Subgroups — #13047028

Order 1296.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₇⋊SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	#88615	—	3	1	no	maximal
3	#88403	C₃	1	1	yes	maximal, characteristic, complemented
4	(C₂₇⋊C₃)⋊C₄	—	4	1	no	maximal
5	C₉×SL(2,𝔽₃)	D₃	1	1	yes	characteristic
6	Q₈×C₂₇	C₂×C₃	1	1	yes	characteristic
7	D₁₀₈	—	3	1	no	—
8	C₄×D₂₇	—	3	1	no	—
9	Q₈⋊C₂₇	—	2	1	no	—
10	C₂×C₂₇⋊C₃	—	4	1	no	—
11	C₃⋊(#41⋊C₃)	—	9	1	no	—
12	#3701	—	3	1	no	—
13	C₃×Dic₉	—	12	1	no	—
14	C₂×D₂₇	—	3	1	no	—
15	C₄×C₂₇	—	3	1	no	—
16	Dic₂₇	A₄	1	1	yes	characteristic
17	C₂₇⋊C₃	—	4	1	no	Sylow
18	C₃×SL(2,𝔽₃)	D₉	1	1	yes	characteristic
19	Q₈×C₉	C₃×D₃	1	1	yes	characteristic
20	D₃₆	—	9	1	no	—
21	C₄×D₉	—	9	1	no	—
22	Q₈⋊C₉	—	2	1	no	—
23	C₂×C₃×C₉	—	4	1	no	—
24	C₂×C₂₇	—	8	1	no	—
25	C₂×C₂₇	C₂×A₄	1	1	yes	characteristic
26	D₂₇	—	6	1	no	—
27	#239	—	9	1	no	—
28	#41⋊C₃	—	27	1	no	—
29	C₃×Dic₃	—	36	1	no	—
30	C₂×D₉	—	9	1	no	—
31	C₄×C₉	—	3	1	no	—
32	Dic₉	—	3	1	no	—
33	C₃×C₉	—	4	1	no	—
34	C₂₇	—	8	1	no	—
35	C₂₇	#41⋊C₃	1	1	yes	characteristic, complemented
36	C₃×Q₈	C₃×D₉	1	1	yes	characteristic
37	D₁₂	—	27	1	no	—
38	C₄×D₃	—	27	1	no	—
39	SL(2,𝔽₃)	—	3	1	no	—
40	C₂×C₃²	—	4	1	no	—
41	C₂×C₉	—	8	1	no	—
42	C₂×C₉	D₃×A₄	1	1	yes	characteristic
43	D₉	—	18	1	no	—
44	#41	—	27	1	no	Sylow
45	C₂×D₃	—	27	1	no	—
46	C₃×C₄	—	3	1	no	—
47	C₃×C₄	—	108	1	no	—
48	Dic₃	—	9	1	no	—
49	C₃²	—	4	1	no	—
50	C₉	—	8	1	no	—
51	C₉	C₃⋊(#41⋊C₃)	1	1	yes	characteristic
52	Q₈	D₂₇⋊C₃	1	1	yes	characteristic
53	D₄	—	81	1	no	—
54	C₂×C₄	—	81	1	no	—
55	C₂×C₃	—	12	1	no	—
56	C₂×C₃	A₄×D₉	1	1	yes	characteristic
57	D₃	—	54	1	no	—
58	C₂²	—	81	1	no	—
59	C₄	—	3	1	no	—
60	C₄	—	27	1	no	—
61	C₃	—	12	1	no	—
62	C₃	#88615	1	1	yes	characteristic
63	C₂	—	162	1	no	—
64	C₂	D₂₇⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	D₂₇⋊A₄
commutator, Fitting, cosocle, hypocenter	Q₈×C₂₇	C₂×C₃
Frattini	C₂×C₉	D₃×A₄
socle	C₂×C₃	A₄×D₉
LCS	G ▷ Q₈×C₂₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂₇⋊SL(2,𝔽₃)	C₂	semidirect
3. #88403	C₃	semidirect
35. C₂₇	#41⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₇⋊SL(2,𝔽₃)	2
#88615	3
#88403	3
(C₂₇⋊C₃)⋊C₄	4