Subgroups — C₃×#88592 #13048276

Order 1296.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1088	C₂	1	2	yes	maximal
2	C₂×C₃×#1088	C₂	1	1	yes	maximal, characteristic
3	#88592	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×#272	C₂²	1	1	yes	characteristic
5	C₃×#1088	—	2	2	no	—
6	C₃×#1088	C₄	1	2	yes	complemented
7	C₂×#1088	C₂×C₃	1	2	yes	—
8	C₂×#1088	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
10	C₃×#272	C₂×C₄	1	1	yes	characteristic
11	C₃×#272	D₄	1	1	yes	characteristic
12	C₃²×C₄⋊C₄	—	9	1	no	maximal
13	C₂×C₃²×D₃	—	12	1	no	—
14	C₂×#272	C₂²×C₃	1	1	yes	characteristic
15	#1088	—	2	2	no	—
16	#1088	C₃×C₄	1	2	yes	complemented
17	C₃×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₃²×C₄	—	9	1	no	—
19	C₂×C₃²×C₄	—	9	2	no	—
20	C₂×C₃³	—	4	1	no	—
21	C₃²×D₃	—	12	1	no	—
22	C₃²×D₃	—	12	1	no	—
23	C₂×H(3,𝔽₃)	—	8	1	no	—
24	C₂×H(3,𝔽₃)	C₃×Q₈	1	1	yes	characteristic
25	#272	C₂×C₃×C₄	1	1	yes	characteristic
26	#272	C₃×D₄	1	1	yes	characteristic
27	C₃×C₄⋊C₄	—	9	1	no	—
28	C₃×C₄⋊C₄	—	9	3	no	—
29	C₂²×C₃²	—	9	1	no	—
30	C₂×C₃×D₃	—	12	1	no	—
31	C₂×C₃×D₃	—	12	3	no	—
32	C₃²×C₄	—	9	2	no	—
33	C₃²×C₄	—	18	2	no	—
34	C₃³	—	4	1	no	—
35	H(3,𝔽₃)	—	8	1	no	—
36	H(3,𝔽₃)	C₃×C₄⋊C₄	1	1	yes	characteristic, complemented
37	C₂×C₃×C₄	—	9	1	no	—
38	C₂×C₃×C₄	—	9	3	no	—
39	C₂×C₃×C₄	—	9	2	no	—
40	C₂×C₃×C₄	—	9	6	no	—
41	C₂×C₃²	—	4	1	no	—
42	C₂×C₃²	—	8	1	no	—
43	C₂×C₃²	—	9	1	no	—
44	C₂×C₃²	—	9	1	no	—
45	C₂×C₃²	—	12	3	no	—
46	C₂×C₃²	PSU(3,𝔽₂)	1	1	yes	characteristic
47	C₃×D₃	—	12	1	no	—
48	C₃×D₃	—	12	1	no	—
49	C₃×D₃	—	12	3	no	—
50	C₃×D₃	—	12	3	no	—
51	C₄⋊C₄	—	9	1	no	Sylow
52	C₂²×C₃	—	9	3	no	—
53	C₂²×C₃	—	9	1	no	—
54	C₂×D₃	—	12	1	no	—
55	C₃×C₄	—	9	2	no	—
56	C₃×C₄	—	9	6	no	—
57	C₃×C₄	—	18	2	no	—
58	C₃×C₄	—	18	6	no	—
59	C₃²	—	4	1	no	—
60	C₃²	—	8	1	no	—
61	C₃²	—	12	3	no	—
62	C₃²	#3777	1	1	yes	characteristic
63	C₂×C₄	—	9	1	no	—
64	C₂×C₄	—	9	2	no	—
65	C₂×C₃	—	9	3	no	—
66	C₂×C₃	—	9	3	no	—
67	C₂×C₃	—	9	1	no	—
68	C₂×C₃	—	9	1	no	—
69	C₂×C₃	—	12	1	no	—
70	C₂×C₃	—	24	1	no	—
71	C₂×C₃	SU(3,𝔽₂)	1	3	yes	—
72	C₂×C₃	C₃×PSU(3,𝔽₂)	1	1	yes	characteristic
73	D₃	—	12	1	no	—
74	D₃	—	12	1	no	—
75	C₂²	—	9	1	no	—
76	C₄	—	9	2	no	—
77	C₄	—	18	2	no	—
78	C₃	—	12	1	no	—
79	C₃	—	24	1	no	—
80	C₃	#88592	1	3	yes	direct product factor
81	C₃	C₃×#3777	1	1	yes	characteristic
82	C₂	—	9	1	no	—
83	C₂	—	9	1	no	—
84	C₂	C₃×SU(3,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂×C₃²	PSU(3,𝔽₂)
commutator	#272	C₂×C₃×C₄
Frattini	C₂×C₃	C₃×PSU(3,𝔽₂)
Fitting	C₂×C₃×H(3,𝔽₃)	Q₈
prime radical	C₂×C₃×#272	C₂²
cosocle	C₂×#272	C₂²×C₃
hypocenter	H(3,𝔽₃)	C₃×C₄⋊C₄
LCS	G ▷ #272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
3. #88592	C₃	direct (N × H)
6. C₃×#1088	C₄	semidirect
16. #1088	C₃×C₄	semidirect
17. C₃×H(3,𝔽₃)	C₄⋊C₄	semidirect
36. H(3,𝔽₃)	C₃×C₄⋊C₄	semidirect
80. C₃	#88592	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#1088	2
#88592	3
C₃²×C₄⋊C₄	9

Subgroups — C3×#88592 #13048276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#88592 #13048276