Subgroups — C₂×#10624689 #13090175

Order 1458.

512 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624689	C₂	1	1	yes	maximal, characteristic, direct product factor, Sylow
2	C₂×C₃²×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃×C₃²⋊C₉	C₃	1	3	yes	maximal, complemented
4	C₂×#6866	C₃	1	9	yes	maximal, complemented
5	C₃²×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
6	C₃×C₃²⋊C₉	C₂×C₃	1	3	yes	complemented
7	#6866	C₂×C₃	1	9	yes	complemented
8	C₂×C₃⁴	C₃²	1	3	yes	—
9	C₂×C₃⁴	C₃²	1	1	yes	characteristic
10	C₂×C₃×H(3,𝔽₃)	—	3	3	no	—
11	C₂×C₃×H(3,𝔽₃)	—	3	6	no	—
12	C₂×C₃×H(3,𝔽₃)	C₉	1	9	yes	complemented
13	C₂×C₃²×C₉	—	3	3	no	—
14	C₂×C₃²⋊C₉	—	3	6	no	—
15	C₂×C₃²⋊C₉	C₃²	1	9	yes	—
16	C₃⁴	C₂×C₃²	1	3	yes	—
17	C₃⁴	C₂×C₃²	1	1	yes	characteristic
18	C₃×H(3,𝔽₃)	—	3	3	no	—
19	C₃×H(3,𝔽₃)	—	3	6	no	—
20	C₃×H(3,𝔽₃)	C₂×C₉	1	9	yes	complemented
21	C₃²×C₉	—	3	3	no	—
22	C₃²⋊C₉	—	3	6	no	—
23	C₃²⋊C₉	C₂×C₃²	1	9	yes	—
24	C₂×C₃³	—	3	3	no	—
25	C₂×C₃³	—	3	6	no	—
26	C₂×C₃³	—	9	3	no	—
27	C₂×C₃³	—	9	9	no	—
28	C₂×C₃³	C₃×C₉	1	3	yes	—
29	C₂×C₃³	C₃×C₉	1	9	yes	complemented
30	C₂×C₃³	H(3,𝔽₃)	1	3	yes	—
31	C₂×C₃³	3₋¹⁺²	1	6	yes	—
32	C₂×C₃³	C₃³	1	1	yes	characteristic
33	C₂×H(3,𝔽₃)	—	3	27	no	—
34	C₂×C₃×C₉	—	3	9	no	—
35	C₂×C₃×C₉	—	9	9	no	—
36	C₃³	—	3	3	no	—
37	C₃³	—	3	6	no	—
38	C₃³	—	9	3	no	—
39	C₃³	—	9	9	no	—
40	C₃³	C₂×C₃×C₉	1	3	yes	—
41	C₃³	C₂×C₃×C₉	1	9	yes	complemented
42	C₃³	C₂×H(3,𝔽₃)	1	3	yes	—
43	C₃³	C₂×3₋¹⁺²	1	6	yes	—
44	C₃³	C₂×C₃³	1	1	yes	characteristic
45	H(3,𝔽₃)	—	3	27	no	—
46	C₃×C₉	—	3	9	no	—
47	C₃×C₉	—	9	9	no	—
48	C₂×C₃²	—	3	3	no	—
49	C₂×C₃²	—	3	3	no	—
50	C₂×C₃²	—	3	6	no	—
51	C₂×C₃²	—	3	9	no	—
52	C₂×C₃²	—	9	3	no	—
53	C₂×C₃²	—	9	6	no	—
54	C₂×C₃²	—	9	18	no	—
55	C₂×C₃²	—	9	9	no	—
56	C₂×C₃²	—	9	9	no	—
57	C₂×C₃²	C₃²⋊C₉	1	9	yes	—
58	C₂×C₃²	C₃²×C₉	1	1	yes	characteristic
59	C₂×C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	C₂×C₃²	C₃×3₋¹⁺²	1	2	yes	—
61	C₂×C₉	—	9	9	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	6	no	—
65	C₃²	—	3	9	no	—
66	C₃²	—	9	3	no	—
67	C₃²	—	9	6	no	—
68	C₃²	—	9	9	no	—
69	C₃²	—	9	18	no	—
70	C₃²	—	9	9	no	—
71	C₃²	C₂×C₃²⋊C₉	1	9	yes	—
72	C₃²	C₂×C₃²×C₉	1	1	yes	characteristic
73	C₃²	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃²	C₂×C₃×3₋¹⁺²	1	2	yes	—
75	C₉	—	9	9	no	—
76	C₂×C₃	—	3	1	no	—
77	C₂×C₃	—	3	2	no	—
78	C₂×C₃	—	3	9	no	—
79	C₂×C₃	—	9	9	no	—
80	C₂×C₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
81	C₂×C₃	#6909	1	3	yes	—
82	C₃	—	3	1	no	—
83	C₃	—	3	2	no	—
84	C₃	—	3	9	no	—
85	C₃	—	9	9	no	—
86	C₃	C₂×C₃×C₃²⋊C₉	1	1	yes	characteristic
87	C₃	C₂×#6909	1	3	yes	—
88	C₂	#10624689	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, socle	C₂×C₃²	C₃×H(3,𝔽₃)
commutator	C₃²	C₂×C₃²×C₉
Frattini, cosocle	C₃³	C₂×C₃³
prime radical	C₂×C₃²×H(3,𝔽₃)	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃⁴ ◁ G

Complements

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

N	H ≅ G/N	type
1. #10624689	C₂	direct (N × H)
3. C₂×C₃×C₃²⋊C₉	C₃	semidirect
4. C₂×#6866	C₃	semidirect
6. C₃×C₃²⋊C₉	C₂×C₃	semidirect
7. #6866	C₂×C₃	semidirect
12. C₂×C₃×H(3,𝔽₃)	C₉	semidirect
20. C₃×H(3,𝔽₃)	C₂×C₉	semidirect
29. C₂×C₃³	C₃×C₉	semidirect
41. C₃³	C₂×C₃×C₉	semidirect
88. C₂	#10624689	direct (N × H)

Maximal subgroups

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

subgroup	index
#10624689	2
C₂×C₃²×H(3,𝔽₃)	3
C₂×C₃×C₃²⋊C₉	3
C₂×#6866	3

Subgroups — C2×#10624689 #13090175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10624689 #13090175