Subgroups — #13048383

Order 1296.

157 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10619912	C₂	1	1	yes	maximal, characteristic, complemented
2	#88611	C₃	1	1	yes	maximal, characteristic
3	C₉×GL(2,𝔽₃)	—	3	3	no	maximal
4	C₉×#215	—	3	1	no	maximal
5	C₂×#4208	—	4	1	no	maximal
6	C₃×Q₈×C₉	D₃	1	1	yes	characteristic, complemented
7	C₉×D₁₂	—	3	1	no	—
8	Q₈⋊H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	#6204	—	2	1	no	—
10	C₉⋊SL(2,𝔽₃)	—	2	3	no	—
11	C₉×SL(2,𝔽₃)	D₃	1	3	yes	complemented
12	C₉×C₃⋊C₈	—	3	1	no	—
13	C₂×#737	—	4	1	no	—
14	#4208	—	4	2	no	—
15	C₃×GL(2,𝔽₃)	—	3	3	no	—
16	C₃×#215	—	3	1	no	—
17	C₉×SD₁₆	—	9	1	no	—
18	C₂×#272	—	4	1	no	—
19	C₂×D₃×C₉	—	6	1	no	—
20	C₂×D₃×C₉	—	12	3	no	—
21	C₃×C₄×C₉	—	3	1	no	—
22	#737	—	4	1	no	Sylow
23	C₃²×Q₈	C₃×D₃	1	1	yes	characteristic
24	C₃×D₁₂	—	3	1	no	—
25	C₃×SL(2,𝔽₃)	C₃×D₃	1	3	yes	—
26	C₃×C₃⋊C₈	—	3	1	no	—
27	Q₈×C₉	—	2	1	no	—
28	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic
29	D₄×C₉	—	9	1	no	—
30	Q₈⋊C₉	—	2	3	no	—
31	C₈×C₉	—	9	1	no	—
32	C₂×3₋¹⁺²	—	8	3	no	—
33	C₂×3₋¹⁺²	—	8	1	no	—
34	C₂×H(3,𝔽₃)	—	4	1	no	—
35	C₂×C₃×C₉	—	4	3	no	—
36	C₂×C₃×C₉	S₄	1	1	yes	characteristic
37	#272	—	4	2	no	—
38	D₃×C₉	—	12	6	no	—
39	D₃×C₉	—	12	1	no	—
40	GL(2,𝔽₃)	—	3	3	no	—
41	C₃×SD₁₆	—	9	1	no	—
42	#215	—	3	1	no	—
43	C₂×C₃×D₃	—	6	1	no	—
44	C₂×C₃×D₃	—	12	3	no	—
45	C₃²×C₄	—	3	1	no	—
46	C₂²×C₉	—	18	1	no	—
47	C₄×C₉	—	3	1	no	—
48	C₄×C₉	—	6	1	no	—
49	3₋¹⁺²	—	8	3	no	—
50	3₋¹⁺²	—	8	1	no	—
51	H(3,𝔽₃)	—	4	1	no	—
52	C₃×C₉	—	4	3	no	—
53	C₃×C₉	GL(2,𝔽₃)	1	1	yes	characteristic, complemented
54	C₃×Q₈	—	3	1	no	—
55	C₃×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
56	C₃×D₄	—	9	1	no	—
57	D₁₂	—	3	1	no	—
58	SL(2,𝔽₃)	—	3	3	no	—
59	C₃×C₈	—	9	1	no	—
60	C₃⋊C₈	—	3	1	no	—
61	C₂×C₃²	—	4	3	no	—
62	C₂×C₃²	C₃×S₄	1	1	yes	characteristic
63	C₃×D₃	—	12	6	no	—
64	C₃×D₃	—	12	1	no	—
65	C₂×C₉	—	2	1	no	—
66	C₂×C₉	—	8	3	no	—
67	C₂×C₉	—	36	1	no	—
68	C₂×C₉	C₃⋊S₄	1	1	yes	characteristic
69	SD₁₆	—	9	1	no	Sylow
70	C₂²×C₃	—	18	1	no	—
71	C₂×D₃	—	6	1	no	—
72	C₂×D₃	—	12	3	no	—
73	C₃×C₄	—	3	1	no	—
74	C₃×C₄	—	3	1	no	—
75	C₃×C₄	—	6	1	no	—
76	C₃²	—	4	3	no	—
77	C₃²	C₃×GL(2,𝔽₃)	1	1	yes	characteristic
78	C₉	—	2	1	no	—
79	C₉	—	8	3	no	—
80	C₉	C₃⋊GL(2,𝔽₃)	1	1	yes	characteristic
81	Q₈	#4208	1	1	yes	characteristic, complemented
82	D₄	—	9	1	no	—
83	C₈	—	9	1	no	—
84	C₂×C₃	—	3	1	no	—
85	C₂×C₃	—	12	3	no	—
86	C₂×C₃	—	36	1	no	—
87	C₂×C₃	C₃×C₃⋊S₄	1	1	yes	characteristic
88	D₃	—	12	1	no	—
89	D₃	—	12	6	no	—
90	C₂²	—	18	1	no	—
91	C₄	—	3	1	no	—
92	C₃	—	3	1	no	—
93	C₃	—	12	3	no	—
94	C₃	C₃×C₃⋊GL(2,𝔽₃)	1	1	yes	characteristic
95	C₂	—	36	1	no	—
96	C₂	#10620044	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₉	C₃⋊S₄
commutator, cosocle, hypocenter	Q₈⋊H(3,𝔽₃)	C₂×C₃
Frattini, socle	C₂×C₃	C₃×C₃⋊S₄
Fitting	C₃×Q₈×C₉	D₃
prime radical	#88611	C₃
LCS	G ▷ Q₈⋊H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₉

Complements

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

N	H ≅ G/N	type
1. #10619912	C₂	semidirect
6. C₃×Q₈×C₉	D₃	semidirect
11. C₉×SL(2,𝔽₃)	D₃	semidirect
53. C₃×C₉	GL(2,𝔽₃)	semidirect
81. Q₈	#4208	semidirect

Maximal subgroups

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

subgroup	index
#10619912	2
#88611	3
C₉×GL(2,𝔽₃)	3
C₉×#215	3
C₂×#4208	4