Finite Group Databasebeta

← C24×C3×#157 #13181863

Subgroups — C24×C3×#157 #13181863

Order 1728.

4488 conjugacy classes of subgroups; only the 1496 normal subgroups are shown.

normal subgroups only

This group has 4488 conjugacy classes of subgroups — too many to draw. Only its 1496 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
1C24×C3×C3⋊D3C211yesmaximal, characteristic
2C23×C3×#157C2130yesmaximal, direct product factor
3C24×#157C311yesmaximal, characteristic, direct product factor
4C24×C33C411yescharacteristic, complemented
5C23×C3×C3⋊D3C4115yescomplemented
6C23×C3×C3⋊D3C22115yes—
7C22×C3×#157C221140yesdirect product factor
8C24×C3⋊D3C2×C311yescharacteristic
9C24×C3×D3—62no—
10C23×#157C2×C3130yesdirect product factor
11C23×C33C2×C4115yescomplemented
12C22×C3×C3⋊D3C2×C41105yescomplemented
13C22×C3×C3⋊D3C23135yes—
14C2×C3×#157C231120yesdirect product factor
15C24×C3×C4—91nomaximal
16C24×C32—22no—
17C24×C32—42no—
18C24×C32C3×C411yescharacteristic, complemented
19C23×C3⋊D3C3×C4115yescomplemented
20C23×C3⋊D3C22×C3115yes—
21C23×C3×D3—630no—
22C23×C3×D3—630no—
23C22×#157C22×C31140yesdirect product factor
24C22×C33C22×C4135yescomplemented
25C2×C3×C3⋊D3C22×C41105yescomplemented
26C2×C3×C3⋊D3C24115yes—
27C3×#157C24116yesdirect product factor
28C25×C3—91no—
29C24×D3—62no—
30C23×C3×C4—930no—
31C23×C32—230no—
32C23×C32—430no—
33C23×C32C2×C3×C4115yescomplemented
34C22×C3⋊D3C2×C3×C41105yescomplemented
35C22×C3⋊D3C23×C3135yes—
36C22×C3×D3—6210no—
37C22×C3×D3—670no—
38C2×#157C23×C31120yesdirect product factor
39C24×C4—91noSylow
40C2×C33C23×C4115yescomplemented
41C3×C3⋊D3C23×C4115yescomplemented
42C3×C3⋊D3C2511yescharacteristic
43C24×C3—22no—
44C24×C3—42no—
45C24×C3—915no—
46C24×C3—915no—
47C24×C3#15711yescharacteristic, direct product factor
48C23×D3—630no—
49C23×D3—630no—
50C22×C3×C4—9140no—
51C22×C32—270no—
52C22×C32—470no—
53C22×C32C22×C3×C4135yescomplemented
54C2×C3⋊D3C22×C3×C41105yescomplemented
55C2×C3⋊D3C24×C3115yes—
56C2×C3×D3—6210no—
57C2×C3×D3—630no—
58#157C24×C3116yesdirect product factor
59C25—91no—
60C23×C4—930no—
61C33C24×C411yescharacteristic, complemented, Sylow
62C23×C3—230no—
63C23×C3—430no—
64C23×C3—935no—
65C23×C3—9105no—
66C23×C3C2×#157115yesdirect product factor
67C22×D3—6210no—
68C22×D3—670no—
69C2×C3×C4—9120no—
70C2×C32—230no—
71C2×C32—430no—
72C2×C32C23×C3×C4115yescomplemented
73C3⋊D3C23×C3×C4115yescomplemented
74C3⋊D3C25×C311yescharacteristic
75C3×D3—630no—
76C3×D3—62no—
77C24—915no—
78C24—915no—
79C24C3×#15711yescharacteristic, direct product factor
80C22×C4—9140no—
81C22×C3—270no—
82C22×C3—470no—
83C22×C3—915no—
84C22×C3—9105no—
85C22×C3C22×#157135yesdirect product factor
86C2×D3—6210no—
87C2×D3—630no—
88C3×C4—916no—
89C32—22no—
90C32—42no—
91C32C24×C3×C411yescharacteristic, complemented
92C23—935no—
93C23—9105no—
94C23C2×C3×#157115yesdirect product factor
95C2×C4—9120no—
96C2×C3—230no—
97C2×C3—430no—
98C2×C3—91no—
99C2×C3—915no—
100C2×C3C23×#157115yesdirect product factor
101D3—62no—
102D3—630no—
103C22—915no—
104C22—9105no—
105C22C22×C3×#157135yesdirect product factor
106C4—916no—
107C3—22no—
108C3—42no—
109C3C24×#15711yescharacteristic, direct product factor
110C2—91no—
111C2—915no—
112C2C23×C3×#157115yesdirect product factor
subgroupG/H
center, hypercenterC24×C3#157
commutator, hypocenterC32C24×C3×C4
FrattiniC1G
Fitting, socleC24×C33C4
prime radicalC24×C3×C3⋊D3C2
cosocleC3⋊D3C25×C3
LCSG ▷ C32
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C23×C3×#157C2direct (N × H)
3. C24×#157C3direct (N × H)
4. C24×C33C4semidirect
5. C23×C3×C3⋊D3C4semidirect
7. C22×C3×#157C22direct (N × H)
10. C23×#157C2×C3direct (N × H)
11. C23×C33C2×C4semidirect
12. C22×C3×C3⋊D3C2×C4semidirect
14. C2×C3×#157C23direct (N × H)
18. C24×C32C3×C4semidirect
19. C23×C3⋊D3C3×C4semidirect
23. C22×#157C22×C3direct (N × H)
24. C22×C33C22×C4semidirect
25. C2×C3×C3⋊D3C22×C4semidirect
27. C3×#157C24direct (N × H)
33. C23×C32C2×C3×C4semidirect
34. C22×C3⋊D3C2×C3×C4semidirect
38. C2×#157C23×C3direct (N × H)
40. C2×C33C23×C4semidirect
41. C3×C3⋊D3C23×C4semidirect
47. C24×C3#157direct (N × H)
53. C22×C32C22×C3×C4semidirect
54. C2×C3⋊D3C22×C3×C4semidirect
58. #157C24×C3direct (N × H)
61. C33C24×C4semidirect
66. C23×C3C2×#157direct (N × H)
72. C2×C32C23×C3×C4semidirect
73. C3⋊D3C23×C3×C4semidirect
79. C24C3×#157direct (N × H)
85. C22×C3C22×#157direct (N × H)
91. C32C24×C3×C4semidirect
94. C23C2×C3×#157direct (N × H)
100. C2×C3C23×#157direct (N × H)
105. C22C22×C3×#157direct (N × H)
109. C3C24×#157direct (N × H)
112. C2C23×C3×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C3⋊D32
C23×C3×#1572
C24×#1573
C24×C3×C49