Subgroups — #14515868

Order 1944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10619912	C₃	1	1	yes	maximal, characteristic
2	Q₈×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#10619570	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×Q₈⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×#6872	—	4	1	no	maximal
6	C₄×C₂₇⋊C₃	—	3	1	no	—
7	#6872	—	4	1	no	Sylow
8	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
9	Q₈⋊H(3,𝔽₃)	C₉	1	1	yes	characteristic
10	#6204	C₉	1	2	yes	—
11	C₉⋊SL(2,𝔽₃)	—	3	2	no	—
12	C₉×SL(2,𝔽₃)	—	3	1	no	—
13	Q₈×C₂₇	—	3	1	no	—
14	Q₈⋊C₂₇	—	3	1	no	—
15	Q₈⋊C₂₇	—	3	1	no	—
16	C₂×#737	—	4	1	no	—
17	C₂×C₂₇⋊C₃	—	4	1	no	—
18	C₂×C₂₇⋊C₃	A₄	1	1	yes	characteristic
19	C₂×C₃×C₂₇	—	4	1	no	—
20	C₃×C₄×C₉	—	3	1	no	—
21	C₄×C₂₇	—	3	3	no	—
22	#737	—	4	1	no	—
23	C₂₇⋊C₃	—	4	1	no	—
24	C₂₇⋊C₃	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
25	C₃×C₂₇	—	4	1	no	—
26	C₃²×Q₈	C₃×C₉	1	1	yes	characteristic
27	C₃×SL(2,𝔽₃)	—	3	1	no	—
28	Q₈×C₉	H(3,𝔽₃)	1	1	yes	characteristic
29	Q₈×C₉	3₋¹⁺²	1	2	yes	—
30	Q₈⋊C₉	—	3	2	no	—
31	C₂×3₋¹⁺²	—	4	2	no	—
32	C₂×3₋¹⁺²	—	12	2	no	—
33	C₂×H(3,𝔽₃)	—	4	1	no	—
34	C₂×C₃×C₉	—	12	1	no	—
35	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
36	C₂×C₂₇	—	3	1	no	—
37	C₂×C₂₇	—	12	1	no	—
38	C₂×C₂₇	—	12	1	no	—
39	C₃²×C₄	—	3	1	no	—
40	C₄×C₉	—	3	2	no	—
41	C₄×C₉	—	3	1	no	—
42	3₋¹⁺²	—	4	2	no	—
43	3₋¹⁺²	—	12	2	no	—
44	H(3,𝔽₃)	—	4	1	no	—
45	C₃×C₉	—	12	1	no	—
46	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
47	C₂₇	—	3	1	no	—
48	C₂₇	—	12	1	no	—
49	C₂₇	—	12	1	no	—
50	C₃×Q₈	—	3	1	no	—
51	C₃×Q₈	C₃²⋊C₉	1	1	yes	characteristic
52	SL(2,𝔽₃)	—	9	1	no	—
53	C₂×C₃²	—	12	1	no	—
54	C₂×C₃²	C₉×A₄	1	1	yes	characteristic
55	C₂×C₉	—	12	2	no	—
56	C₂×C₉	C₉⋊A₄	1	2	yes	—
57	C₂×C₉	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
58	C₃×C₄	—	3	1	no	—
59	C₃×C₄	—	9	1	no	—
60	C₃²	—	12	1	no	—
61	C₃²	C₉×SL(2,𝔽₃)	1	1	yes	characteristic
62	C₉	—	12	2	no	—
63	C₉	C₉⋊SL(2,𝔽₃)	1	2	yes	—
64	C₉	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
65	Q₈	#6872	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	3	1	no	—
67	C₂×C₃	—	36	1	no	—
68	C₂×C₃	(C₃×A₄)⋊C₉	1	1	yes	characteristic
69	C₄	—	3	1	no	—
70	C₃	—	3	1	no	—
71	C₃	—	36	1	no	—
72	C₃	(C₃×SL(2,𝔽₃))⋊C₉	1	1	yes	characteristic
73	C₂	#11757433	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂²⋊H(3,𝔽₃)
commutator	C₃²×Q₈	C₃×C₉
Frattini	C₂×C₃×C₉	C₃×A₄
Fitting	Q₈×C₂₇⋊C₃	C₃
socle	C₂×C₃	(C₃×A₄)⋊C₉
prime radical	Q₈⋊H(3,𝔽₃)	C₉
cosocle	C₃×Q₈×C₉	C₃²
hypercenter	C₂×C₂₇⋊C₃	A₄
hypocenter	Q₈	#6872
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₃×C₉ ◁ C₂×C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
2. Q₈×C₂₇⋊C₃	C₃	semidirect
3. #10619570	C₃	semidirect
4. C₃×Q₈⋊C₂₇	C₃	semidirect
24. C₂₇⋊C₃	SL(2,𝔽₃)	semidirect
65. Q₈	#6872	semidirect

Maximal subgroups

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

subgroup	index
#10619912	3
Q₈×C₂₇⋊C₃	3
#10619570	3
C₃×Q₈⋊C₂₇	3
C₂×#6872	4