Subgroups — #14517204

Order 1944.

177 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	C₂₇⋊SL(2,𝔽₃)	C₃	1	6	yes	maximal, complemented
6	SL(2,𝔽₃)×C₂₇	C₃	1	3	yes	maximal, complemented
7	C₂×#6903	—	4	1	no	maximal
8	C₄×C₂₇⋊C₃	—	3	1	no	—
9	#6903	—	4	1	no	Sylow
10	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
11	Q₈⋊H(3,𝔽₃)	C₉	1	1	yes	characteristic
12	#6204	C₉	1	2	yes	—
13	C₉⋊SL(2,𝔽₃)	C₉	1	6	yes	—
14	C₉×SL(2,𝔽₃)	C₃²	1	3	yes	—
15	Q₈×C₂₇	C₃²	1	3	yes	—
16	Q₈⋊C₂₇	C₃²	1	1	yes	characteristic
17	Q₈⋊C₂₇	C₃²	1	3	yes	—
18	Q₈⋊C₂₇	C₃²	1	2	yes	—
19	C₂×#737	—	4	1	no	—
20	C₂×C₂₇⋊C₃	—	4	1	no	—
21	C₂×C₂₇⋊C₃	—	4	6	no	—
22	C₂×C₂₇⋊C₃	A₄	1	1	yes	characteristic
23	C₂×C₃×C₂₇	—	4	1	no	—
24	C₂×C₃×C₂₇	—	4	3	no	—
25	C₃×C₄×C₉	—	3	1	no	—
26	C₄×C₂₇	—	3	3	no	—
27	#737	—	4	1	no	—
28	C₂₇⋊C₃	—	4	1	no	—
29	C₂₇⋊C₃	—	4	6	no	—
30	C₂₇⋊C₃	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
31	C₃×C₂₇	—	4	1	no	—
32	C₃×C₂₇	—	4	3	no	—
33	C₃²×Q₈	C₃×C₉	1	1	yes	characteristic
34	C₃×SL(2,𝔽₃)	C₃×C₉	1	3	yes	—
35	Q₈×C₉	C₃×C₉	1	2	yes	—
36	Q₈×C₉	C₃³	1	1	yes	characteristic
37	Q₈⋊C₉	C₃×C₉	1	6	yes	—
38	C₂×3₋¹⁺²	—	4	2	no	—
39	C₂×3₋¹⁺²	—	4	6	no	—
40	C₂×H(3,𝔽₃)	—	4	1	no	—
41	C₂×C₃×C₉	—	4	3	no	—
42	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
43	C₂×C₂₇	—	4	1	no	—
44	C₂×C₂₇	—	4	3	no	—
45	C₂×C₂₇	—	4	2	no	—
46	C₂×C₂₇	C₃×A₄	1	3	yes	—
47	C₃²×C₄	—	3	1	no	—
48	C₄×C₉	—	3	2	no	—
49	C₄×C₉	—	3	1	no	—
50	3₋¹⁺²	—	4	2	no	—
51	3₋¹⁺²	—	4	6	no	—
52	H(3,𝔽₃)	—	4	1	no	—
53	C₃×C₉	—	4	3	no	—
54	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
55	C₂₇	—	4	1	no	—
56	C₂₇	—	4	2	no	—
57	C₂₇	—	4	3	no	—
58	C₂₇	C₃×SL(2,𝔽₃)	1	3	yes	—
59	C₃×Q₈	—	3	1	no	—
60	C₃×Q₈	C₃²×C₉	1	1	yes	characteristic
61	SL(2,𝔽₃)	—	3	3	no	—
62	C₂×C₃²	—	4	3	no	—
63	C₂×C₃²	C₉×A₄	1	1	yes	characteristic
64	C₂×C₉	—	4	6	no	—
65	C₂×C₉	C₉×A₄	1	2	yes	—
66	C₂×C₉	C₃²×A₄	1	1	yes	characteristic
67	C₃×C₄	—	3	1	no	—
68	C₃×C₄	—	9	1	no	—
69	C₃²	—	4	3	no	—
70	C₃²	C₉×SL(2,𝔽₃)	1	1	yes	characteristic
71	C₉	—	4	6	no	—
72	C₉	C₉×SL(2,𝔽₃)	1	2	yes	—
73	C₉	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
74	Q₈	#6903	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃	—	3	1	no	—
76	C₂×C₃	—	12	3	no	—
77	C₂×C₃	C₃×C₉×A₄	1	1	yes	characteristic
78	C₄	—	3	1	no	—
79	C₃	—	3	1	no	—
80	C₃	—	12	3	no	—
81	C₃	C₃×C₉×SL(2,𝔽₃)	1	1	yes	characteristic
82	C₂	#11757797	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₉	C₃²×A₄
commutator	C₃×Q₈	C₃²×C₉
Fitting	Q₈×C₂₇⋊C₃	C₃
socle	C₂×C₃	C₃×C₉×A₄
prime radical	Q₈⋊H(3,𝔽₃)	C₉
cosocle	Q₈×C₉	C₃³
hypercenter	C₂×C₂₇⋊C₃	A₄
hypocenter	Q₈	#6903
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×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
5. C₂₇⋊SL(2,𝔽₃)	C₃	semidirect
6. SL(2,𝔽₃)×C₂₇	C₃	semidirect
30. C₂₇⋊C₃	SL(2,𝔽₃)	semidirect
74. Q₈	#6903	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
C₂₇⋊SL(2,𝔽₃)	3
SL(2,𝔽₃)×C₂₇	3
C₂×#6903	4