Finite Group Databasebeta

← #10592553

Subgroups — #10592553

Order 576.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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#64097C211yesmaximal, characteristic
2#64096C211yesmaximal, characteristic
3C32×#106C211yesmaximal, characteristic
4#63643C211yesmaximal, characteristic
5#63641C211yesmaximal, characteristic
6#63637C211yesmaximal, characteristic
7#63635C211yesmaximal, characteristic
8#4854—34nomaximal
9C2×#634—21no—
10C2×#634C2211yescharacteristic
11C2×C32×C8C2211yescharacteristic
12C32×C4⋊C4C2211yescharacteristic
13C32×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15#3750—21no—
16#3750—21no—
17C4×C3⋊Dic3—21no—
18C4×C3⋊Dic3C2211yescharacteristic
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20#888—34no—
21#887—34no—
22C3×#106D314yes—
23#816—34no—
24Dic3⋊C8—34no—
25#810—34no—
26#808—34no—
27C2×C32×C4—21no—
28C2×C32×C4—21no—
29C2×C32×C4C2311yescharacteristic
30C2×C3⋊Dic3—21no—
31C2×C3⋊Dic3—21no—
32C2×C3⋊Dic3D411yescharacteristic
33C2×C3⋊Dic3D411yescharacteristic
34#634—21no—
35#634—42no—
36#634Q812yescomplemented
37C32×C8—21no—
38C3⋊(C3⋊C8)—21no—
39#475—91noSylow
40C2×Dic6—34no—
41C2×Dic6—64no—
42C2×C3×C8C2×D314yes—
43C3×C4⋊C4C2×D314yes—
44C3×C4⋊C4C2×D314yes—
45C4⋊Dic3—34no—
46Dic3⋊C4—64no—
47Dic3⋊C4—64no—
48C4×Dic3—34no—
49C4×Dic3—64no—
50C2×C3⋊C8—34no—
51C22×C32C2×D411yescharacteristic
52C32×C4—41no—
53C32×C4—41no—
54C32×C4C2×Q811yescharacteristic
55C32×C4#4111yescharacteristic
56C3⋊Dic3—22no—
57C3⋊Dic3—21no—
58C3⋊Dic3—41no—
59C3⋊Dic3SD1612yes—
60C4⋊Q8—91no—
61C4×Q8—91no—
62#106—91no—
63#106C3⋊D311yescharacteristic
64C4⋊C8—91no—
65#103—91no—
66#103—91no—
67C2×C3×C4—24no—
68C2×C3×C4—24no—
69C2×C3×C4C22×D314yes—
70C2×Dic3—34no—
71C2×Dic3—34no—
72C2×Dic3—64no—
73C2×Dic3—64no—
74Dic6—38no—
75Dic6—64no—
76Dic6—128no—
77C3×C8—24no—
78C3⋊C8—64no—
79C2×C32C22⋊Q811yescharacteristic
80C2×C32C2×SD1611yescharacteristic
81C2×C32#13711yescharacteristic
82C2×Q8—91no—
83C2×Q8—181no—
84C2×C8—91no—
85C2×C8C2×C3⋊D311yescharacteristic
86C4⋊C4—91no—
87C4⋊C4—181no—
88C4⋊C4—181no—
89C4⋊C4C2×C3⋊D311yescharacteristic
90C4⋊C4C2×C3⋊D311yescharacteristic
91C42—91no—
92C42—181no—
93C22×C3D3×D414yes—
94C3×C4—44no—
95C3×C4—44no—
96C3×C4#23514yes—
97C3×C4D3×Q814yes—
98Dic3—38no—
99Dic3—68no—
100Dic3—64no—
101Dic3—124no—
102C32#47511yescharacteristic, complemented, Sylow
103Q8—92no—
104Q8—181no—
105Q8—362no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—91no—
109C2×C4—91no—
110C2×C4—181no—
111C2×C4—181no—
112C2×C4C22×C3⋊D311yescharacteristic
113C8—21no—
114C8—181no—
115C2×C3#89614yes—
116C2×C3D3×SD1614yes—
117C2×C3#91514yes—
118C22D4×C3⋊D311yescharacteristic
119C4—41no—
120C4—41no—
121C4—92no—
122C4—182no—
123C4—181no—
124C4—361no—
125C4#382811yescharacteristic
126C4Q8×C3⋊D311yescharacteristic
127C3#485414yescomplemented
128C2#6410511yescharacteristic
129C2SD16×C3⋊D311yescharacteristic
130C2#6412411yescharacteristic
subgroupG/H
centerC22D4×C3⋊D3
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C22×C3⋊D3
FittingC32×#106C2
socle, prime radicalC22×C32C2×D4
hypercenter#106C3⋊D3
hypocenterC32#475
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
36. #634Q8semidirect
102. C32#475semidirect
127. C3#4854semidirect

Maximal subgroups

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

subgroupindex
#640972
#640962
C32×#1062
#636432
#636412
#636372
#636352
#48543