Finite Group Databasebeta

← #11521

Subgroups — #11521

Order 256.

165 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 49 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#2920C211yesmaximal, characteristic
2#2165C212yesmaximal, complemented
3#1327C212yesmaximal, complemented
4#1320C212yesmaximal
5C22×C42C411yescharacteristic
6#432—22no—
7#431—22no—
8#423C411yescharacteristic
9#423C2211yescharacteristic
10#407C411yescharacteristic
11#407C2211yescharacteristic
12C4×M16C411yescharacteristic
13C4×M16C2211yescharacteristic
14#364—22no—
15#349C2214yes—
16#347—22no—
17C23×C4—21no—
18C23×C4C2×C411yescharacteristic
19C2×M16C2×C411yescharacteristic
20C2×M16C2×C411yescharacteristic
21C2×M16C2×C411yescharacteristic
22C2×M16C2311yescharacteristic
23C2×C42—21no—
24C2×C42—21no—
25C2×C42—21no—
26C2×C42—21no—
27C2×C42—21no—
28C2×C42C2×C411yescharacteristic
29C2×C42C2×C411yescharacteristic
30M32—24no—
31C4⋊C8—22no—
32C4⋊C8—22no—
33C22⋊C8—22no—
34C22⋊C8—22no—
35#97—21no—
36#97D412yes—
37C4×C8—21no—
38C4×C8D412yes—
39C24M1611yescharacteristic
40C22×C4—21no—
41C22×C4—21no—
42C22×C4—21no—
43C22×C4—21no—
44C22×C4—21no—
45C22×C4—21no—
46C22×C4—21no—
47C22×C4—21no—
48C22×C4—44no—
49C22×C4—41no—
50C22×C4—41no—
51C22×C4M1611yescharacteristic
52C22×C4C22×C411yescharacteristic
53M16—41no—
54M16—41no—
55M16—41no—
56M16—41no—
57C2×C8—21no—
58C2×C8—21no—
59C2×C8C22⋊C412yes—
60C2×C8C2×D412yes—
61C42—21no—
62C42—21no—
63C42—21no—
64C42—21no—
65C42—21no—
66C42—21no—
67C42C22⋊C411yescharacteristic
68C42C22⋊C411yescharacteristic
69C42M1611yescharacteristic
70C42M1611yescharacteristic
71C16—44no—
72C23—21no—
73C23—42no—
74C23—41no—
75C23C2×M1611yescharacteristic
76C2×C4—24no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—42no—
85C2×C4—42no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—41no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4C2×C22⋊C411yescharacteristic
94C2×C4C2×M1611yescharacteristic
95C8—22no—
96C8—22no—
97C8—41no—
98C8—41no—
99C22—22no—
100C22—21no—
101C22—41no—
102C22—41no—
103C22—42no—
104C22—42no—
105C22—41no—
106C22#40711yescharacteristic
107C4—22no—
108C4—22no—
109C4—21no—
110C4—41no—
111C4—41no—
112C4—41no—
113C4#41011yescharacteristic
114C4#41311yescharacteristic
115C2—21no—
116C2—42no—
117C2—41no—
118C2#146411yescharacteristic
subgroupG/H
centerC4#410
commutatorC22×C4C22×C4
Frattini, cosocleC2×M16C23
socleC2#1464
prime radicalC24M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2165C2semidirect
3. #1327C2semidirect

Maximal subgroups

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

subgroupindex
#29202
#21652
#13272
#13202