Finite Group Databasebeta

← #11534

Subgroups — #11534

Order 256.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#2326C211yesmaximal, characteristic
2#1467C212yesmaximal, complemented
3#1325C212yesmaximal, complemented
4#1318C212yesmaximal
5C2×#124C411yescharacteristic
6C2×C4×Q8C2211yescharacteristic
7C2×C4×D4C411yescharacteristic
8#432—22no—
9#391C411yescharacteristic
10#390C2211yescharacteristic
11#388—22no—
12#385—22no—
13#384C2211yescharacteristic
14#376C411yescharacteristic
15#324—22no—
16#324C2214yes—
17C22×Q8C2×C411yescharacteristic
18C22×D4C2×C411yescharacteristic
19C23×C4—21no—
20#124—22no—
21#124—22no—
22C4×Q8—42no—
23C4×D4—42no—
24C2×C4⋊C4—22no—
25C2×C4⋊C4C2×C411yescharacteristic
26C2×C4⋊C4D412yes—
27C2×C4⋊C4C2311yescharacteristic
28C2×C22⋊C4—22no—
29C2×C22⋊C4—21no—
30C2×C22⋊C4—22no—
31C2×C42C2×C411yescharacteristic
32C2×C42C2×C411yescharacteristic
33C2×C42C2×C411yescharacteristic
34C2×C42D412yes—
35C4⋊C8—42no—
36C22⋊C8—24no—
37#97—42no—
38#95—21no—
39#95—22no—
40#95—21no—
41C24—21no—
42C2×Q8—21no—
43C2×Q8—21no—
44C2×Q8—41no—
45C2×Q8—41no—
46C2×D4—21no—
47C2×D4—41no—
48C2×D4—41no—
49C2×D4M1612yes—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—42no—
54C22×C4—82no—
55C22×C4C22⋊C412yes—
56C22×C4C22⋊C411yescharacteristic
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22×C411yescharacteristic
59C22×C4C2×D412yes—
60C2×C8—44no—
61C4⋊C4—42no—
62C4⋊C4—41no—
63C4⋊C4—41no—
64C4⋊C4—82no—
65C22⋊C4—28no—
66C22⋊C4—42no—
67C22⋊C4—82no—
68C42—22no—
69C42—22no—
70C42—22no—
71C42—21no—
72C42—41no—
73C42M1612yes—
74C23—22no—
75C23—42no—
76C23—42no—
77C23—41no—
78C23—41no—
79C23C2×C22⋊C411yescharacteristic
80Q8—42no—
81Q8—42no—
82D4—42no—
83D4—42no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—22no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—42no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—42no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—84no—
103C2×C4#9912yes—
104C2×C4C2×M1611yescharacteristic
105C2×C4C2×M1611yescharacteristic
106C8—84no—
107C22—21no—
108C22—21no—
109C22—44no—
110C22—42no—
111C22—82no—
112C22—81no—
113C22—81no—
114C22#40711yescharacteristic
115C22C2×#9911yescharacteristic
116C22#41311yescharacteristic
117C4—22no—
118C4—21no—
119C4—21no—
120C4—42no—
121C4—42no—
122C4—41no—
123C4—42no—
124C4—81no—
125C4—81no—
126C2—21no—
127C2—21no—
128C2—82no—
129C2#146511yescharacteristic
130C2#212811yescharacteristic
131C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1467C2semidirect
3. #1325C2semidirect

Maximal subgroups

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

subgroupindex
#23262
#14672
#13252
#13182