Finite Group Databasebeta

← #8108

Subgroups — #8108

Order 256.

457 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#407C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic
3#2280C211yesmaximal, characteristic
4#1305C214yesmaximal, complemented
5C24×C4C2211yescharacteristic
6C22×M16—21no—
7C22×M16C412yescomplemented
8C2×#117—21no—
9#407—21no—
10#407—21no—
11#407—21no—
12#407—21no—
13#407C412yescomplemented
14#407C412yescomplemented
15#407C2212yes—
16#407C2212yes—
17C2×C22⋊C8—21no—
18C2×C22⋊C8—21no—
19C2×C22⋊C8—21no—
20C2×C22⋊C8C412yescomplemented
21#381—21no—
22#380—21no—
23#380C2212yes—
24#379C412yes—
25C4×C22⋊C4—21no—
26C4×C22⋊C4C412yes—
27C25C2×C411yescharacteristic
28C23×C4—44no—
29C23×C4C2×C411yescharacteristic
30C23×C4C2×C411yescharacteristic
31C23×C4C2×C411yescharacteristic
32C23×C4C2×C411yescharacteristic
33C23×C4C2×C411yescharacteristic
34C23×C4D411yescharacteristic
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4D411yescharacteristic
39C23×C4D411yescharacteristic
40C23×C4Q811yescharacteristic
41C23×C4Q811yescharacteristic
42C23×C4C2311yescharacteristic
43C2×M16—22no—
44C2×M16—22no—
45C2×M16—22no—
46C2×M16—44no—
47C2×M16—44no—
48C2×M16C2×C414yes—
49C22×C8—22no—
50C22×C8—22no—
51#117—44no—
52C2×C4⋊C4—22no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C22⋊C4C2×C414yes—
56C2×C42—22no—
57C22⋊C8—22no—
58C22⋊C8—22no—
59C22⋊C8—22no—
60C22⋊C8—22no—
61C22⋊C8—22no—
62C22⋊C8—22no—
63C22⋊C8—22no—
64C22⋊C8C2×C414yes—
65#95—22no—
66#95—22no—
67C24—42no—
68C24—42no—
69C24—42no—
70C24C4211yescharacteristic
71C24C4211yescharacteristic
72C24C22⋊C411yescharacteristic
73C24C22⋊C411yescharacteristic
74C24C4⋊C411yescharacteristic
75C24C4⋊C411yescharacteristic
76C24C22×C411yescharacteristic
77C22×C4—24no—
78C22×C4—24no—
79C22×C4—24no—
80C22×C4—24no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4C4211yescharacteristic
93C22×C4C4211yescharacteristic
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22⋊C411yescharacteristic
98C22×C4C22⋊C411yescharacteristic
99C22×C4C22⋊C411yescharacteristic
100C22×C4C22⋊C411yescharacteristic
101C22×C4C22⋊C411yescharacteristic
102C22×C4C22⋊C411yescharacteristic
103C22×C4C22⋊C411yescharacteristic
104C22×C4C4⋊C411yescharacteristic
105C22×C4C4⋊C411yescharacteristic
106C22×C4C4⋊C411yescharacteristic
107C22×C4C4⋊C411yescharacteristic
108C22×C4C4⋊C411yescharacteristic
109C22×C4C4⋊C411yescharacteristic
110C22×C4C4⋊C411yescharacteristic
111C22×C4C4⋊C411yescharacteristic
112C22×C4C4⋊C411yescharacteristic
113C22×C4C4⋊C411yescharacteristic
114C22×C4C22×C411yescharacteristic
115C22×C4C22×C411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×Q811yescharacteristic
120M16—44no—
121M16—44no—
122M16—44no—
123M16—44no—
124C2×C8—24no—
125C2×C8—24no—
126C2×C8—44no—
127C2×C8—44no—
128C4⋊C4—44no—
129C22⋊C4—44no—
130C22⋊C4—44no—
131C22⋊C4—44no—
132C42—44no—
133C23—22no—
134C23—22no—
135C23—22no—
136C23—22no—
137C23—22no—
138C23—22no—
139C23—22no—
140C23—22no—
141C23—22no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—416no—
155C23#9511yescharacteristic
156C23#9511yescharacteristic
157C23#9511yescharacteristic
158C23#9511yescharacteristic
159C23C2×C4211yescharacteristic
160C23C2×C22⋊C411yescharacteristic
161C23C2×C4⋊C411yescharacteristic
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4#9511yescharacteristic
193C2×C4#9511yescharacteristic
194C2×C4#9511yescharacteristic
195C2×C4#9511yescharacteristic
196C2×C4#9914yes—
197C2×C4C2×C22⋊C411yescharacteristic
198C2×C4C2×C22⋊C411yescharacteristic
199C2×C4C2×C4⋊C411yescharacteristic
200C2×C4C2×C4⋊C411yescharacteristic
201C8—44no—
202C8—44no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—44no—
222C22—44no—
223C22—44no—
224C22—44no—
225C22—44no—
226C22—44no—
227C22—44no—
228C22#34212yes—
229C22#34412yes—
230C22C2×#9511yescharacteristic
231C22#41312yes—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—21no—
236C4—21no—
237C4—21no—
238C4—84no—
239C4#34212yes—
240C4C2×#9912yes—
241C2—22no—
242C2—22no—
243C2—22no—
244C2—41no—
245C2—41no—
246C2—41no—
247C2—41no—
248C2C2×#34211yescharacteristic
249C2#174211yescharacteristic
250C2C2×#34411yescharacteristic
subgroupG/H
center, socleC22C2×#95
commutatorC23C2×C42
Frattini, cosocleC23×C4C23
prime radicalC25C2×C4
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1305C2semidirect
7. C22×M16C4semidirect
13. #407C4semidirect
14. #407C4semidirect
20. C2×C22⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4072
#22802
#13052