Finite Group Databasebeta

← #10592416

Subgroups — #10592416

Order 576.

312 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#64113C211yesmaximal, characteristic
2#64110C212yesmaximal, complemented
3C32×#97C211yesmaximal, characteristic, complemented
4#63646C211yesmaximal, characteristic, complemented
5#63632C212yesmaximal, complemented
6#4717—34nomaximal
7#3828C2212yes—
8C2×C32×C8C2211yescharacteristic
9C2×C32×C8C2211yescharacteristic
10C32×C42C2211yescharacteristic
11#3748C2212yes—
12#3746C411yescharacteristic, complemented
13#3745C411yescharacteristic, complemented
14#3744C412yescomplemented
15#3743—22no—
16C8×C3⋊D3—22no—
17#904—34no—
18#901—38no—
19C3×#97D314yescomplemented
20#819—34no—
21#805—38no—
22C2×C32×C4—21no—
23C2×C32×C4C2311yescharacteristic
24#638—22no—
25C4⋊(C3⋊D3)C2×C412yes—
26C4×C3⋊D3—22no—
27#634C2×C412yes—
28C32×C8C2×C411yescharacteristic
29C32×C8C2×C411yescharacteristic
30C32×C8D412yes—
31C3⋊(C3⋊C8)—22no—
32#444—91noSylow
33#235—38no—
34C2×C3×C8C2×D314yes—
35C2×C3×C8C2×D314yes—
36C3×C42C2×D314yes—
37#208—38no—
38Dic12—34no—
39D24—34no—
40#204—38no—
41ΓL(1,𝔽25)—68no—
42D3×C8—68no—
43C22×C32#4111yescharacteristic
44C2×C3⋊D3—22no—
45C32×C4—41no—
46C32×C4C22×C411yescharacteristic
47C32×C4C2×D411yescharacteristic
48C3⋊Dic3—22no—
49#135—91no—
50#131—92no—
51#108—91no—
52#104—92no—
53#97C3⋊D311yescharacteristic, complemented
54C2×C3×C4—24no—
55C2×C3×C4C22×D314yes—
56#67—68no—
57D12—38no—
58C4×D3—68no—
59Dic6—38no—
60C3×C8C4×D314yes—
61C3×C8C4×D314yes—
62C3×C8D1218yes—
63C3⋊C8—68no—
64C2×C32—21no—
65C2×C32C4×D411yescharacteristic
66C3⋊D3—42no—
67#41—92no—
68Q16—91no—
69SD16—92no—
70D8—91no—
71M16—92no—
72M16—182no—
73C2×C8—182no—
74C2×C8C2×C3⋊D311yescharacteristic
75C2×C8C2×C3⋊D311yescharacteristic
76C42C2×C3⋊D311yescharacteristic
77C22×C3#23514yes—
78C2×D3—68no—
79C3×C4—44no—
80C3×C4C2×C4×D314yes—
81C3×C4C2×D1214yes—
82Dic3—68no—
83C32#44411yescharacteristic, complemented, Sylow
84Q8—92no—
85D4—92no—
86D4—182no—
87C2×C4—21no—
88C2×C4—182no—
89C2×C4C22×C3⋊D311yescharacteristic
90C8—182no—
91C8C4×C3⋊D311yescharacteristic
92C8C4×C3⋊D311yescharacteristic
93C8C4⋊(C3⋊D3)12yes—
94C2×C3—24no—
95C2×C3C4×D1214yes—
96D3—128no—
97C22—182no—
98C22#382811yescharacteristic
99C4—41no—
100C4—182no—
101C4C2×C4×C3⋊D311yescharacteristic
102C4C2×C4⋊(C3⋊D3)11yescharacteristic
103C3#471714yescomplemented
104C2—21no—
105C2—362no—
106C2C4×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC4C2×C4⋊(C3⋊D3)
commutatorC32×C4C22×C4
FrattiniC2×C4C22×C3⋊D3
FittingC32×#97C2
socleC2×C32C4×D4
prime radical#64113C2
cosocleC2×C32×C4C23
hypercenter#97C3⋊D3
hypocenterC32#444
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C8 ◁ #97

Complements

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

NH ≅ G/Ntype
2. #64110C2semidirect
3. C32×#97C2semidirect
4. #63646C2semidirect
5. #63632C2semidirect
12. #3746C4semidirect
13. #3745C4semidirect
14. #3744C4semidirect
19. C3×#97D3semidirect
53. #97C3⋊D3semidirect
83. C32#444semidirect
103. C3#4717semidirect

Maximal subgroups

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

subgroupindex
#641132
#641102
C32×#972
#636462
#636322
#47173