Finite Group Databasebeta

← #10596065

Subgroups — #10596065

Order 576.

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

normal subgroups only

This group has 780 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×#120C211yesmaximal, characteristic, complemented
2#64162C212yesmaximal, complemented
3#64146C213yesmaximal, complemented
4#64145C213yesmaximal, complemented
5#64094C216yesmaximal, complemented
6#5594—34nomaximal
7C24×C32C2211yescharacteristic, complemented
8C23×C3⋊D3C2211yescharacteristic, complemented
9C2×C32×D4C2213yes—
10C2×#638C2216yes—
11C2×#634C2213yescomplemented
12C32×C22⋊C4C2213yescomplemented
13#3757C2216yescomplemented
14#3757C2213yescomplemented
15#3752C2216yescomplemented
16C4×C3⋊Dic3C2213yescomplemented
17C3×#120D314yescomplemented
18#953—38no—
19#937—312no—
20#936—312no—
21#885—324no—
22C23×C32—43no—
23C23×C32C2311yescharacteristic
24C23×C32C2313yes—
25C22×C3⋊D3—43no—
26C22×C3⋊D3C2312yes—
27C32×D4—43no—
28C2×C32×C4C2313yes—
29#638—46no—
30C2×C3⋊Dic3C2316yes—
31#634—43no—
32H(3,𝔽4)—91noSylow
33C24×C3C2×D314yescomplemented
34C23×D3—34no—
35C2×C3×D4C2×D3112yes—
36C2×#67—324no—
37C2×Dic6—312no—
38C3×C22⋊C4C2×D3112yescomplemented
39#217—324no—
40#217—312no—
41(C2×D3)⋊C4—324no—
42C4×Dic3—312no—
43C22×C32—26no—
44C22×C32—23no—
45C22×C32—23no—
46C22×C32—44no—
47C22×C32C2411yescharacteristic
48C2×C3⋊D3—26no—
49C2×C3⋊D3—44no—
50C32×C4—23no—
51C3⋊Dic3—26no—
52#124—93no—
53#124—96no—
54#120—92no—
55#120—93no—
56#120C3⋊D311yescharacteristic, complemented
57C23×C3—412no—
58C23×C3C22×D314yes—
59C23×C3C22×D3112yes—
60C22×D3—38no—
61C22×D3—1212no—
62C3×D4—412no—
63C2×C3×C4C22×D3112yes—
64#67—1224no—
65C2×Dic3—324no—
66Dic6—1212no—
67C2×C32—41no—
68C2×C32—43no—
69C2×C32H(5,𝔽2)13yes—
70C3⋊D3—42no—
71C24—91no—
72C24C2×C3⋊D311yescharacteristic, complemented
73C2×Q8—93no—
74C2×D4—96no—
75C2×D4C2×C3⋊D313yes—
76C22⋊C4—93no—
77C22⋊C4—96no—
78C22⋊C4—96no—
79C22⋊C4C2×C3⋊D313yescomplemented
80C42—93no—
81C22×C3—224no—
82C22×C3—212no—
83C22×C3—212no—
84C22×C3—416no—
85C22×C3C23×D314yes—
86C2×D3—624no—
87C2×D3—1216no—
88C3×C4—212no—
89Dic3—624no—
90C32H(3,𝔽4)11yescharacteristic, complemented, Sylow
91C23—43no—
92C23—92no—
93C23—363no—
94C23C22×C3⋊D311yescharacteristic
95C23C22×C3⋊D313yes—
96Q8—363no—
97D4—43no—
98D4—366no—
99C2×C4—96no—
100C2×C4C22×C3⋊D313yes—
101C2×C3—44no—
102C2×C3—412no—
103C2×C3#1004112yes—
104D3—128no—
105C22—23no—
106C22—23no—
107C22—26no—
108C22—44no—
109C22—186no—
110C22—364no—
111C22C23×C3⋊D311yescharacteristic
112C4—23no—
113C4—186no—
114C3#559414yescomplemented
115C2—43no—
116C2—41no—
117C2—362no—
118C2#6436113yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#120C2
hypercenter#120C3⋊D3
hypocenterC32H(3,𝔽4)
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C32×#120C2semidirect
2. #64162C2semidirect
3. #64146C2semidirect
4. #64145C2semidirect
5. #64094C2semidirect
7. C24×C32C22semidirect
8. C23×C3⋊D3C22semidirect
11. C2×#634C22semidirect
12. C32×C22⋊C4C22semidirect
13. #3757C22semidirect
14. #3757C22semidirect
15. #3752C22semidirect
16. C4×C3⋊Dic3C22semidirect
17. C3×#120D3semidirect
33. C24×C3C2×D3semidirect
38. C3×C22⋊C4C2×D3semidirect
56. #120C3⋊D3semidirect
72. C24C2×C3⋊D3semidirect
79. C22⋊C4C2×C3⋊D3semidirect
90. C32H(3,𝔽4)semidirect
114. C3#5594semidirect

Maximal subgroups

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

subgroupindex
C32×#1202
#641622
#641462
#641452
#640942
#55943