Finite Group Databasebeta

← #11448

Subgroups — #11448

Order 256.

227 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 71 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#2317C211yesmaximal, characteristic
2C2×#324C212yesmaximal
3#1325C212yesmaximal, complemented
4#1324C212yesmaximal, complemented
5C2×#117C411yescharacteristic
6C2×#117C411yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8C2×C22⋊C8—22no—
9#389—22no—
10#385—22no—
11#384C411yescharacteristic
12#384C2211yescharacteristic
13#382C411yescharacteristic
14#382C2211yescharacteristic
15#324—22no—
16#324C2214yes—
17C23×C4C2×C411yescharacteristic
18C23×C4D412yes—
19C22×C8—42no—
20#117—22no—
21#117—42no—
22#117C814yescomplemented
23C2×C4⋊C4—21no—
24C2×C4⋊C4—22no—
25C2×C4⋊C4—22no—
26C2×C4⋊C4—42no—
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4D412yes—
31C2×C4⋊C4C2311yescharacteristic
32C2×C22⋊C4—21no—
33C2×C22⋊C4—21no—
34C2×C42—21no—
35C2×C42C2×C411yescharacteristic
36C2×C42C2×C411yescharacteristic
37C22⋊C8—24no—
38C22⋊C8—42no—
39#95—22no—
40#95—22no—
41C24C22⋊C411yescharacteristic
42C22×C4—22no—
43C22×C4—22no—
44C22×C4—22no—
45C22×C4—21no—
46C22×C4—21no—
47C22×C4—22no—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—82no—
51C22×C4C22⋊C411yescharacteristic
52C22×C4C22⋊C412yes—
53C22×C4C2×C812yes—
54C22×C4M1612yes—
55C22×C4C22×C411yescharacteristic
56C22×C4C2×D412yes—
57C2×C8—44no—
58C2×C8—84no—
59C4⋊C4—24no—
60C4⋊C4—41no—
61C4⋊C4—42no—
62C4⋊C4—41no—
63C4⋊C4—41no—
64C4⋊C4—41no—
65C4⋊C4—82no—
66C4⋊C4—82no—
67C22⋊C4—24no—
68C22⋊C4—42no—
69C42—21no—
70C42—21no—
71C42—42no—
72C42C2×C812yescomplemented
73C42C2×C812yescomplemented
74C23—41no—
75C23—41no—
76C23C22⋊C812yes—
77C23#9912yes—
78C23#10112yes—
79C23C2×C22⋊C411yescharacteristic
80C2×C4—24no—
81C2×C4—24no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—22no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—41no—
91C2×C4—41no—
92C2×C4—44no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—82no—
102C2×C4C22⋊C812yes—
103C2×C4C22×C811yescharacteristic
104C2×C4C2×M1611yescharacteristic
105C8—84no—
106C22—21no—
107C22—22no—
108C22—21no—
109C22—22no—
110C22—42no—
111C22—42no—
112C22#32414yes—
113C22C2×C22⋊C811yescharacteristic
114C22C2×#9911yescharacteristic
115C22C2×#10111yescharacteristic
116C4—22no—
117C4—21no—
118C4—21no—
119C4—42no—
120C4—42no—
121C4—42no—
122C4—42no—
123C4—81no—
124C4—81no—
125C2—21no—
126C2—24no—
127C2—21no—
128C2C2×#32411yescharacteristic
129C2#212711yescharacteristic
130C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC2×C4C22×C8
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1325C2semidirect
4. #1324C2semidirect
22. #117C8semidirect
72. C42C2×C8semidirect
73. C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
#23172
C2×#3242
#13252
#13242