Finite Group Databasebeta

← C24×C4×#64408 #34032689

Subgroups — C24×C4×#64408 #34032689

Order 18688.

24030 conjugacy classes of subgroups; only the 12696 normal subgroups are shown.

normal subgroups only

This group has 24030 conjugacy classes of subgroups — too many to draw. Only its 12696 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
1C25×#64408C212yesmaximal
2C23×C4×#64408C2160yesmaximal, direct product factor
3C24×C4×D73C211yesmaximal, characteristic
4C24×#64408C4164yesdirect product factor
5C24×#64408C22160yes—
6C22×C4×#64408C221560yesdirect product factor
7C25×D73C2211yescharacteristic
8C24×Dic73C411yescharacteristic, complemented
9C23×C4×D73C4130yescomplemented
10C23×C4×D73C22130yes—
11C24×C4×C73C411yescharacteristic, complemented
12C23×#64408C2×C41960yesdirect product factor
13C23×#64408C231280yes—
14C2×C4×#64408C231960yesdirect product factor
15C24×D73C2×C4132yes—
16C24×D73C2×C4115yes—
17C24×D73C23115yes—
18C23×Dic73C2×C4130yescomplemented
19C22×C4×D73C2×C41420yescomplemented
20C22×C4×D73C231140yes—
21C25×C73C2×C411yescharacteristic
22C23×C4×C73C2×C4130yescomplemented
23C22×#64408C22×C412240yesdirect product factor
24C22×#64408C241240yes—
25C4×#64408C241256yesdirect product factor
26C23×D73C421240yescomplemented
27C23×D73C22×C41105yes—
28C23×D73C22×C41240yes—
29C23×D73C24135yes—
30C22×Dic73C22×C41140yescomplemented
31C2×C4×D73C22×C41840yescomplemented
32C2×C4×D73C241120yes—
33C24×C73C42116yescomplemented
34C24×C73C22×C4115yes—
35C22×C4×C73C22×C41140yescomplemented
36C23×C73C2×C421120yescomplemented
37C23×C73C23×C4135yes—
38C22×D73C2×C421840yescomplemented
39C22×D73C23×C41280yes—
40C22×D73C23×C41105yes—
41C22×D73C25115yes—
42C2×#64408C23×C41960yesdirect product factor
43C2×#64408C25132yes—
44C2×C4×C73C23×C41120yescomplemented
45C2×Dic73C23×C41120yescomplemented
46C4×D73C23×C41240yescomplemented
47C4×D73C25116yes—
48C22×C73C22×C421140yescomplemented
49C22×C73C24×C4115yes—
50C2×D73C22×C421420yescomplemented
51C2×D73C24×C4160yes—
52C2×D73C24×C4115yes—
53C2×D73C2611yescharacteristic
54#64408C24×C4164yesdirect product factor
55C4×C73C24×C4116yescomplemented
56Dic73C24×C4116yescomplemented
57C24×C42—731nomaximal, Sylow
58C2×C73C23×C42130yescomplemented
59C2×C73C25×C411yescharacteristic
60D73C23×C42130yescomplemented
61D73C25×C412yes—
62C25×C4—732no—
63C25×C4—731no—
64C23×C42—7360no—
65C73C24×C4211yescharacteristic, complemented, Sylow
66C26—731no—
67C24×C4—7364no—
68C24×C4—7360no—
69C24×C4—731no—
70C24×C4—7330no—
71C24×C4—7330no—
72C24×C4#6440811yescharacteristic, direct product factor
73C22×C42—73560no—
74C25—7332no—
75C25—7315no—
76C25—7315no—
77C25C2×#6440811yescharacteristic
78C23×C4—73960no—
79C23×C4—73280no—
80C23×C4—7330no—
81C23×C4—73420no—
82C23×C4—73140no—
83C23×C4C2×#64408130yesdirect product factor
84C2×C42—73960no—
85C24—73105no—
86C24—7335no—
87C24—73240no—
88C24—73240no—
89C24C4×#64408116yesdirect product factor
90C24C22×#64408115yes—
91C22×C4—732240no—
92C22×C4—73240no—
93C22×C4—73140no—
94C22×C4—73840no—
95C22×C4—73120no—
96C22×C4C22×#644081140yesdirect product factor
97C42—73256no—
98C23—73280no—
99C23—73840no—
100C23—7315no—
101C23—73105no—
102C23C2×C4×#644081120yesdirect product factor
103C23C23×#64408135yes—
104C2×C4—7332no—
105C2×C4—73960no—
106C2×C4—73120no—
107C2×C4—7316no—
108C2×C4—73240no—
109C2×C4C23×#644081120yesdirect product factor
110C22—7360no—
111C22—73420no—
112C22—731no—
113C22—7315no—
114C22C22×C4×#644081140yesdirect product factor
115C22C24×#64408115yes—
116C4—7364no—
117C4—7316no—
118C4C24×#64408116yesdirect product factor
119C2—732no—
120C2—7330no—
121C2C23×C4×#64408130yesdirect product factor
122C2C25×#6440811yescharacteristic
subgroupG/H
center, hypercenterC24×C4#64408
commutator, hypocenterC73C24×C42
FrattiniC2C25×#64408
FittingC24×C4×C73C4
socleC25×C73C2×C4
prime radicalC25×D73C22
cosocleC2×D73C26
LCSG ▷ C73
UCSC1 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
2. C23×C4×#64408C2direct (N × H)
4. C24×#64408C4direct (N × H)
6. C22×C4×#64408C22direct (N × H)
8. C24×Dic73C4semidirect
9. C23×C4×D73C4semidirect
11. C24×C4×C73C4semidirect
12. C23×#64408C2×C4direct (N × H)
14. C2×C4×#64408C23direct (N × H)
18. C23×Dic73C2×C4semidirect
19. C22×C4×D73C2×C4semidirect
22. C23×C4×C73C2×C4semidirect
23. C22×#64408C22×C4direct (N × H)
25. C4×#64408C24direct (N × H)
26. C23×D73C42semidirect
30. C22×Dic73C22×C4semidirect
31. C2×C4×D73C22×C4semidirect
33. C24×C73C42semidirect
35. C22×C4×C73C22×C4semidirect
36. C23×C73C2×C42semidirect
38. C22×D73C2×C42semidirect
42. C2×#64408C23×C4direct (N × H)
44. C2×C4×C73C23×C4semidirect
45. C2×Dic73C23×C4semidirect
46. C4×D73C23×C4semidirect
48. C22×C73C22×C42semidirect
50. C2×D73C22×C42semidirect
54. #64408C24×C4direct (N × H)
55. C4×C73C24×C4semidirect
56. Dic73C24×C4semidirect
58. C2×C73C23×C42semidirect
60. D73C23×C42semidirect
65. C73C24×C42semidirect
72. C24×C4#64408direct (N × H)
83. C23×C4C2×#64408direct (N × H)
89. C24C4×#64408direct (N × H)
96. C22×C4C22×#64408direct (N × H)
102. C23C2×C4×#64408direct (N × H)
109. C2×C4C23×#64408direct (N × H)
114. C22C22×C4×#64408direct (N × H)
118. C4C24×#64408direct (N × H)
121. C2C23×C4×#64408direct (N × H)

Maximal subgroups

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

subgroupindex
C25×#644082
C23×C4×#644082
C24×C4×D732
C24×C4273