Subgroups — H(3,𝔽₃)×Q₈⋊C₉ #14517238

Order 1944.

359 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×Q₈⋊C₉	C₃	1	4	yes	maximal, complemented
3	#10619584	C₃	1	8	yes	maximal, complemented
4	C₂×C₉×H(3,𝔽₃)	—	4	1	no	maximal
5	C₃×C₄×H(3,𝔽₃)	—	3	1	no	—
6	C₉×H(3,𝔽₃)	—	4	1	no	Sylow
7	C₃³×Q₈	C₃²	1	4	yes	—
8	Q₈×H(3,𝔽₃)	C₉	1	9	yes	complemented
9	C₃×Q₈⋊C₉	—	3	12	no	—
10	C₃×Q₈⋊C₉	C₃²	1	8	yes	—
11	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
12	C₂×C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
13	C₂×C₃²×C₉	—	4	4	no	—
14	C₂×C₃²⋊C₉	—	4	8	no	—
15	C₃³×C₄	—	3	4	no	—
16	C₄×H(3,𝔽₃)	—	3	9	no	—
17	C₃×H(3,𝔽₃)	SL(2,𝔽₃)	1	1	yes	characteristic
18	C₃²×C₉	—	4	4	no	—
19	C₃²⋊C₉	—	4	8	no	—
20	C₃²×Q₈	—	3	4	no	—
21	C₃²×Q₈	—	3	8	no	—
22	C₃²×Q₈	C₃×C₉	1	12	yes	complemented
23	C₃²×Q₈	C₃³	1	1	yes	characteristic
24	Q₈⋊C₉	—	3	8	no	—
25	Q₈⋊C₉	H(3,𝔽₃)	1	3	yes	direct product factor
26	C₂×C₃³	C₃×A₄	1	4	yes	—
27	C₂×H(3,𝔽₃)	C₂²⋊C₉	1	9	yes	—
28	C₂×C₃×C₉	—	4	1	no	—
29	C₂×C₃×C₉	—	4	8	no	—
30	C₂×C₃×C₉	—	12	12	no	—
31	C₃²×C₄	—	3	1	no	—
32	C₃²×C₄	—	3	12	no	—
33	C₃²×C₄	—	9	8	no	—
34	C₃²×C₄	—	9	4	no	—
35	C₃³	C₃×SL(2,𝔽₃)	1	4	yes	—
36	H(3,𝔽₃)	Q₈⋊C₉	1	9	yes	direct product factor
37	C₃×C₉	—	4	1	no	—
38	C₃×C₉	—	4	8	no	—
39	C₃×C₉	—	12	12	no	—
40	C₃×Q₈	—	3	12	no	—
41	C₃×Q₈	C₃²×C₉	1	1	yes	characteristic
42	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₃×Q₈	#737	1	2	yes	—
44	C₂×C₃²	—	3	8	no	—
45	C₂×C₃²	—	3	4	no	—
46	C₂×C₃²	C₃×C₂²⋊C₉	1	12	yes	—
47	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
48	C₂×C₉	—	4	3	no	—
49	C₂×C₉	—	12	8	no	—
50	C₃×C₄	—	3	2	no	—
51	C₃×C₄	—	3	1	no	—
52	C₃×C₄	—	3	1	no	—
53	C₃×C₄	—	9	12	no	—
54	C₃²	—	3	8	no	—
55	C₃²	—	3	4	no	—
56	C₃²	C₃×Q₈⋊C₉	1	12	yes	complemented
57	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
58	C₉	—	4	3	no	—
59	C₉	—	12	8	no	—
60	Q₈	C₉×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₃	—	3	12	no	—
62	C₂×C₃	#66437	1	2	yes	—
63	C₂×C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
64	C₂×C₃	C₃²×C₂²⋊C₉	1	1	yes	characteristic
65	C₄	—	3	1	no	—
66	C₃	—	3	12	no	—
67	C₃	#10619913	1	2	yes	—
68	C₃	SL(2,𝔽₃)×H(3,𝔽₃)	1	1	yes	characteristic
69	C₃	C₃²×Q₈⋊C₉	1	1	yes	characteristic
70	C₂	H(3,𝔽₃)×C₂²⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator	C₃×Q₈	C₃²×C₉
Fitting	C₃×Q₈×H(3,𝔽₃)	C₃
prime radical, hypercenter	C₂×C₃×H(3,𝔽₃)	A₄
cosocle	C₃²×Q₈	C₃³
hypocenter	Q₈	C₉×H(3,𝔽₃)
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. C₃²×Q₈⋊C₉	C₃	semidirect
3. #10619584	C₃	semidirect
8. Q₈×H(3,𝔽₃)	C₉	semidirect
22. C₃²×Q₈	C₃×C₉	semidirect
25. Q₈⋊C₉	H(3,𝔽₃)	direct (N × H)
36. H(3,𝔽₃)	Q₈⋊C₉	direct (N × H)
56. C₃²	C₃×Q₈⋊C₉	semidirect
60. Q₈	C₉×H(3,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×H(3,𝔽₃)	3
C₃²×Q₈⋊C₉	3
#10619584	3
C₂×C₉×H(3,𝔽₃)	4

Subgroups — H(3,𝔽3)×Q8⋊C9 #14517238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×Q₈⋊C₉ #14517238