Finite Group Databasebeta

← #10591403

Subgroups — #10591403

Order 576.

230 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 58 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#64145C211yesmaximal, characteristic
2#63872C211yesmaximal, characteristic, complemented
3C3×#832C212yesmaximal
4#63567C212yesmaximal, complemented
5#63553C211yesmaximal, characteristic, complemented
6#4768—32nomaximal
7C2×C32×D4C2211yescharacteristic
8C2×#634C2211yescharacteristic
9#3757—22no—
10C4×C3⋊Dic3C2211yescharacteristic
11C3×C4⋊Dic3C2212yes—
12C2×C3×C3⋊C8C2212yes—
13Dic3⋊Dic3—22no—
14#936—32no—
15#936—61no—
16#890—32no—
17C3×#102—32no—
18#832D312yes—
19#816—32no—
20C8×Dic3—32no—
21#810—32no—
22C23×C32—21no—
23C32×D4—22no—
24C2×C32×C4C2311yescharacteristic
25C2×C3⋊Dic3—21no—
26C2×C3⋊Dic3D412yes—
27#634—22no—
28C2×C3×Dic3—22no—
29C3×C3⋊C8—22no—
30#488—91noSylow
31C2×C3×D4—21no—
32C2×C3×D4C2×D312yes—
33C2×Dic6—32no—
34C2×Dic6—61no—
35C2×C3×C8—32no—
36C3×C4⋊C4—32no—
37#217—62no—
38#217—62no—
39#217—64no—
40C4⋊Dic3—62no—
41C4⋊Dic3C2×D312yes—
42Dic3⋊C4—62no—
43C4×Dic3—32no—
44C4×Dic3—61no—
45C2×C3⋊C8C2×D312yes—
46C22×C32—41no—
47C22×C32—42no—
48C22×C32C2×D411yescharacteristic
49C32×C4#4112yes—
50C3⋊Dic3—22no—
51C3⋊Dic3—41no—
52C3×Dic3—42no—
53#125—91no—
54#124—91no—
55#103—92no—
56#102—32no—
57C4×C8—91no—
58C23×C3—22no—
59C23×C3—22no—
60C3×D4—22no—
61C3×D4—22no—
62C3×D4—42no—
63C2×C3×C4—21no—
64C2×C3×C4—62no—
65C2×C3×C4C22×D312yes—
66C2×Dic3—22no—
67C2×Dic3—32no—
68C2×Dic3—32no—
69C2×Dic3—62no—
70C2×Dic3—62no—
71C2×Dic3—62no—
72Dic6—62no—
73Dic6—62no—
74Dic6—122no—
75C3×C8—62no—
76C3⋊C8—22no—
77C2×C32—81no—
78C2×C32#12411yescharacteristic
79C2×C32#13512yes—
80C2×Q8—91no—
81C2×D4D3211yescharacteristic
82C2×C8—32no—
83C4⋊C4—32no—
84C4⋊C4—182no—
85C22⋊C4—182no—
86C42—91no—
87C22×C3—21no—
88C22×C3—42no—
89C22×C3—44no—
90C22×C3—42no—
91C22×C3—42no—
92C22×C3—42no—
93C22×C3D3×D412yes—
94C3×C4—22no—
95C3×C4—122no—
96C3×C4#23512yes—
97C3×C4#23712yes—
98Dic3—42no—
99Dic3—62no—
100Dic3—62no—
101Dic3—122no—
102Dic3—122no—
103Dic3—122no—
104C32#48811yescharacteristic, complemented, Sylow
105C23—21no—
106Q8—182no—
107D4—22no—
108C2×C4—62no—
109C2×C4—92no—
110C2×C4—181no—
111C2×C4C2×D3211yescharacteristic
112C8—62no—
113C2×C3—22no—
114C2×C3—21no—
115C2×C3—82no—
116C2×C3—82no—
117C2×C3#88512yes—
118C2×C3#91212yes—
119C2×C3#91612yes—
120C22—42no—
121C22—41no—
122C22#381111yescharacteristic
123C4—122no—
124C4—182no—
125C4—361no—
126C4#380412yes—
127C3—21no—
128C3#476812yescomplemented
129C2—81no—
130C2#6393612yes—
131C2#6395311yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
Fitting, prime radicalC2×C32×D4C22
socleC22×C32C2×D4
hypercenterC2×D4D32
hypocenterC32#488
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #63872C2semidirect
4. #63567C2semidirect
5. #63553C2semidirect
104. C32#488semidirect
128. C3#4768semidirect

Maximal subgroups

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

subgroupindex
#641452
#638722
C3×#8322
#635672
#635532
#47683