Finite Group Databasebeta

← #11535

Subgroups — #11535

Order 256.

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

normal subgroups only

This group has 220 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#1465C212yesmaximal
3#1325C212yesmaximal, complemented
4#1319C212yesmaximal, complemented
5C2×#124C411yescharacteristic
6C2×C4×Q8C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8#407—22no—
9#391C2211yescharacteristic
10#390C411yescharacteristic
11#388—22no—
12#385—22no—
13#384C2211yescharacteristic
14#376C411yescharacteristic
15#324C2214yes—
16#323—22no—
17C22×Q8C2×C411yescharacteristic
18C22×D4C2×C411yescharacteristic
19C23×C4D412yes—
20C2×M16—42no—
21#124—22no—
22#124—22no—
23C4×Q8—42no—
24C4×D4—42no—
25C2×C4⋊C4—21no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C4⋊C4C2311yescharacteristic
29C2×C22⋊C4—22no—
30C2×C22⋊C4—22no—
31C2×C22⋊C4D412yes—
32C2×C42—21no—
33C2×C42C2×C411yescharacteristic
34C2×C42C2×C411yescharacteristic
35C2×C42C2×C411yescharacteristic
36C22⋊C8—24no—
37C22⋊C8—42no—
38#95—21no—
39#95—21no—
40#95—22no—
41C24C22⋊C412yes—
42C2×Q8—21no—
43C2×Q8—41no—
44C2×Q8—41no—
45C2×Q8M1612yescomplemented
46C2×D4—21no—
47C2×D4—21no—
48C2×D4—41no—
49C2×D4—41no—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—82no—
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22×C411yescharacteristic
60C22×C4C2×D412yes—
61M16—84no—
62C2×C8—44no—
63C4⋊C4—41no—
64C4⋊C4—41no—
65C4⋊C4—42no—
66C4⋊C4—82no—
67C22⋊C4—28no—
68C22⋊C4—42no—
69C22⋊C4—82no—
70C42—22no—
71C42—21no—
72C42—41no—
73C42—42no—
74C42M1612yescomplemented
75C23—22no—
76C23—22no—
77C23—22no—
78C23—42no—
79C23—42no—
80C23C2×C22⋊C411yescharacteristic
81Q8—42no—
82Q8—42no—
83D4—42no—
84D4—42no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—41no—
92C2×C4—44no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—42no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—82no—
105C2×C4#9912yes—
106C2×C4C2×M1611yescharacteristic
107C2×C4C2×M1611yescharacteristic
108C8—84no—
109C22—21no—
110C22—21no—
111C22—44no—
112C22—42no—
113C22—42no—
114C22—42no—
115C22—82no—
116C22#40711yescharacteristic
117C22C2×#9911yescharacteristic
118C22#41311yescharacteristic
119C4—22no—
120C4—21no—
121C4—21no—
122C4—42no—
123C4—42no—
124C4—41no—
125C4—42no—
126C4—81no—
127C4—81no—
128C2—21no—
129C2—21no—
130C2—82no—
131C2#146511yescharacteristic
132C2#212711yescharacteristic
133C2#213511yescharacteristic
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×D4 ◁ 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. #1319C2semidirect
45. C2×Q8M16semidirect
74. C42M16semidirect

Maximal subgroups

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

subgroupindex
#23262
#14652
#13252
#13192