Finite Group Databasebeta

← #11537

Subgroups — #11537

Order 256.

174 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 51 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#2929C211yesmaximal, characteristic
2#2166C212yesmaximal, complemented
3#1321C212yesmaximal, complemented
4#1320C212yesmaximal, complemented
5C2×#131C411yescharacteristic
6C23×C8C411yescharacteristic
7#525C411yescharacteristic
8#435—22no—
9C8×D4—22no—
10#424C411yescharacteristic
11#408C2211yescharacteristic
12#407C2211yescharacteristic
13#405C2211yescharacteristic
14#364—22no—
15#349C2214yes—
16#347—22no—
17C2×#41C2×C411yescharacteristic
18C23×C4C2×C411yescharacteristic
19#131—42no—
20C2×M16—21no—
21C2×M16C2×C411yescharacteristic
22C2×M16C2311yescharacteristic
23C22×C8—21no—
24C22×C8—21no—
25C22×C8—21no—
26C22×C8—42no—
27C22×C8C2×C411yescharacteristic
28C22×C8C2×C411yescharacteristic
29#123—21no—
30C22⋊Q8—21no—
31#121—21no—
32#120—21no—
33C4×D4—22no—
34#117C2×C411yescharacteristic
35M32—24no—
36C4⋊C8—22no—
37C22⋊C8—22no—
38C22⋊C8—22no—
39#97D412yes—
40C4×C8D412yes—
41C24M1611yescharacteristic
42#41—41no—
43#41—41no—
44C2×Q8M1611yescharacteristic
45C2×D4—22no—
46C2×D4—21no—
47C2×D4M1611yescharacteristic
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—21no—
51C22×C4—41no—
52C22×C4—41no—
53C22×C4M1611yescharacteristic
54C22×C4C22×C411yescharacteristic
55M16—41no—
56M16—42no—
57M16—41no—
58C2×C8—22no—
59C2×C8—21no—
60C2×C8—21no—
61C2×C8—42no—
62C2×C8—42no—
63C2×C8—42no—
64C2×C8—41no—
65C2×C8C22⋊C411yescharacteristic
66C2×C8C22⋊C411yescharacteristic
67C2×C8C2×D412yes—
68C4⋊C4—22no—
69C4⋊C4—21no—
70C22⋊C4—22no—
71C22⋊C4—22no—
72C22⋊C4—21no—
73C42C22⋊C412yes—
74C16—44no—
75C23—21no—
76C23—21no—
77C23—41no—
78C23—42no—
79C23C2×M1611yescharacteristic
80Q8—41no—
81D4—41no—
82D4—44no—
83D4—42no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—22no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—42no—
90C2×C4—41no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4C2×C22⋊C411yescharacteristic
96C2×C4C2×M1611yescharacteristic
97C8—22no—
98C8—21no—
99C8—41no—
100C8—41no—
101C8#10012yes—
102C22—22no—
103C22—21no—
104C22—41no—
105C22—41no—
106C22—42no—
107C22—42no—
108C22—42no—
109C22—41no—
110C22—41no—
111C22#40711yescharacteristic
112C4—21no—
113C4—42no—
114C4—41no—
115C4—41no—
116C4#41011yescharacteristic
117C4C2×#10011yescharacteristic
118C2—21no—
119C2—42no—
120C2—41no—
121C2—81no—
122C2#146611yescharacteristic
subgroupG/H
centerC4#410
commutatorC22×C4C22×C4
Frattini, cosocleC2×M16C23
socleC2#1466
prime radical#525C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2166C2semidirect
3. #1321C2semidirect
4. #1320C2semidirect

Maximal subgroups

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

subgroupindex
#29292
#21662
#13212
#13202