Subgroups — C₉⋊(Q₈⋊SL(2,𝔽₃)) #13152246

Order 1728.

218 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈⋊SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	Q₈²×C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	Q₈⋊(Q₈⋊C₉)	C₃	1	2	yes	maximal, complemented
4	C₂×C₉⋊SL(2,𝔽₃)	—	4	3	no	maximal
5	C₂×C₉⋊SL(2,𝔽₃)	—	4	2	no	maximal
6	C₉×C₄⋊Q₈	—	3	3	no	—
7	C₄×Q₈×C₉	—	3	2	no	—
8	C₉⋊SL(2,𝔽₃)	—	4	3	no	—
9	C₉⋊SL(2,𝔽₃)	—	4	2	no	—
10	C₃×Q₈²	C₃²	1	1	yes	characteristic
11	Q₈⋊SL(2,𝔽₃)	—	3	1	no	—
12	C₂×C₃×SL(2,𝔽₃)	—	4	3	no	—
13	C₂×C₃×SL(2,𝔽₃)	—	4	2	no	—
14	C₂×Q₈×C₉	—	3	1	no	—
15	C₂×Q₈×C₉	A₄	1	2	yes	—
16	C₂×Q₈×C₉	A₄	1	3	yes	—
17	C₂×Q₈⋊C₉	—	4	6	no	—
18	C₂×Q₈⋊C₉	—	4	4	no	—
19	C₉×C₄⋊C₄	—	3	6	no	—
20	C₄²×C₉	—	3	3	no	—
21	C₂²×3₋¹⁺²	—	16	1	no	—
22	C₃×C₄⋊Q₈	—	3	3	no	—
23	C₃×C₄×Q₈	—	3	2	no	—
24	C₃×SL(2,𝔽₃)	—	4	3	no	—
25	C₃×SL(2,𝔽₃)	—	4	2	no	—
26	Q₈×C₉	—	3	2	no	—
27	Q₈×C₉	—	4	3	no	—
28	Q₈×C₉	—	12	1	no	—
29	Q₈×C₉	SL(2,𝔽₃)	1	2	yes	complemented
30	C₂×C₄×C₉	—	3	2	no	—
31	C₂×C₄×C₉	—	3	3	no	—
32	Q₈⋊C₉	—	4	6	no	—
33	Q₈⋊C₉	—	4	4	no	—
34	Q₈²	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
35	C₂×3₋¹⁺²	—	16	2	no	—
36	C₂×3₋¹⁺²	—	16	1	no	—
37	C₂×C₃×Q₈	—	3	1	no	—
38	C₂×C₃×Q₈	C₃×A₄	1	2	yes	—
39	C₂×C₃×Q₈	C₃×A₄	1	3	yes	—
40	C₂×SL(2,𝔽₃)	—	12	2	no	—
41	C₂×SL(2,𝔽₃)	—	12	3	no	—
42	C₃×C₄⋊C₄	—	3	6	no	—
43	C₃×C₄²	—	3	3	no	—
44	C₂²×C₃²	—	16	1	no	—
45	C₂²×C₉	—	16	2	no	—
46	C₂²×C₉	C₂²⋊A₄	1	1	yes	characteristic
47	C₄×C₉	—	3	2	no	—
48	C₄×C₉	—	3	2	no	—
49	C₄×C₉	—	6	3	no	—
50	C₄⋊Q₈	—	3	3	no	—
51	C₄×Q₈	—	3	2	no	—
52	3₋¹⁺²	—	16	1	no	Sylow
53	C₃×Q₈	—	3	2	no	—
54	C₃×Q₈	—	4	3	no	—
55	C₃×Q₈	—	12	1	no	—
56	C₃×Q₈	C₃×SL(2,𝔽₃)	1	2	yes	—
57	C₂×C₃×C₄	—	3	2	no	—
58	C₂×C₃×C₄	—	3	3	no	—
59	SL(2,𝔽₃)	—	12	3	no	—
60	SL(2,𝔽₃)	—	12	2	no	—
61	C₂×C₃²	—	16	2	no	—
62	C₂×C₃²	—	16	1	no	—
63	C₂×C₉	—	16	4	no	—
64	C₂×C₉	—	16	2	no	—
65	C₂×C₉	C₂²⋊SL(2,𝔽₃)	1	2	yes	—
66	C₂×C₉	#997	1	1	yes	characteristic
67	C₂×Q₈	—	3	1	no	—
68	C₂×Q₈	C₉⋊A₄	1	2	yes	—
69	C₂×Q₈	C₉⋊A₄	1	3	yes	—
70	C₄⋊C₄	—	3	6	no	—
71	C₄²	—	3	3	no	—
72	C₂²×C₃	—	48	1	no	—
73	C₂²×C₃	C₃×C₂²⋊A₄	1	1	yes	characteristic
74	C₃×C₄	—	3	2	no	—
75	C₃×C₄	—	3	2	no	—
76	C₃×C₄	—	6	3	no	—
77	C₃²	—	16	1	no	—
78	C₉	—	16	2	no	—
79	C₉	Q₈⋊SL(2,𝔽₃)	1	1	yes	characteristic, complemented
80	Q₈	—	3	2	no	—
81	Q₈	—	4	3	no	—
82	Q₈	—	12	1	no	—
83	Q₈	C₉⋊SL(2,𝔽₃)	1	2	yes	complemented
84	C₂×C₄	—	3	2	no	—
85	C₂×C₄	—	3	3	no	—
86	C₂×C₃	—	48	2	no	—
87	C₂×C₃	—	48	1	no	—
88	C₂×C₃	C₃×C₂²⋊SL(2,𝔽₃)	1	2	yes	—
89	C₂×C₃	C₃×#997	1	1	yes	characteristic
90	C₂²	C₉⋊(C₂²⋊A₄)	1	1	yes	characteristic
91	C₄	—	3	2	no	—
92	C₄	—	3	2	no	—
93	C₄	—	6	3	no	—
94	C₃	—	48	1	no	—
95	C₃	C₃×Q₈⋊SL(2,𝔽₃)	1	1	yes	characteristic
96	C₂	C₉⋊(C₂²⋊SL(2,𝔽₃))	1	2	yes	—
97	C₂	C₉⋊#997	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂²×C₃	C₃×C₂²⋊A₄
commutator, cosocle	C₃×Q₈²	C₃²
Fitting	Q₈²×C₉	C₃
prime radical	C₃×Q₈⋊SL(2,𝔽₃)	C₃
hypercenter	C₂²×C₉	C₂²⋊A₄
hypocenter	Q₈²	3₋¹⁺²
LCS	G ▷ C₃×Q₈² ▷ Q₈²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₉

Complements

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

N	H ≅ G/N	type
2. Q₈²×C₉	C₃	semidirect
3. Q₈⋊(Q₈⋊C₉)	C₃	semidirect
29. Q₈×C₉	SL(2,𝔽₃)	semidirect
34. Q₈²	3₋¹⁺²	semidirect
79. C₉	Q₈⋊SL(2,𝔽₃)	semidirect
83. Q₈	C₉⋊SL(2,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈⋊SL(2,𝔽₃)	3
Q₈²×C₉	3
Q₈⋊(Q₈⋊C₉)	3
C₂×C₉⋊SL(2,𝔽₃)	4

Subgroups — C9⋊(Q8⋊SL(2,𝔽3)) #13152246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉⋊(Q₈⋊SL(2,𝔽₃)) #13152246