Finite Group Databasebeta

← #11501

Subgroups — #11501

Order 256.

331 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 55 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#2280C211yesmaximal, characteristic
2#1464C212yesmaximal, complemented
3#1317C212yesmaximal, complemented
4#1316C212yesmaximal
5C24×C4C411yescharacteristic
6C2×#117C411yescharacteristic
7C2×#117C2211yescharacteristic
8#432—22no—
9#381C411yescharacteristic
10#381C2211yescharacteristic
11#380—22no—
12#379C411yescharacteristic
13#379C2211yescharacteristic
14C4×C22⋊C4—22no—
15#324—22no—
16#323C2214yes—
17C25C2×C411yescharacteristic
18C23×C4—22no—
19C23×C4—21no—
20C23×C4—21no—
21C23×C4—22no—
22C23×C4—42no—
23C23×C4—42no—
24C23×C4C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26#117—42no—
27#117—42no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4D412yes—
30C2×C22⋊C4—22no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4C2×C411yescharacteristic
33C2×C22⋊C4C2×C411yescharacteristic
34C2×C22⋊C4C2×C411yescharacteristic
35C2×C22⋊C4C2311yescharacteristic
36C2×C42—21no—
37C2×C42D412yes—
38C4⋊C8—42no—
39C22⋊C8—24no—
40#97—42no—
41#95—22no—
42#95—22no—
43C24—21no—
44C24—21no—
45C24—41no—
46C24—41no—
47C24—44no—
48C24M1612yes—
49C24C22×C411yescharacteristic
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—24no—
55C22×C4—24no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—48no—
61C22×C4—48no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4—42no—
65C22×C4—42no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C412yes—
70C22×C4C22⋊C411yescharacteristic
71C22×C4M1612yes—
72C22×C4C2×D412yes—
73C2×C8—44no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C22⋊C4—41no—
77C22⋊C4—41no—
78C22⋊C4—44no—
79C22⋊C4—44no—
80C22⋊C4—41no—
81C22⋊C4—41no—
82C42—22no—
83C42—22no—
84C42—42no—
85C23—24no—
86C23—22no—
87C23—21no—
88C23—21no—
89C23—42no—
90C23—44no—
91C23—42no—
92C23—42no—
93C23—42no—
94C23—44no—
95C23—44no—
96C23—48no—
97C23—44no—
98C23—41no—
99C23—41no—
100C23C2×C22⋊C411yescharacteristic
101C23C2×M1611yescharacteristic
102C23C2×M1611yescharacteristic
103C2×C4—24no—
104C2×C4—24no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—48no—
113C2×C4—48no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4#9912yes—
125C2×C4#10012yes—
126C8—84no—
127C22—22no—
128C22—24no—
129C22—21no—
130C22—21no—
131C22—41no—
132C22—41no—
133C22—44no—
134C22—42no—
135C22—42no—
136C22—44no—
137C22—44no—
138C22—48no—
139C22—44no—
140C22—42no—
141C22—42no—
142C22#40711yescharacteristic
143C22C2×#9911yescharacteristic
144C22C2×#10011yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—42no—
149C4—42no—
150C4—81no—
151C4—81no—
152C2—21no—
153C2—21no—
154C2—44no—
155C2—41no—
156C2—41no—
157C2#146211yescharacteristic
158C2#212012yes—
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC25C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1464C2semidirect
3. #1317C2semidirect

Maximal subgroups

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

subgroupindex
#22802
#14642
#13172
#13162