Subgroups — #14515869

Order 1944.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#732	C₃	1	1	yes	maximal, characteristic, complemented
2	#10619575	C₃	1	2	yes	maximal, complemented
3	C₉×Q₈⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×#6879	—	4	1	no	maximal
5	C₄×#732	—	3	1	no	—
6	#6879	—	4	1	no	Sylow
7	Q₈×H(3,𝔽₃)	—	3	1	no	—
8	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
9	Q₈⋊H(3,𝔽₃)	—	3	2	no	—
10	C₃×Q₈⋊C₉	—	3	1	no	—
11	C₂×#732	—	4	2	no	—
12	C₂×#732	A₄	1	1	yes	characteristic
13	C₂×C₉²	—	4	1	no	—
14	C₄×H(3,𝔽₃)	—	3	3	no	—
15	C₃×C₄×C₉	—	3	1	no	—
16	#732	—	4	2	no	—
17	#732	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
18	C₉²	—	4	1	no	—
19	C₃²×Q₈	—	9	1	no	—
20	C₃²×Q₈	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×SL(2,𝔽₃)	—	9	2	no	—
22	Q₈×C₉	—	3	1	no	—
23	Q₈⋊C₉	—	3	3	no	—
24	C₂×H(3,𝔽₃)	—	3	1	no	—
25	C₂×H(3,𝔽₃)	—	12	2	no	—
26	C₂×C₃×C₉	—	12	1	no	—
27	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
28	C₃²×C₄	—	3	1	no	—
29	C₃²×C₄	—	9	3	no	—
30	C₄×C₉	—	9	1	no	—
31	H(3,𝔽₃)	—	3	1	no	—
32	H(3,𝔽₃)	—	12	2	no	—
33	C₃×C₉	—	12	1	no	—
34	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
35	C₃×Q₈	—	3	1	no	—
36	C₃×Q₈	—	27	1	no	—
37	C₃×Q₈	#732	1	1	yes	characteristic
38	SL(2,𝔽₃)	—	27	2	no	—
39	C₂×C₃²	—	9	1	no	—
40	C₂×C₃²	—	36	2	no	—
41	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
42	C₂×C₉	—	3	1	no	—
43	C₂×C₉	—	12	3	no	—
44	C₃×C₄	—	3	1	no	—
45	C₃×C₄	—	9	1	no	—
46	C₃×C₄	—	27	3	no	—
47	C₃²	—	9	1	no	—
48	C₃²	—	36	2	no	—
49	C₃²	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
50	C₉	—	3	1	no	—
51	C₉	—	12	3	no	—
52	Q₈	#6879	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃	—	3	1	no	—
54	C₂×C₃	—	27	1	no	—
55	C₂×C₃	—	108	2	no	—
56	C₂×C₃	#66362	1	1	yes	characteristic
57	C₄	—	3	1	no	—
58	C₃	—	3	1	no	—
59	C₃	—	27	1	no	—
60	C₃	—	108	2	no	—
61	C₃	#10619579	1	1	yes	characteristic
62	C₂	#11757434	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#66362
commutator, cosocle	C₃×Q₈×C₉	C₃²
Frattini	C₂×C₃×C₉	C₃×A₄
Fitting	Q₈×#732	C₃
hypercenter	C₂×#732	A₄
hypocenter	Q₈	#6879
LCS	G ▷ C₃×Q₈×C₉ ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ C₂×C₃×C₉ ◁ C₂×#732

Complements

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

N	H ≅ G/N	type
1. Q₈×#732	C₃	semidirect
2. #10619575	C₃	semidirect
3. C₉×Q₈⋊C₉	C₃	semidirect
17. #732	SL(2,𝔽₃)	semidirect
52. Q₈	#6879	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#732	3
#10619575	3
C₉×Q₈⋊C₉	3
C₂×#6879	4