Finite Group Databasebeta

← C23×C3×AGL(1,𝔽9) #13181844

Subgroups — C23×C3×AGL(1,𝔽9) #13181844

Order 1728.

842 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 370 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×#157C211yesmaximal, characteristic
2C22×C3×AGL(1,𝔽9)C2114yesmaximal, direct product factor
3C23×AGL(1,𝔽9)C311yesmaximal, characteristic, direct product factor
4C23×C3×C3⋊D3C411yescharacteristic
5C22×C3×#157C417yes—
6C22×C3×#157C2217yes—
7C2×C3×AGL(1,𝔽9)C22128yesdirect product factor
8C23×#157C2×C311yescharacteristic
9C22×AGL(1,𝔽9)C2×C3114yesdirect product factor
10C23×C33C811yescharacteristic, complemented
11C22×C3×C3⋊D3C817yescomplemented
12C22×C3×C3⋊D3C2×C417yes—
13C2×C3×#157C2×C4121yes—
14C2×C3×#157C2317yes—
15C3×AGL(1,𝔽9)C2318yesdirect product factor
16C23×C3×C8—91nomaximal
17C23×C3⋊D3C3×C411yescharacteristic
18C23×C3×D3—121no—
19C22×#157C3×C417yes—
20C22×#157C22×C317yes—
21C2×AGL(1,𝔽9)C22×C3128yesdirect product factor
22C22×C33C2×C817yescomplemented
23C2×C3×C3⋊D3C2×C8121yescomplemented
24C2×C3×C3⋊D3C22×C417yes—
25C3×#157C22×C417yes—
26C3×#157C2411yescharacteristic
27C23×C3×C4—91no—
28C22×C3×C8—914no—
29C23×C32—41no—
30C23×C32—81no—
31C23×C32C3×C811yescharacteristic, complemented
32C22×C3⋊D3C3×C817yescomplemented
33C22×C3⋊D3C2×C3×C417yes—
34C22×C3×D3—127no—
35C22×C3×D3—127no—
36C2×#157C2×C3×C4121yes—
37C2×#157C23×C317yes—
38AGL(1,𝔽9)C23×C318yesdirect product factor
39C23×C8—91noSylow
40C2×C33C22×C817yescomplemented
41C3×C3⋊D3C22×C817yescomplemented
42C3×C3⋊D3C23×C411yescharacteristic
43C24×C3—91no—
44C23×D3—121no—
45C22×C3×C4—97no—
46C22×C3×C4—97no—
47C2×C3×C8—928no—
48C22×C32—47no—
49C22×C32—87no—
50C22×C32C2×C3×C817yescomplemented
51C2×C3⋊D3C2×C3×C8121yescomplemented
52C2×C3⋊D3C22×C3×C417yes—
53C2×C3×D3—1221no—
54C2×C3×D3—127no—
55#157C22×C3×C417yes—
56#157C24×C311yescharacteristic
57C23×C4—91no—
58C22×C8—914no—
59C33C23×C811yescharacteristic, complemented, Sylow
60C23×C3—41no—
61C23×C3—81no—
62C23×C3—97no—
63C23×C3—97no—
64C23×C3AGL(1,𝔽9)11yescharacteristic, direct product factor
65C22×D3—127no—
66C22×D3—127no—
67C2×C3×C4—921no—
68C2×C3×C4—97no—
69C3×C8—98no—
70C2×C32—47no—
71C2×C32—87no—
72C2×C32C22×C3×C817yescomplemented
73C3⋊D3C22×C3×C817yescomplemented
74C3⋊D3C23×C3×C411yescharacteristic
75C3×D3—127no—
76C3×D3—121no—
77C24—91no—
78C22×C4—97no—
79C22×C4—97no—
80C2×C8—928no—
81C22×C3—47no—
82C22×C3—87no—
83C22×C3—97no—
84C22×C3—921no—
85C22×C3C2×AGL(1,𝔽9)17yesdirect product factor
86C2×D3—1221no—
87C2×D3—127no—
88C3×C4—91no—
89C3×C4—97no—
90C32—41no—
91C32—81no—
92C32C23×C3×C811yescharacteristic, complemented
93C23—97no—
94C23—97no—
95C23C3×AGL(1,𝔽9)11yescharacteristic, direct product factor
96C2×C4—97no—
97C2×C4—921no—
98C8—98no—
99C2×C3—47no—
100C2×C3—87no—
101C2×C3—91no—
102C2×C3—97no—
103C2×C3C22×AGL(1,𝔽9)17yesdirect product factor
104D3—127no—
105D3—121no—
106C22—97no—
107C22—921no—
108C22C2×C3×AGL(1,𝔽9)17yesdirect product factor
109C4—91no—
110C4—97no—
111C3—41no—
112C3—81no—
113C3C23×AGL(1,𝔽9)11yescharacteristic, direct product factor
114C2—91no—
115C2—97no—
116C2C22×C3×AGL(1,𝔽9)17yesdirect product factor
subgroupG/H
center, hypercenterC23×C3AGL(1,𝔽9)
commutator, hypocenterC32C23×C3×C8
FrattiniC1G
Fitting, socleC23×C33C8
prime radicalC23×C3×C3⋊D3C4
cosocle#157C24×C3
LCSG ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×AGL(1,𝔽9)C2direct (N × H)
3. C23×AGL(1,𝔽9)C3direct (N × H)
7. C2×C3×AGL(1,𝔽9)C22direct (N × H)
9. C22×AGL(1,𝔽9)C2×C3direct (N × H)
10. C23×C33C8semidirect
11. C22×C3×C3⋊D3C8semidirect
15. C3×AGL(1,𝔽9)C23direct (N × H)
21. C2×AGL(1,𝔽9)C22×C3direct (N × H)
22. C22×C33C2×C8semidirect
23. C2×C3×C3⋊D3C2×C8semidirect
31. C23×C32C3×C8semidirect
32. C22×C3⋊D3C3×C8semidirect
38. AGL(1,𝔽9)C23×C3direct (N × H)
40. C2×C33C22×C8semidirect
41. C3×C3⋊D3C22×C8semidirect
50. C22×C32C2×C3×C8semidirect
51. C2×C3⋊D3C2×C3×C8semidirect
59. C33C23×C8semidirect
64. C23×C3AGL(1,𝔽9)direct (N × H)
72. C2×C32C22×C3×C8semidirect
73. C3⋊D3C22×C3×C8semidirect
85. C22×C3C2×AGL(1,𝔽9)direct (N × H)
92. C32C23×C3×C8semidirect
95. C23C3×AGL(1,𝔽9)direct (N × H)
103. C2×C3C22×AGL(1,𝔽9)direct (N × H)
108. C22C2×C3×AGL(1,𝔽9)direct (N × H)
113. C3C23×AGL(1,𝔽9)direct (N × H)
116. C2C22×C3×AGL(1,𝔽9)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C22×C3×AGL(1,𝔽9)2
C23×AGL(1,𝔽9)3
C23×C3×C89