Finite Group Databasebeta

← #11522

Subgroups — #11522

Order 256.

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

normal subgroups only

This group has 142 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#2921C211yesmaximal, characteristic
2#2165C212yesmaximal, complemented
3#1327C212yesmaximal, complemented
4#1320C212yesmaximal
5C2×#117C411yescharacteristic
6#433—22no—
7#433—22no—
8#423C411yescharacteristic
9#423C2211yescharacteristic
10#407C411yescharacteristic
11#407C2211yescharacteristic
12C4×M16C411yescharacteristic
13C4×M16C2211yescharacteristic
14#364—22no—
15#349C2214yes—
16#347—22no—
17C23×C4C2×C411yescharacteristic
18C2×M16C2×C411yescharacteristic
19C2×M16C2×C411yescharacteristic
20C2×M16C2×C411yescharacteristic
21C2×M16C2311yescharacteristic
22#117—21no—
23#117—21no—
24#117—22no—
25C2×C4⋊C4—21no—
26C2×C22⋊C4—21no—
27C2×C42C2×C411yescharacteristic
28C2×C42C2×C411yescharacteristic
29M32—24no—
30C4⋊C8—22no—
31C4⋊C8—22no—
32C22⋊C8—22no—
33C22⋊C8—22no—
34#97—21no—
35#97D412yes—
36C4×C8—21no—
37C4×C8D412yes—
38C24M1611yescharacteristic
39C22×C4—21no—
40C22×C4—21no—
41C22×C4—21no—
42C22×C4—21no—
43C22×C4—41no—
44C22×C4—41no—
45C22×C4M1611yescharacteristic
46C22×C4C22×C411yescharacteristic
47M16—41no—
48M16—41no—
49M16—41no—
50M16—41no—
51C2×C8—21no—
52C2×C8—21no—
53C2×C8C22⋊C412yes—
54C2×C8C2×D412yes—
55C4⋊C4—22no—
56C4⋊C4—22no—
57C22⋊C4—22no—
58C22⋊C4—22no—
59C42—21no—
60C42—21no—
61C42C22⋊C411yescharacteristic
62C42C22⋊C411yescharacteristic
63C42M1612yes—
64C16—44no—
65C23—21no—
66C23—41no—
67C23—81no—
68C23C2×M1611yescharacteristic
69C2×C4—21no—
70C2×C4—21no—
71C2×C4—21no—
72C2×C4—21no—
73C2×C4—22no—
74C2×C4—21no—
75C2×C4—42no—
76C2×C4—41no—
77C2×C4—41no—
78C2×C4—41no—
79C2×C4—81no—
80C2×C4—81no—
81C2×C4C2×C22⋊C411yescharacteristic
82C2×C4C2×M1611yescharacteristic
83C8—22no—
84C8—22no—
85C8—41no—
86C8—41no—
87C22—21no—
88C22—41no—
89C22—42no—
90C22—41no—
91C22—82no—
92C22#40711yescharacteristic
93C4—21no—
94C4—41no—
95C4—41no—
96C4—42no—
97C4—41no—
98C4#41011yescharacteristic
99C4#41311yescharacteristic
100C2—21no—
101C2—41no—
102C2—81no—
103C2#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
#29212
#21652
#13272
#13202