Subgroups — #14515870

Order 1944.

79 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	#10619576	C₃	1	1	yes	maximal, characteristic, complemented
3	#10619575	C₃	1	1	yes	maximal, characteristic, complemented
4	C₉⋊(Q₈⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×#6881	—	4	1	no	maximal
6	C₄×#732	—	3	1	no	—
7	#6881	—	4	1	no	Sylow
8	Q₈×H(3,𝔽₃)	—	3	1	no	—
9	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
10	Q₈⋊H(3,𝔽₃)	—	3	1	no	—
11	#6204	—	3	1	no	—
12	C₃×Q₈⋊C₉	—	3	1	no	—
13	C₂×#733	—	4	1	no	—
14	C₂×#732	—	4	1	no	—
15	C₂×#732	A₄	1	1	yes	characteristic
16	C₂×C₉⋊C₉	—	4	1	no	—
17	C₄×H(3,𝔽₃)	—	3	3	no	—
18	C₃×C₄×C₉	—	3	1	no	—
19	#733	—	4	1	no	—
20	#732	—	4	1	no	—
21	#732	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
22	C₉⋊C₉	—	4	1	no	—
23	C₃²×Q₈	—	9	1	no	—
24	C₃²×Q₈	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×SL(2,𝔽₃)	—	9	1	no	—
26	Q₈×C₉	—	3	1	no	—
27	Q₈⋊C₉	—	9	1	no	—
28	Q₈⋊C₉	—	9	1	no	—
29	C₂×3₋¹⁺²	—	12	1	no	—
30	C₂×H(3,𝔽₃)	—	3	1	no	—
31	C₂×H(3,𝔽₃)	—	12	1	no	—
32	C₂×C₃×C₉	—	12	1	no	—
33	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
34	C₃²×C₄	—	3	1	no	—
35	C₃²×C₄	—	9	3	no	—
36	C₄×C₉	—	9	1	no	—
37	3₋¹⁺²	—	12	1	no	—
38	H(3,𝔽₃)	—	3	1	no	—
39	H(3,𝔽₃)	—	12	1	no	—
40	C₃×C₉	—	12	1	no	—
41	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
42	C₃×Q₈	—	3	1	no	—
43	C₃×Q₈	—	27	1	no	—
44	C₃×Q₈	#732	1	1	yes	characteristic
45	SL(2,𝔽₃)	—	27	1	no	—
46	C₂×C₃²	—	9	1	no	—
47	C₂×C₃²	—	36	1	no	—
48	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
49	C₂×C₉	—	3	1	no	—
50	C₂×C₉	—	36	1	no	—
51	C₂×C₉	—	36	1	no	—
52	C₃×C₄	—	3	1	no	—
53	C₃×C₄	—	9	1	no	—
54	C₃×C₄	—	27	3	no	—
55	C₃²	—	9	1	no	—
56	C₃²	—	36	1	no	—
57	C₃²	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
58	C₉	—	3	1	no	—
59	C₉	—	36	1	no	—
60	C₉	—	36	1	no	—
61	Q₈	#6881	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₃	—	3	1	no	—
63	C₂×C₃	—	27	1	no	—
64	C₂×C₃	—	108	1	no	—
65	C₂×C₃	#66362	1	1	yes	characteristic
66	C₄	—	3	1	no	—
67	C₃	—	3	1	no	—
68	C₃	—	27	1	no	—
69	C₃	—	108	1	no	—
70	C₃	#10619579	1	1	yes	characteristic
71	C₂	#11757435	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	#6881
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. #10619576	C₃	semidirect
3. #10619575	C₃	semidirect
4. C₉⋊(Q₈⋊C₉)	C₃	semidirect
21. #732	SL(2,𝔽₃)	semidirect
61. Q₈	#6881	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
#10619576	3
#10619575	3
C₉⋊(Q₈⋊C₉)	3
C₂×#6881	4