Finite Group Databasebeta

← #64308

Subgroups — #64308

Order 288.

359 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C2×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#235C212yesmaximal, complemented
3#3811C214yesmaximal, complemented
4D3×D12C214yesmaximal, complemented
5#3799C214yesmaximal, complemented
6#1009—32nomaximal
7C22×C3⋊D3C2212yescomplemented
8C2×D32C2214yescomplemented
9C2×C32×C4C2211yescharacteristic, complemented
10C4⋊(C3⋊D3)C2214yescomplemented
11C3×#67C2214yescomplemented
12C3×D12C2212yescomplemented
13C3×C4×D3C2214yescomplemented
14C3×Dic6C2212yescomplemented
15#626C2218yescomplemented
16#624C2214yescomplemented
17C3×#41—32no—
18#239—34no—
19D3×D4—38no—
20D3×D4—34no—
21#235—34no—
22#235D312yescomplemented
23C2×D12—32no—
24C2×D12—34no—
25C2×D12—61no—
26C22×C32C2311yescharacteristic
27C2×C3⋊D3—42no—
28C2×C3⋊D3C2314yes—
29C2×C3×D3C2314yes—
30D32—44no—
31C32×C4C2312yes—
32C3×Dic3C2314yes—
33H(5,𝔽2)—91noSylow
34C22×D3—34no—
35C22×D3—38no—
36C22×D3—62no—
37C3×Q8—32no—
38C3×D4—34no—
39C3×D4—32no—
40C2×C3×C4—21no—
41C2×C3×C4—34no—
42C2×C3×C4C2×D312yescomplemented
43#67—38no—
44#67C2×D314yescomplemented
45D12—38no—
46D12—34no—
47D12—34no—
48D12—64no—
49D12C2×D312yescomplemented
50C4×D3—38no—
51C4×D3C2×D314yescomplemented
52Dic6C2×D312yescomplemented
53C2×C32—21no—
54C2×C32C2411yescharacteristic
55C3⋊D3—24no—
56C3×D3—24no—
57#41—32no—
58#41—94no—
59C2×D4—94no—
60C2×D4—94no—
61C2×D4—91no—
62C22×C3—21no—
63C22×C3—34no—
64C22×C3C22×D312yes—
65C2×D3—38no—
66C2×D3—64no—
67C2×D3—128no—
68C2×D3—124no—
69C2×D3—124no—
70C2×D3C22×D314yes—
71C3×C4—22no—
72C3×C4—34no—
73C3×C4C22×D312yes—
74C3×C4C22×D312yes—
75Dic3C22×D314yes—
76C32H(5,𝔽2)11yescharacteristic, complemented, Sylow
77C23—94no—
78C23—92no—
79Q8—32no—
80D4—34no—
81D4—32no—
82D4—98no—
83D4—94no—
84C2×C4—34no—
85C2×C4—94no—
86C2×C4D3211yescharacteristic, complemented
87C2×C3—22no—
88C2×C3—22no—
89C2×C3—21no—
90C2×C3—64no—
91C2×C3C23×D312yes—
92D3—24no—
93D3—68no—
94D3—124no—
95C22—34no—
96C22—94no—
97C22—362no—
98C22—364no—
99C22C2×D3211yescharacteristic
100C4—34no—
101C4C2×D3212yes—
102C3—21no—
103C3#100912yescomplemented
104C2—21no—
105C2—64no—
106C2—184no—
107C2C22×D3211yescharacteristic
subgroupG/H
center, FrattiniC2C22×D32
commutator, socle, cosocleC2×C32C24
FittingC2×C32×C4C22
hypercenterC2×C4D32
hypocenterC32H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C3⋊D3)C2semidirect
2. C3×#235C2semidirect
3. #3811C2semidirect
4. D3×D12C2semidirect
5. #3799C2semidirect
7. C22×C3⋊D3C22semidirect
8. C2×D32C22semidirect
9. C2×C32×C4C22semidirect
10. C4⋊(C3⋊D3)C22semidirect
11. C3×#67C22semidirect
12. C3×D12C22semidirect
13. C3×C4×D3C22semidirect
14. C3×Dic6C22semidirect
15. #626C22semidirect
16. #624C22semidirect
22. #235D3semidirect
42. C2×C3×C4C2×D3semidirect
44. #67C2×D3semidirect
49. D12C2×D3semidirect
51. C4×D3C2×D3semidirect
52. Dic6C2×D3semidirect
76. C32H(5,𝔽2)semidirect
86. C2×C4D32semidirect
103. C3#1009semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊D3)2
C3×#2352
#38112
D3×D122
#37992
#10093