Finite Group Databasebeta

← #10592450

Subgroups — #10592450

Order 576.

450 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3832C211yesmaximal, characteristic
2M16×C3⋊D3C212yesmaximal, complemented
3C32×#101C211yesmaximal, characteristic, complemented
4#63663C211yesmaximal, characteristic, complemented
5#63650C212yesmaximal, complemented
6#4751—34nomaximal
7C2×C32×Q8C2211yescharacteristic
8#3832—22no—
9#3832—22no—
10C2×C4⋊(C3⋊D3)C412yes—
11C2×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C4×C3⋊D3C412yes—
13C2×C4×C3⋊D3C2211yescharacteristic
14C32×M16C2212yes—
15#3748C2212yes—
16#3743—22no—
17C8×C3⋊D3—22no—
18C2×#239—34no—
19D3×M16—38no—
20C3×#101D314yescomplemented
21#836—34no—
22#823—38no—
23C22×C3⋊D3C2×C412yes—
24C22×C3⋊D3C2×C411yescharacteristic
25C32×Q8—22no—
26C2×C32×C4C2×C412yes—
27C2×C32×C4C2311yescharacteristic
28C2×C3⋊Dic3C2×C411yescharacteristic
29C4⋊(C3⋊D3)—24no—
30C4⋊(C3⋊D3)—22no—
31C4×C3⋊D3—22no—
32C4×C3⋊D3—22no—
33C4×C3⋊D3D412yes—
34C4×C3⋊D3D412yes—
35C32×C8—22no—
36C3⋊(C3⋊C8)—22no—
37#413—91noSylow
38C2×C3×Q8C2×D314yes—
39#239—68no—
40#239—68no—
41C2×D12—38no—
42C2×D12—34no—
43C2×C4×D3—34no—
44C2×C4×D3—38no—
45C3×M16C2×D318yes—
46#208—38no—
47ΓL(1,𝔽25)—68no—
48D3×C8—68no—
49C22×C32C22×C411yescharacteristic
50C2×C3⋊D3—22no—
51C2×C3⋊D3—41no—
52C2×C3⋊D3—42no—
53C2×C3⋊D3C22⋊C412yes—
54C32×C4—22no—
55C32×C4C2×D412yes—
56C3⋊Dic3C22⋊C411yescharacteristic
57C3⋊Dic3C22⋊C411yescharacteristic
58C2×#41—91no—
59C2×M16—92no—
60#101—91no—
61#101C3⋊D311yescharacteristic, complemented
62#100—92no—
63C22×D3—34no—
64C22×D3—38no—
65C3×Q8—28no—
66C2×C3×C4C4×D318yes—
67C2×C3×C4C22×D314yes—
68C2×Dic3—34no—
69D12—616no—
70D12—68no—
71C4×D3—38no—
72C4×D3—38no—
73C4×D3—68no—
74C4×D3—68no—
75C3×C8—28no—
76C3⋊C8—68no—
77C2×C32—21no—
78C2×C32C2×C22⋊C411yescharacteristic
79C3⋊D3—22no—
80C3⋊D3—42no—
81#41—182no—
82#41—182no—
83C2×Q8C2×C3⋊D311yescharacteristic
84C2×D4—92no—
85C2×D4—91no—
86C22×C4—91no—
87C22×C4—92no—
88M16—92no—
89M16—182no—
90M16C2×C3⋊D312yes—
91C2×C8—182no—
92C22×C3C2×C4×D314yes—
93C2×D3—38no—
94C2×D3—68no—
95C2×D3—124no—
96C2×D3—128no—
97C3×C4—28no—
98C3×C4D3×D418yes—
99Dic3—34no—
100Dic3—34no—
101C32#41311yescharacteristic, complemented, Sylow
102C23—91no—
103C23—92no—
104Q8—22no—
105D4—184no—
106D4—182no—
107C2×C4—91no—
108C2×C4—92no—
109C2×C4—92no—
110C2×C4—182no—
111C2×C4—182no—
112C2×C4C4×C3⋊D312yes—
113C2×C4C22×C3⋊D311yescharacteristic
114C8—22no—
115C8—182no—
116C2×C3—24no—
117C2×C3D3×C22⋊C414yes—
118D3—68no—
119D3—128no—
120C22—92no—
121C22—182no—
122C22—361no—
123C22—362no—
124C22C2×C4×C3⋊D311yescharacteristic
125C4—22no—
126C4—91no—
127C4—91no—
128C4D4×C3⋊D312yes—
129C3#475114yescomplemented
130C2—21no—
131C2—182no—
132C2—362no—
133C2C22⋊C4×C3⋊D311yescharacteristic
subgroupG/H
centerC2C22⋊C4×C3⋊D3
commutatorC22×C32C22×C4
FrattiniC2×C4C22×C3⋊D3
FittingC32×#101C2
socleC2×C32C2×C22⋊C4
prime radicalC2×#3832C2
cosocleC2×C32×C4C23
hypercenter#101C3⋊D3
hypocenterC32#413
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. M16×C3⋊D3C2semidirect
3. C32×#101C2semidirect
4. #63663C2semidirect
5. #63650C2semidirect
20. C3×#101D3semidirect
61. #101C3⋊D3semidirect
101. C32#413semidirect
129. C3#4751semidirect

Maximal subgroups

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

subgroupindex
C2×#38322
M16×C3⋊D32
C32×#1012
#636632
#636502
#47513