Subgroups — C₂³×#6249 #13180075

Order 1728.

641 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1088	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6249	C₂	1	14	yes	maximal, direct product factor
3	C₂³×#272	C₄	1	1	yes	characteristic
4	C₂²×#1088	C₄	1	7	yes	—
5	C₂²×#1088	C₂²	1	7	yes	—
6	C₂×#6249	C₂²	1	28	yes	direct product factor
7	C₂³×H(3,𝔽₃)	C₈	1	1	yes	characteristic, complemented
8	C₂²×#272	C₈	1	7	yes	complemented
9	C₂²×#272	C₂×C₄	1	7	yes	—
10	C₂×#1088	C₂×C₄	1	21	yes	—
11	C₂×#1088	C₂³	1	7	yes	—
12	#6249	C₂³	1	8	yes	direct product factor
13	C₂³×C₃⋊C₈	—	9	1	no	maximal
14	C₂³×C₃×D₃	—	12	1	no	—
15	C₂²×H(3,𝔽₃)	C₂×C₈	1	7	yes	complemented
16	C₂×#272	C₂×C₈	1	21	yes	complemented
17	C₂×#272	C₂²×C₄	1	7	yes	—
18	#1088	C₂²×C₄	1	7	yes	—
19	#1088	C₂⁴	1	1	yes	characteristic
20	C₂³×C₃×C₄	—	9	1	no	—
21	C₂²×C₃⋊C₈	—	9	14	no	—
22	C₂³×C₃²	—	4	1	no	—
23	C₂²×C₃×D₃	—	12	7	no	—
24	C₂²×C₃×D₃	—	12	7	no	—
25	C₂³×C₈	—	27	1	no	Sylow
26	C₂×H(3,𝔽₃)	C₂²×C₈	1	7	yes	complemented
27	#272	C₂²×C₈	1	7	yes	complemented
28	#272	C₂³×C₄	1	1	yes	characteristic
29	C₂⁴×C₃	—	9	1	no	—
30	C₂³×D₃	—	12	1	no	—
31	C₂²×C₃×C₄	—	9	7	no	—
32	C₂²×C₃×C₄	—	9	7	no	—
33	C₂×C₃⋊C₈	—	9	28	no	—
34	C₂²×C₃²	—	4	7	no	—
35	C₂×C₃×D₃	—	12	21	no	—
36	C₂×C₃×D₃	—	12	7	no	—
37	C₂³×C₄	—	9	1	no	—
38	C₂²×C₈	—	27	14	no	—
39	H(3,𝔽₃)	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
40	C₂³×C₃	—	9	7	no	—
41	C₂³×C₃	—	9	7	no	—
42	C₂³×C₃	—	12	1	no	—
43	C₂³×C₃	AGL(1,𝔽₉)	1	1	yes	characteristic
44	C₂²×D₃	—	12	7	no	—
45	C₂²×D₃	—	12	7	no	—
46	C₂×C₃×C₄	—	9	21	no	—
47	C₂×C₃×C₄	—	9	7	no	—
48	C₃⋊C₈	—	9	8	no	—
49	C₂×C₃²	—	4	7	no	—
50	C₃×D₃	—	12	7	no	—
51	C₃×D₃	—	12	1	no	—
52	C₂⁴	—	9	1	no	—
53	C₂²×C₄	—	9	7	no	—
54	C₂²×C₄	—	9	7	no	—
55	C₂×C₈	—	27	28	no	—
56	C₂²×C₃	—	9	7	no	—
57	C₂²×C₃	—	9	21	no	—
58	C₂²×C₃	—	12	7	no	—
59	C₂²×C₃	C₂×AGL(1,𝔽₉)	1	7	yes	—
60	C₂×D₃	—	12	7	no	—
61	C₂×D₃	—	12	21	no	—
62	C₃×C₄	—	9	1	no	—
63	C₃×C₄	—	9	7	no	—
64	C₃²	—	4	1	no	—
65	C₂³	—	9	7	no	—
66	C₂³	—	9	7	no	—
67	C₂³	#6249	1	1	yes	characteristic, direct product factor
68	C₂×C₄	—	9	7	no	—
69	C₂×C₄	—	9	21	no	—
70	C₈	—	27	8	no	—
71	C₂×C₃	—	9	1	no	—
72	C₂×C₃	—	9	7	no	—
73	C₂×C₃	—	12	7	no	—
74	C₂×C₃	C₂²×AGL(1,𝔽₉)	1	7	yes	—
75	D₃	—	12	1	no	—
76	D₃	—	12	7	no	—
77	C₂²	—	9	7	no	—
78	C₂²	—	9	21	no	—
79	C₂²	C₂×#6249	1	7	yes	direct product factor
80	C₄	—	9	1	no	—
81	C₄	—	9	7	no	—
82	C₃	—	12	1	no	—
83	C₃	C₂³×AGL(1,𝔽₉)	1	1	yes	characteristic
84	C₂	—	9	1	no	—
85	C₂	—	9	7	no	—
86	C₂	C₂²×#6249	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	#6249
commutator, hypocenter	H(3,𝔽₃)	C₂³×C₈
Frattini	C₃	C₂³×AGL(1,𝔽₉)
Fitting	C₂³×H(3,𝔽₃)	C₈
socle	C₂³×C₃	AGL(1,𝔽₉)
prime radical	C₂³×#272	C₄
cosocle	#1088	C₂⁴
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²×#6249	C₂	direct (N × H)
6. C₂×#6249	C₂²	direct (N × H)
7. C₂³×H(3,𝔽₃)	C₈	semidirect
8. C₂²×#272	C₈	semidirect
12. #6249	C₂³	direct (N × H)
15. C₂²×H(3,𝔽₃)	C₂×C₈	semidirect
16. C₂×#272	C₂×C₈	semidirect
26. C₂×H(3,𝔽₃)	C₂²×C₈	semidirect
27. #272	C₂²×C₈	semidirect
39. H(3,𝔽₃)	C₂³×C₈	semidirect
67. C₂³	#6249	direct (N × H)
79. C₂²	C₂×#6249	direct (N × H)
86. C₂	C₂²×#6249	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#1088	2
C₂²×#6249	2
C₂³×C₃⋊C₈	9

Subgroups — C23×#6249 #13180075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6249 #13180075