Subgroups — #13047989

Order 1296.

201 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#737	C₂	1	1	yes	maximal, characteristic
2	#10619966	C₂	1	1	yes	maximal, characteristic
3	#10619725	C₂	1	1	yes	maximal, characteristic
4	#88550	C₃	1	1	yes	maximal, characteristic
5	C₉×#216	—	3	4	no	maximal
6	C₄×#737	—	2	1	no	—
7	C₄×#737	C₂²	1	1	yes	characteristic
8	#66410	—	2	1	no	—
9	Q₈×3₋¹⁺²	—	2	4	no	—
10	Q₈×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
11	C₃×Q₈×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	D₄	1	1	yes	characteristic
17	C₃×#216	—	3	4	no	—
18	C₉×Q₁₆	—	9	1	no	—
19	C₄×3₋¹⁺²	—	2	8	no	—
20	C₄×3₋¹⁺²	—	2	4	no	—
21	C₄×H(3,𝔽₃)	—	2	1	no	—
22	C₄×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
23	C₃×C₄×C₉	—	2	4	no	—
24	C₃×C₄×C₉	C₂×D₃	1	4	yes	—
25	#1084	—	2	1	no	—
26	C₉×Dic₃	—	6	4	no	—
27	#737	Q₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₃²×Q₈	C₃×D₃	1	4	yes	—
29	C₃×Dic₆	—	3	4	no	—
30	C₃×C₃⋊C₈	—	3	4	no	—
31	Q₈×C₉	—	2	4	no	—
32	Q₈×C₉	—	9	1	no	—
33	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic
34	C₈×C₉	—	9	1	no	—
35	C₂×3₋¹⁺²	—	2	4	no	—
36	C₂×H(3,𝔽₃)	C₃×D₄	1	1	yes	characteristic
37	C₂×C₃×C₉	#67	1	4	yes	—
38	C₃×Q₁₆	—	9	1	no	—
39	#216	—	3	4	no	—
40	C₃²×C₄	—	2	4	no	—
41	C₃²×C₄	C₂×C₃×D₃	1	4	yes	—
42	C₃×Dic₃	—	6	4	no	—
43	C₄×C₉	—	2	8	no	—
44	C₄×C₉	—	2	1	no	—
45	C₄×C₉	—	2	4	no	—
46	C₄×C₉	—	18	1	no	—
47	C₄×C₉	C₂×C₃⋊D₃	1	1	yes	characteristic
48	3₋¹⁺²	—	2	4	no	—
49	H(3,𝔽₃)	C₃×Q₁₆	1	1	yes	characteristic
50	C₃×C₉	#216	1	4	yes	complemented
51	C₃×Q₈	—	3	4	no	—
52	C₃×Q₈	—	9	1	no	—
53	C₃×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
54	Dic₆	—	3	4	no	—
55	C₃×C₈	—	9	1	no	—
56	C₃⋊C₈	—	3	4	no	—
57	C₂×C₃²	C₃×#67	1	4	yes	—
58	C₂×C₉	—	2	4	no	—
59	C₂×C₉	#638	1	1	yes	characteristic
60	Q₁₆	—	9	1	no	Sylow
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	—	3	4	no	—
63	C₃×C₄	—	6	4	no	—
64	C₃×C₄	—	18	1	no	—
65	C₃×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
66	Dic₃	—	6	4	no	—
67	C₃²	C₃×#216	1	4	yes	—
68	C₉	—	2	4	no	—
69	C₉	#3756	1	1	yes	characteristic
70	Q₈	—	9	1	no	—
71	Q₈	#4208	1	1	yes	characteristic
72	C₈	—	9	1	no	—
73	C₂×C₃	—	3	4	no	—
74	C₂×C₃	C₃×#638	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	18	1	no	—
77	C₄	C₂×#4208	1	1	yes	characteristic
78	C₃	—	3	4	no	—
79	C₃	C₃×#3756	1	1	yes	characteristic
80	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	Q₈×#737	C₂
socle	C₂×C₃	C₃×#638
prime radical	C₂×H(3,𝔽₃)	C₃×D₄
hypercenter	Q₈×C₉	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₃×Q₁₆
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ Q₈×C₉

Complements

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

N	H ≅ G/N	type
27. #737	Q₁₆	semidirect
50. C₃×C₉	#216	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#737	2
#10619966	2
#10619725	2
#88550	3
C₉×#216	3