C₃²
hypercenter	C₃²×C₁₆	A₄
hypocenter	Q₈	C₈×3₋¹⁺²
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₃×C₁₆ ◁ C₃²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #504⋊C₉	C₃	semidirect
35. #504	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
#11721359	2
C₃²×#504	3
#504⋊C₉	3
C₁₆×3₋¹⁺²	4