Finite Group Databasebeta

← #135504

Subgroups — #135504

Order 512.

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

normal subgroups only

This group has 264 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#21432C211yesmaximal, characteristic
2#8745C211yesmaximal, characteristic
3#8741C211yesmaximal, characteristic
4#7216C212yesmaximal
5#7214C212yesmaximal
6#2298C2211yescharacteristic
7#1923—22no—
8#1916—22no—
9#1822C2211yescharacteristic
10#1820C2211yescharacteristic
11#1768C412yescomplemented
12#1750C2212yes—
13#1745C411yescharacteristic, complemented
14C42×C8C411yescharacteristic, complemented
15#1464C2212yes—
16#1285—22no—
17#1284—22no—
18C2×#117C2311yescharacteristic
19#432—22no—
20C2×C4⋊C8—22no—
21#407—21no—
22C2×C22⋊C8—22no—
23C2×#97C2×C412yes—
24C2×C4×C8—22no—
25C2×C4×C8C2×C412yes—
26#388—22no—
27#387—22no—
28#381—21no—
29C4×C4⋊C4D412yes—
30C4×C22⋊C4D412yes—
31#376C2×C411yescharacteristic
32C43C2×C411yescharacteristic
33#343—22no—
34#342—22no—
35#336—22no—
36#336—22no—
37#336—22no—
38#324—22no—
39#323—22no—
40C23×C4C2×D411yescharacteristic
41C2×M16—41no—
42C22×C8—22no—
43C22×C8—22no—
44#117—42no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4#4112yes—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—41no—
49C2×C22⋊C4#4112yes—
50C2×C42—22no—
51C2×C42C22⋊C412yes—
52C2×C42C22⋊C412yes—
53C2×C42C22×C411yescharacteristic
54C2×C42C2×D411yescharacteristic
55C4⋊C8—42no—
56C4⋊C8—44no—
57C22⋊C8—44no—
58C22⋊C8—42no—
59C22⋊C8—42no—
60#97—42no—
61#97—42no—
62C4×C8—42no—
63C4×C8—44no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95—42no—
68C24#12311yescharacteristic
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—42no—
73C22×C4—81no—
74C22×C4C2×C22⋊C411yescharacteristic
75C22×C4#11712yes—
76C22×C4#12311yescharacteristic
77C22×C4#12312yes—
78M16—82no—
79C2×C8—24no—
80C2×C8—42no—
81C2×C8—42no—
82C2×C8—42no—
83C2×C8—42no—
84C2×C8—42no—
85C2×C8—82no—
86C2×C8—81no—
87C4⋊C4—42no—
88C4⋊C4—44no—
89C22⋊C4—42no—
90C22⋊C4—44no—
91C22⋊C4—82no—
92C42—22no—
93C42—22no—
94C42—42no—
95C42—42no—
96C42—44no—
97C23—21no—
98C23—22no—
99C23—42no—
100C23#38111yescharacteristic
101C2×C4—22no—
102C2×C4—24no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—21no—
106C2×C4—22no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—82no—
115C2×C4—161no—
116C2×C4—162no—
117C2×C4#41012yes—
118C8—44no—
119C8—82no—
120C8—162no—
121C8—161no—
122C22—22no—
123C22—41no—
124C22—42no—
125C22—82no—
126C22#182011yescharacteristic
127C22#182712yes—
128C4—22no—
129C4—44no—
130C4—41no—
131C4—42no—
132C4—42no—
133C4—82no—
134C4—321no—
135C2—22no—
136C2—81no—
137C2#874111yescharacteristic
138C2#1286412yes—
subgroupG/H
center, socleC22#1820
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2298 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #1768C4semidirect
13. #1745C4semidirect
14. C42×C8C4semidirect

Maximal subgroups

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

subgroupindex
#214322
#87452
#87412
#72162
#72142