Finite Group Databasebeta

← #8757

Subgroups — #8757

Order 256.

219 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 59 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#2316C211yesmaximal, characteristic
2#1463C212yesmaximal
3#1305C214yesmaximal
4C2×#117C411yescharacteristic
5C2×#117C411yescharacteristic
6C22×C4⋊C4C2211yescharacteristic
7#407C2214yes—
8#389—22no—
9#389—22no—
10#380C412yes—
11#380C2212yes—
12#324—22no—
13#324—22no—
14C23×C4C2×C411yescharacteristic
15C23×C4C2×C411yescharacteristic
16C23×C4C2311yescharacteristic
17C2×M16—24no—
18#117—42no—
19#117—42no—
20C2×C4⋊C4—24no—
21C2×C4⋊C4—21no—
22C2×C4⋊C4—21no—
23C2×C4⋊C4—22no—
24C2×C4⋊C4—22no—
25C2×C4⋊C4D412yes—
26C2×C4⋊C4D412yes—
27C2×C22⋊C4C2×C412yes—
28C2×C22⋊C4C2×C412yes—
29C2×C42—21no—
30C2×C42—21no—
31C22⋊C8—24no—
32C22⋊C8—24no—
33#95—22no—
34#95—22no—
35C24C22×C411yescharacteristic
36C22×C4—22no—
37C22×C4—22no—
38C22×C4—22no—
39C22×C4—22no—
40C22×C4—22no—
41C22×C4—22no—
42C22×C4—81no—
43C22×C4—81no—
44C22×C4C22⋊C411yescharacteristic
45C22×C4C22⋊C411yescharacteristic
46C22×C4C22⋊C412yes—
47C22×C4C2×D412yes—
48C22×C4#4114yes—
49M16—84no—
50C2×C8—44no—
51C4⋊C4—24no—
52C4⋊C4—24no—
53C4⋊C4—42no—
54C4⋊C4—42no—
55C4⋊C4—82no—
56C4⋊C4—82no—
57C22⋊C4—42no—
58C22⋊C4—42no—
59C42—42no—
60C42—42no—
61C23—21no—
62C23—21no—
63C23—41no—
64C23—41no—
65C23C2×C22⋊C411yescharacteristic
66C23#11712yes—
67C2×C4—28no—
68C2×C4—21no—
69C2×C4—21no—
70C2×C4—21no—
71C2×C4—21no—
72C2×C4—22no—
73C2×C4—42no—
74C2×C4—44no—
75C2×C4—42no—
76C2×C4—42no—
77C2×C4—42no—
78C2×C4—42no—
79C2×C4—42no—
80C2×C4—42no—
81C2×C4—42no—
82C2×C4—82no—
83C2×C4#10012yes—
84C2×C4#10112yes—
85C2×C4#12314yes—
86C8—84no—
87C22—21no—
88C22—21no—
89C22—22no—
90C22—22no—
91C22—22no—
92C22—42no—
93C22—42no—
94C22#38111yescharacteristic
95C22C2×#10011yescharacteristic
96C22C2×#10111yescharacteristic
97C4—22no—
98C4—22no—
99C4—24no—
100C4—42no—
101C4—82no—
102C4—82no—
103C2—21no—
104C2—21no—
105C2—42no—
106C2#182111yescharacteristic
107C2#182811yescharacteristic
108C2#183011yescharacteristic
subgroupG/H
center, socleC22#381
commutator, prime radicalC24C22×C4
Frattini, cosocleC23×C4C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#23162
#14632
#13052