Subgroups — C₃×C₉×#368 #13136105

Order 1728.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×#108	C₂	1	2	yes	maximal
3	C₃²×#368	C₃	1	1	yes	maximal, characteristic
4	C₉×#368	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₉×M₁₆	—	2	2	no	—
6	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₉×C₁₆	C₄	1	2	yes	—
8	C₂×C₃²×C₁₆	C₂×C₃	1	1	yes	characteristic
9	C₃²×#108	C₂×C₃	1	2	yes	—
10	C₂×C₉×C₁₆	C₂×C₃	1	3	yes	—
11	C₉×#108	C₂×C₃	1	6	yes	—
12	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
13	C₃×C₈×C₉	—	4	2	no	—
14	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
15	C₃×C₈×C₉	Q₈	1	1	yes	characteristic
16	C₃×#368	C₉	1	3	yes	direct product factor
17	C₃×#368	C₃²	1	1	yes	characteristic
18	C₃²×M₁₆	—	2	2	no	—
19	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
20	C₃²×C₁₆	C₃×C₄	1	2	yes	—
21	C₉×M₁₆	—	2	6	no	—
22	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
23	C₉×C₁₆	C₃×C₄	1	6	yes	—
24	C₂²×C₃×C₉	Q₁₆	1	1	yes	characteristic
25	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
26	C₃×C₄×C₉	D₈	1	1	yes	characteristic
27	C₂×C₃×C₁₆	C₂×C₉	1	3	yes	—
28	C₂×C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic
29	C₃×#108	C₂×C₉	1	6	yes	—
30	C₃×#108	C₂×C₃²	1	2	yes	—
31	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
32	C₃²×C₈	—	4	2	no	—
33	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
34	C₃²×C₈	C₃×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
36	C₈×C₉	—	4	6	no	—
37	C₈×C₉	C₂×C₃×C₄	1	3	yes	—
38	C₈×C₉	C₃×Q₈	1	3	yes	—
39	#368	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₃×C₉	—	2	1	no	—
41	C₂×C₃×C₉	#107	1	1	yes	characteristic
42	C₃×M₁₆	—	2	6	no	—
43	C₃×M₁₆	—	2	2	no	—
44	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
45	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
46	C₃×C₁₆	C₄×C₉	1	6	yes	—
47	C₃×C₁₆	C₃²×C₄	1	2	yes	—
48	C₂²×C₃²	C₃×Q₁₆	1	1	yes	characteristic
49	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
50	C₃²×C₄	C₃×D₈	1	1	yes	characteristic
51	C₂²×C₉	C₃×Q₁₆	1	3	yes	—
52	C₄×C₉	C₃×C₄⋊C₄	1	3	yes	—
53	C₄×C₉	C₃×D₈	1	3	yes	—
54	C₂×C₁₆	C₂×C₃×C₉	1	1	yes	characteristic
55	#108	C₂×C₃×C₉	1	2	yes	—
56	C₃×C₉	#368	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
58	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
59	C₃×C₈	—	4	6	no	—
60	C₃×C₈	—	4	2	no	—
61	C₃×C₈	C₂×C₄×C₉	1	3	yes	—
62	C₃×C₈	Q₈×C₉	1	3	yes	—
63	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
64	C₃×C₈	C₃²×Q₈	1	1	yes	characteristic
65	C₂×C₃²	—	2	1	no	—
66	C₂×C₃²	C₃×#107	1	1	yes	characteristic
67	C₂×C₉	—	2	3	no	—
68	C₂×C₉	C₃×#107	1	3	yes	—
69	M₁₆	—	2	2	no	—
70	C₂×C₈	C₂²×C₃×C₉	1	1	yes	characteristic
71	C₁₆	C₃×C₄×C₉	1	2	yes	—
72	C₂²×C₃	C₉×Q₁₆	1	3	yes	—
73	C₂²×C₃	C₃²×Q₁₆	1	1	yes	characteristic
74	C₃×C₄	C₉×C₄⋊C₄	1	3	yes	—
75	C₃×C₄	C₉×D₈	1	3	yes	—
76	C₃×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
77	C₃×C₄	C₃²×D₈	1	1	yes	characteristic
78	C₃²	C₃×#368	1	1	yes	characteristic
79	C₉	C₃×#368	1	3	yes	direct product factor
80	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
81	C₈	—	4	2	no	—
82	C₈	C₂×C₃×C₄×C₉	1	1	yes	characteristic
83	C₈	C₃×Q₈×C₉	1	1	yes	characteristic
84	C₂×C₃	—	2	1	no	—
85	C₂×C₃	—	2	3	no	—
86	C₂×C₃	C₉×#107	1	3	yes	—
87	C₂×C₃	C₃²×#107	1	1	yes	characteristic
88	C₂²	C₃×C₉×Q₁₆	1	1	yes	characteristic
89	C₄	C₃×C₉×C₄⋊C₄	1	1	yes	characteristic
90	C₄	C₃×C₉×D₈	1	1	yes	characteristic
91	C₃	C₉×#368	1	3	yes	direct product factor
92	C₃	C₃²×#368	1	1	yes	characteristic
93	C₂	—	2	1	no	—
94	C₂	C₃×C₉×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₉	D₈
commutator	C₈	C₂×C₃×C₄×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂²×C₃²
socle	C₂×C₃²	C₃×#107
prime radical	C₂²×C₃²	C₃×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₉ ◁ C₂×C₃×C₄×C₉ ◁ C₂×C₃×C₈×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×#368	C₃	direct (N × H)
16. C₃×#368	C₉	direct (N × H)
39. #368	C₃×C₉	direct (N × H)
56. C₃×C₉	#368	direct (N × H)
79. C₉	C₃×#368	direct (N × H)
91. C₃	C₉×#368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₁₆	2
C₃×C₉×#108	2
C₃²×#368	3
C₉×#368	3

Subgroups — C3×C9×#368 #13136105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#368 #13136105