Subgroups — #13047975

Order 1296.

234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#737	C₂	1	1	yes	maximal, characteristic
2	#10619966	C₂	1	1	yes	maximal, characteristic, complemented
3	#10619725	C₂	1	1	yes	maximal, characteristic, complemented
4	#88546	C₃	1	1	yes	maximal, characteristic
5	C₉×#214	—	3	4	no	maximal
6	C₂²×#737	—	2	1	no	—
7	C₄×#737	C₂²	1	1	yes	characteristic
8	#66410	—	2	1	no	—
9	D₄×3₋¹⁺²	—	2	4	no	—
10	D₄×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
11	C₃×D₄×C₉	D₃	1	4	yes	—
12	#6229	C₂×C₃	1	1	yes	characteristic
13	C₉×Dic₆	—	3	4	no	—
14	#6180	C₂×C₃	1	1	yes	characteristic
15	C₉×C₃⋊C₈	—	3	4	no	—
16	C₂×#737	—	4	1	no	—
17	C₂×#737	D₄	1	1	yes	characteristic
18	C₃×#214	—	3	4	no	—
19	C₉×SD₁₆	—	9	1	no	—
20	C₂²×3₋¹⁺²	—	2	8	no	—
21	C₂²×H(3,𝔽₃)	—	2	1	no	—
22	C₂²×C₃×C₉	—	2	4	no	—
23	C₄×3₋¹⁺²	—	2	4	no	—
24	C₄×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
25	C₃×C₄×C₉	C₂×D₃	1	4	yes	—
26	#1084	—	2	1	no	—
27	C₉×Dic₃	—	6	4	no	—
28	#737	SD₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃²×D₄	C₃×D₃	1	4	yes	—
30	C₃×Dic₆	—	3	4	no	—
31	C₃×C₃⋊C₈	—	3	4	no	—
32	Q₈×C₉	—	9	1	no	—
33	D₄×C₉	—	2	4	no	—
34	D₄×C₉	C₃⋊D₃	1	1	yes	characteristic
35	C₈×C₉	—	9	1	no	—
36	C₂×3₋¹⁺²	—	2	4	no	—
37	C₂×3₋¹⁺²	—	4	8	no	—
38	C₂×H(3,𝔽₃)	—	4	1	no	—
39	C₂×H(3,𝔽₃)	C₃×D₄	1	1	yes	characteristic
40	C₂×C₃×C₉	—	4	4	no	—
41	C₂×C₃×C₉	#67	1	4	yes	—
42	C₃×SD₁₆	—	9	1	no	—
43	#214	—	3	4	no	—
44	C₂²×C₃²	—	2	4	no	—
45	C₃²×C₄	C₂×C₃×D₃	1	4	yes	—
46	C₃×Dic₃	—	6	4	no	—
47	C₂²×C₉	—	2	8	no	—
48	C₂²×C₉	—	2	1	no	—
49	C₄×C₉	—	2	4	no	—
50	C₄×C₉	—	18	1	no	—
51	C₄×C₉	C₂×C₃⋊D₃	1	1	yes	characteristic
52	3₋¹⁺²	—	2	4	no	—
53	H(3,𝔽₃)	C₃×SD₁₆	1	1	yes	characteristic
54	C₃×C₉	#214	1	4	yes	complemented
55	C₃×Q₈	—	9	1	no	—
56	C₃×D₄	—	3	4	no	—
57	C₃×D₄	C₃×C₃⋊D₃	1	1	yes	characteristic
58	Dic₆	—	3	4	no	—
59	C₃×C₈	—	9	1	no	—
60	C₃⋊C₈	—	3	4	no	—
61	C₂×C₃²	—	4	4	no	—
62	C₂×C₃²	C₃×#67	1	4	yes	—
63	C₂×C₉	—	2	4	no	—
64	C₂×C₉	—	4	8	no	—
65	C₂×C₉	—	4	1	no	—
66	C₂×C₉	#638	1	1	yes	characteristic
67	SD₁₆	—	9	1	no	Sylow
68	C₂²×C₃	—	2	1	no	—
69	C₂²×C₃	—	6	4	no	—
70	C₃×C₄	—	3	4	no	—
71	C₃×C₄	—	18	1	no	—
72	C₃×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
73	Dic₃	—	6	4	no	—
74	C₃²	C₃×#214	1	4	yes	—
75	C₉	—	2	4	no	—
76	C₉	#3754	1	1	yes	characteristic
77	Q₈	—	9	1	no	—
78	D₄	#4208	1	1	yes	characteristic
79	C₈	—	9	1	no	—
80	C₂×C₃	—	3	4	no	—
81	C₂×C₃	—	4	1	no	—
82	C₂×C₃	—	12	4	no	—
83	C₂×C₃	C₃×#638	1	1	yes	characteristic
84	C₂²	—	2	1	no	—
85	C₄	—	18	1	no	—
86	C₄	C₂×#4208	1	1	yes	characteristic
87	C₃	—	3	4	no	—
88	C₃	C₃×#3754	1	1	yes	characteristic
89	C₂	—	4	1	no	—
90	C₂	#10619984	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#638
commutator, cosocle	C₄×H(3,𝔽₃)	C₂²×C₃
Frattini	C₃×C₄	C₂×C₃×C₃⋊D₃
Fitting	D₄×#737	C₂
socle	C₂×C₃	C₃×#638
prime radical	D₄×H(3,𝔽₃)	C₂×C₃
hypercenter	D₄×C₉	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₃×SD₁₆
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ D₄×C₉

Complements

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

N	H ≅ G/N	type
2. #10619966	C₂	semidirect
3. #10619725	C₂	semidirect
28. #737	SD₁₆	semidirect
54. C₃×C₉	#214	semidirect

Maximal subgroups

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

subgroup	index
D₄×#737	2
#10619966	2
#10619725	2
#88546	3
C₉×#214	3