Finite Group Databasebeta

← #10596191

Subgroups — #10596191

Order 576.

756 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C32×#127C211yesmaximal, characteristic, complemented
2#64148C216yesmaximal, complemented
3#64146C213yesmaximal, complemented
4#64144C213yesmaximal, complemented
5#64085C212yesmaximal, complemented
6#5720—34nomaximal
7C23×C3⋊D3C2211yescharacteristic
8C2×C32×D4C2213yes—
9C2×C32×D4C2213yescomplemented
10C2×#638C2216yes—
11C22×C3⋊Dic3C2213yescomplemented
12C32×C42C2211yescharacteristic, complemented
13#3757C2213yes—
14#3757C2213yescomplemented
15#3752C2216yes—
16#3750C2216yescomplemented
17C3×#127D314yescomplemented
18#939—324no—
19#937—312no—
20#935—312no—
21#876—38no—
22C23×C32C2311yescharacteristic
23C23×C32C2313yes—
24C22×C3⋊D3—43no—
25C22×C3⋊D3C2312yes—
26C32×D4—43no—
27C32×D4—43no—
28C2×C32×C4C2313yes—
29#638—46no—
30C2×C3⋊Dic3—43no—
31C2×C3⋊Dic3C2316yes—
32#560—91noSylow
33C23×D3—34no—
34C2×C3×D4C2×D3112yes—
35C2×C3×D4C2×D3112yescomplemented
36C2×#67—324no—
37C22×Dic3—312no—
38C3×C42C2×D314yescomplemented
39#217—312no—
40#217—312no—
41(C2×D3)⋊C4—324no—
42Dic3⋊C4—324no—
43C22×C32—23no—
44C22×C32—26no—
45C22×C32—23no—
46C22×C32C2411yescharacteristic
47C2×C3⋊D3—26no—
48C2×C3⋊D3—44no—
49C32×C4—23no—
50C3⋊Dic3—26no—
51#127C3⋊D311yescharacteristic, complemented
52#126—92no—
53#123—93no—
54#121—96no—
55#120—93no—
56C23×C3C22×D314yes—
57C23×C3C22×D3112yes—
58C22×D3—38no—
59C22×D3—1212no—
60C3×D4—412no—
61C3×D4—412no—
62C2×C3×C4C22×D3112yes—
63#67—1224no—
64C2×Dic3—324no—
65C2×Dic3—1212no—
66C2×C32—41no—
67C2×C32—43no—
68C2×C32H(5,𝔽2)13yes—
69C3⋊D3—42no—
70C24—91no—
71C2×D4—96no—
72C2×D4C2×C3⋊D313yes—
73C2×D4C2×C3⋊D313yescomplemented
74C22×C4—93no—
75C4⋊C4—96no—
76C22⋊C4—93no—
77C22⋊C4—93no—
78C22⋊C4—96no—
79C42C2×C3⋊D311yescharacteristic, complemented
80C22×C3—212no—
81C22×C3—224no—
82C22×C3—212no—
83C22×C3C23×D314yes—
84C2×D3—624no—
85C2×D3—1216no—
86C3×C4—212no—
87Dic3—624no—
88C32#56011yescharacteristic, complemented, Sylow
89C23—92no—
90C23—363no—
91C23C22×C3⋊D311yescharacteristic
92C23C22×C3⋊D313yes—
93D4—43no—
94D4—43no—
95D4—366no—
96C2×C4—96no—
97C2×C4—363no—
98C2×C4C22×C3⋊D313yes—
99C2×C3—44no—
100C2×C3—412no—
101C2×C3#1004112yes—
102D3—128no—
103C22—23no—
104C22—26no—
105C22—23no—
106C22—186no—
107C22—364no—
108C22C23×C3⋊D311yescharacteristic
109C4—23no—
110C4—186no—
111C3#572014yescomplemented
112C2—41no—
113C2—43no—
114C2—362no—
115C2#6436113yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#127C2
hypercenter#127C3⋊D3
hypocenterC32#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C32×#127C2semidirect
2. #64148C2semidirect
3. #64146C2semidirect
4. #64144C2semidirect
5. #64085C2semidirect
9. C2×C32×D4C22semidirect
11. C22×C3⋊Dic3C22semidirect
12. C32×C42C22semidirect
14. #3757C22semidirect
16. #3750C22semidirect
17. C3×#127D3semidirect
35. C2×C3×D4C2×D3semidirect
38. C3×C42C2×D3semidirect
51. #127C3⋊D3semidirect
73. C2×D4C2×C3⋊D3semidirect
79. C42C2×C3⋊D3semidirect
88. C32#560semidirect
111. C3#5720semidirect

Maximal subgroups

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

subgroupindex
C32×#1272
#641482
#641462
#641442
#640852
#57203