Finite Group Databasebeta

← C23×C3×PSU(3,𝔽2) #13181850

Subgroups — C23×C3×PSU(3,𝔽2) #13181850

Order 1728.

1866 conjugacy classes of subgroups; only the 882 normal subgroups are shown.

normal subgroups only

This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 882 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×#157C213yesmaximal
2C22×C3×PSU(3,𝔽2)C2128yesmaximal, direct product factor
3C23×PSU(3,𝔽2)C311yesmaximal, characteristic, direct product factor
4C23×C3×C3⋊D3C2211yescharacteristic
5C22×C3×#157C22142yes—
6C2×C3×PSU(3,𝔽2)C221112yesdirect product factor
7C23×#157C2×C313yes—
8C22×PSU(3,𝔽2)C2×C3128yesdirect product factor
9C23×C33Q811yescharacteristic, complemented
10C22×C3×C3⋊D3Q817yescomplemented
11C22×C3×C3⋊D3C2317yes—
12C2×C3×#157C23184yes—
13C3×PSU(3,𝔽2)C23164yesdirect product factor
14C23×C3×Q8—91nomaximal
15C23×C3⋊D3C22×C311yescharacteristic
16C23×C3×D3—121no—
17C22×#157C22×C3142yes—
18C2×PSU(3,𝔽2)C22×C31112yesdirect product factor
19C22×C33C2×Q817yescomplemented
20C2×C3×C3⋊D3C2×Q8121yescomplemented
21C2×C3×C3⋊D3C2417yes—
22C3×#157C24124yes—
23C22×C3×Q8—928no—
24C23×C3×C4—93no—
25C23×C32—41no—
26C23×C32—81no—
27C23×C32C3×Q811yescharacteristic, complemented
28C22×C3⋊D3C3×Q817yescomplemented
29C22×C3⋊D3C23×C317yes—
30C22×C3×D3—127no—
31C22×C3×D3—127no—
32C2×#157C23×C3184yes—
33PSU(3,𝔽2)C23×C3164yesdirect product factor
34C23×Q8—91noSylow
35C2×C33C22×Q817yescomplemented
36C3×C3⋊D3C22×Q817yescomplemented
37C3×C3⋊D3C2511yescharacteristic
38C24×C3—91no—
39C23×D3—121no—
40C2×C3×Q8—9112no—
41C22×C3×C4—942no—
42C22×C32—47no—
43C22×C32—87no—
44C22×C32C2×C3×Q817yescomplemented
45C2×C3⋊D3C2×C3×Q8121yescomplemented
46C2×C3⋊D3C24×C317yes—
47C2×C3×D3—1221no—
48C2×C3×D3—127no—
49#157C24×C3124yes—
50C22×Q8—928no—
51C23×C4—93no—
52C33C23×Q811yescharacteristic, complemented, Sylow
53C23×C3—41no—
54C23×C3—81no—
55C23×C3—97no—
56C23×C3—97no—
57C23×C3PSU(3,𝔽2)11yescharacteristic, direct product factor
58C22×D3—127no—
59C22×D3—127no—
60C3×Q8—964no—
61C2×C3×C4—984no—
62C2×C32—47no—
63C2×C32—87no—
64C2×C32C22×C3×Q817yescomplemented
65C3⋊D3C22×C3×Q817yescomplemented
66C3⋊D3C25×C311yescharacteristic
67C3×D3—127no—
68C3×D3—121no—
69C24—91no—
70C2×Q8—9112no—
71C22×C4—942no—
72C22×C3—47no—
73C22×C3—87no—
74C22×C3—97no—
75C22×C3—921no—
76C22×C3C2×PSU(3,𝔽2)17yesdirect product factor
77C2×D3—1221no—
78C2×D3—127no—
79C3×C4—924no—
80C32—41no—
81C32—81no—
82C32C23×C3×Q811yescharacteristic, complemented
83C23—97no—
84C23—97no—
85C23C3×PSU(3,𝔽2)11yescharacteristic, direct product factor
86Q8—964no—
87C2×C4—984no—
88C2×C3—47no—
89C2×C3—87no—
90C2×C3—91no—
91C2×C3—97no—
92C2×C3C22×PSU(3,𝔽2)17yesdirect product factor
93D3—121no—
94D3—127no—
95C22—97no—
96C22—921no—
97C22C2×C3×PSU(3,𝔽2)17yesdirect product factor
98C4—924no—
99C3—41no—
100C3—81no—
101C3C23×PSU(3,𝔽2)11yescharacteristic, direct product factor
102C2—91no—
103C2—97no—
104C2C22×C3×PSU(3,𝔽2)17yesdirect product factor
subgroupG/H
center, hypercenterC23×C3PSU(3,𝔽2)
commutator, cosocleC3⋊D3C25×C3
FrattiniC1G
Fitting, socleC23×C33Q8
prime radicalC23×C3×C3⋊D3C22
hypocenterC32C23×C3×Q8
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×PSU(3,𝔽2)C2direct (N × H)
3. C23×PSU(3,𝔽2)C3direct (N × H)
6. C2×C3×PSU(3,𝔽2)C22direct (N × H)
8. C22×PSU(3,𝔽2)C2×C3direct (N × H)
9. C23×C33Q8semidirect
10. C22×C3×C3⋊D3Q8semidirect
13. C3×PSU(3,𝔽2)C23direct (N × H)
18. C2×PSU(3,𝔽2)C22×C3direct (N × H)
19. C22×C33C2×Q8semidirect
20. C2×C3×C3⋊D3C2×Q8semidirect
27. C23×C32C3×Q8semidirect
28. C22×C3⋊D3C3×Q8semidirect
33. PSU(3,𝔽2)C23×C3direct (N × H)
35. C2×C33C22×Q8semidirect
36. C3×C3⋊D3C22×Q8semidirect
44. C22×C32C2×C3×Q8semidirect
45. C2×C3⋊D3C2×C3×Q8semidirect
52. C33C23×Q8semidirect
57. C23×C3PSU(3,𝔽2)direct (N × H)
64. C2×C32C22×C3×Q8semidirect
65. C3⋊D3C22×C3×Q8semidirect
76. C22×C3C2×PSU(3,𝔽2)direct (N × H)
82. C32C23×C3×Q8semidirect
85. C23C3×PSU(3,𝔽2)direct (N × H)
92. C2×C3C22×PSU(3,𝔽2)direct (N × H)
97. C22C2×C3×PSU(3,𝔽2)direct (N × H)
101. C3C23×PSU(3,𝔽2)direct (N × H)
104. C2C22×C3×PSU(3,𝔽2)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×PSU(3,𝔽2)2
C23×PSU(3,𝔽2)3
C23×C3×Q89