Finite Group Databasebeta

← #11425

Subgroups — #11425

Order 256.

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

normal subgroups only

This group has 339 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#2280C211yesmaximal, characteristic
2#1458C212yesmaximal, complemented
3#1317C212yesmaximal, complemented
4#1316C212yesmaximal
5C24×C4C411yescharacteristic
6C2×#117C411yescharacteristic
7C2×#117C2211yescharacteristic
8#431—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—21no—
19C23×C4—21no—
20C23×C4—22no—
21C23×C4—41no—
22C23×C4—41no—
23C23×C4—42no—
24C23×C4C814yescomplemented
25C23×C4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27#117—41no—
28#117—41no—
29#117—42no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4D412yes—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4C2×C411yescharacteristic
35C2×C22⋊C4C2×C411yescharacteristic
36C2×C22⋊C4C2×C411yescharacteristic
37C2×C22⋊C4C2311yescharacteristic
38C2×C42—21no—
39C2×C42D412yes—
40C4⋊C8—42no—
41C22⋊C8—24no—
42C4×C8—42no—
43#95—22no—
44#95—22no—
45C24—21no—
46C24—21no—
47C24—41no—
48C24—41no—
49C24—44no—
50C24C2×C812yes—
51C24C22×C411yescharacteristic
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—22no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—24no—
58C22×C4—24no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—42no—
62C22×C4—42no—
63C22×C4—44no—
64C22×C4—44no—
65C22×C4—44no—
66C22×C4—44no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—42no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C412yes—
74C22×C4C22⋊C411yescharacteristic
75C22×C4C2×C812yes—
76C22×C4C2×C812yes—
77C22×C4M1612yes—
78C22×C4C2×D412yes—
79C2×C8—44no—
80C4⋊C4—42no—
81C4⋊C4—42no—
82C22⋊C4—41no—
83C22⋊C4—41no—
84C22⋊C4—44no—
85C22⋊C4—44no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C42—22no—
89C42—22no—
90C42—42no—
91C23—24no—
92C23—22no—
93C23—21no—
94C23—21no—
95C23—42no—
96C23—44no—
97C23—42no—
98C23—42no—
99C23—42no—
100C23—44no—
101C23—44no—
102C23—48no—
103C23—44no—
104C23—41no—
105C23—41no—
106C23C2×C22⋊C411yescharacteristic
107C23C22×C811yescharacteristic
108C23C2×M1611yescharacteristic
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—48no—
119C2×C4—42no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—44no—
123C2×C4—44no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4C22⋊C812yes—
132C2×C4C22⋊C812yes—
133C2×C4#9911yescharacteristic
134C2×C4#9911yescharacteristic
135C2×C4#10012yes—
136C8—84no—
137C22—22no—
138C22—24no—
139C22—21no—
140C22—21no—
141C22—41no—
142C22—41no—
143C22—44no—
144C22—42no—
145C22—42no—
146C22—44no—
147C22—44no—
148C22—48no—
149C22—44no—
150C22—42no—
151C22—42no—
152C22C2×C22⋊C811yescharacteristic
153C22C2×#9911yescharacteristic
154C22C2×#10011yescharacteristic
155C4—22no—
156C4—42no—
157C4—42no—
158C4—42no—
159C4—81no—
160C4—81no—
161C4#32312yes—
162C4#32312yes—
163C2—21no—
164C2—21no—
165C2—44no—
166C2—41no—
167C2—41no—
168C2C2×#32311yescharacteristic
169C2#212012yes—
subgroupG/H
centerC2×C4#99
commutatorC23C22×C8
Frattini, cosocleC2×C22⋊C4C23
socleC22C2×#99
prime radicalC25C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #1458C2semidirect
3. #1317C2semidirect
24. C23×C4C8semidirect

Maximal subgroups

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

subgroupindex
#22802
#14582
#13172
#13162