Finite Group Databasebeta

← #10592729

Subgroups — #10592729

Order 576.

324 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C32×C4×Q8C211yesmaximal, characteristic
2C2×#3756C211yesmaximal, characteristic
3C4×#634C211yesmaximal, characteristic
4#63662C211yesmaximal, characteristic
5#63637C211yesmaximal, characteristic
6#63635C211yesmaximal, characteristic
7#63630C211yesmaximal, characteristic
8#5030—34nomaximal
9C2×C32×Q8C2211yescharacteristic
10C2×#634C2211yescharacteristic
11C32×C4⋊C4—21no—
12C32×C4⋊C4C2211yescharacteristic
13C32×C42—21no—
14C32×C42C2211yescharacteristic
15#3756C414yescomplemented
16C4⋊(C3⋊Dic3)C2211yescharacteristic
17#3750—21no—
18C4×C3⋊Dic3—21no—
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×Q8D314yes—
22C2×#216—34no—
23C4×Dic6—34no—
24#835—34no—
25#810—34no—
26#808—34no—
27#803—34no—
28C32×Q8—21no—
29C32×Q8C2×C412yes—
30C2×C32×C4—21no—
31C2×C32×C4—21no—
32C2×C32×C4D411yescharacteristic
33C2×C32×C4D411yescharacteristic
34C2×C32×C4C2311yescharacteristic
35C2×C3⋊Dic3—21no—
36C2×C3⋊Dic3—21no—
37#634—21no—
38#634C2×C412yes—
39C3⋊(C3⋊C8)—21no—
40C3⋊(C3⋊C8)C2×C412yes—
41#441—91noSylow
42C2×C3×Q8C2×D314yes—
43C2×Dic6—34no—
44C3×C4⋊C4—24no—
45C3×C4⋊C4C2×D314yes—
46C3×C42—24no—
47C3×C42C2×D314yes—
48#216—316no—
49C4⋊Dic3—34no—
50Dic3⋊C4—64no—
51C4×Dic3—64no—
52C2×C3⋊C8—34no—
53C2×C3⋊C8—34no—
54C22×C32C2×D411yescharacteristic
55C32×C4—22no—
56C32×C4—21no—
57C32×C4—21no—
58C32×C4—41no—
59C32×C4C22×C411yescharacteristic
60C32×C4#4111yescharacteristic
61C3⋊Dic3—22no—
62C3⋊Dic3—41no—
63C2×Q16—91no—
64C4×Q8—91no—
65C4×Q8C3⋊D311yescharacteristic
66#106—91no—
67#103—91no—
68#103—91no—
69#97—91no—
70C3×Q8—24no—
71C3×Q8C4×D318yes—
72C2×C3×C4—24no—
73C2×C3×C4—24no—
74C2×C3×C4#6714yes—
75C2×C3×C4#6714yes—
76C2×C3×C4C22×D314yes—
77C2×Dic3—64no—
78C2×Dic3—64no—
79Dic6—38no—
80Dic6—64no—
81C3⋊C8—38no—
82C3⋊C8—64no—
83C2×C32C4×D411yescharacteristic
84C2×C32#13711yescharacteristic
85C2×C32#13711yescharacteristic
86C2×Q8—91no—
87C2×Q8C2×C3⋊D311yescharacteristic
88Q16—94no—
89C2×C8—91no—
90C2×C8—91no—
91C4⋊C4—21no—
92C4⋊C4—91no—
93C4⋊C4—181no—
94C4⋊C4C2×C3⋊D311yescharacteristic
95C42—21no—
96C42—181no—
97C42C2×C3⋊D311yescharacteristic
98C22×C3C2×#6714yes—
99C3×C4—28no—
100C3×C4—24no—
101C3×C4—24no—
102C3×C4—44no—
103C3×C4C2×C4×D314yes—
104C3×C4#23514yes—
105Dic3—68no—
106Dic3—124no—
107C32#44111yescharacteristic, complemented, Sylow
108Q8—21no—
109Q8—92no—
110Q8—181no—
111Q8C4×C3⋊D312yes—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—181no—
115C2×C4—181no—
116C2×C4#63811yescharacteristic
117C2×C4#63811yescharacteristic
118C2×C4C22×C3⋊D311yescharacteristic
119C8—92no—
120C8—181no—
121C2×C3C4×#6714yes—
122C2×C3#94214yes—
123C2×C3#95114yes—
124C22C2×#63811yescharacteristic
125C4—22no—
126C4—21no—
127C4—21no—
128C4—41no—
129C4—182no—
130C4—361no—
131C4C2×C4×C3⋊D311yescharacteristic
132C4#382811yescharacteristic
133C3#503014yescomplemented
134C2C4×#63811yescharacteristic
135C2#6415111yescharacteristic
136C2#6416011yescharacteristic
subgroupG/H
centerC22C2×#638
commutatorC32×C4C22×C4
FrattiniC2×C4C22×C3⋊D3
FittingC32×C4×Q8C2
socle, prime radicalC22×C32C2×D4
cosocleC2×C32×C4C23
hypercenterC4×Q8C3⋊D3
hypocenterC32#441
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
15. #3756C4semidirect
107. C32#441semidirect
133. C3#5030semidirect

Maximal subgroups

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

subgroupindex
C32×C4×Q82
C2×#37562
C4×#6342
#636622
#636372
#636352
#636302
#50303