Finite Group Databasebeta

← #24025

Subgroups — #24025

Order 256.

643 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 643 conjugacy classes of subgroups — too many to draw. Only its 127 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#3485C211yesmaximal, characteristic
2#3092C212yesmaximal, complemented
3#2311C211yesmaximal, characteristic, complemented
4#2038C214yesmaximal, complemented
5#1897C212yesmaximal, complemented
6#1856C212yesmaximal, complemented
7#1792C212yesmaximal, complemented
8#1785C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)C2211yescharacteristic
10C22×#41C2211yescharacteristic
11#547—22no—
12#547—22no—
13#546—22no—
14#546—22no—
15#536C2212yes—
16#534—22no—
17#525C2212yes—
18C2×#123—22no—
19#518C2211yescharacteristic
20C2×#117C2211yescharacteristic, complemented
21C22×C4⋊C4C2211yescharacteristic, complemented
22#484—22no—
23#483—22no—
24#481—22no—
25#480—22no—
26#428C2214yescomplemented
27C2×#104C2214yes—
28#417C2214yes—
29C2×#101C2212yescomplemented
30C2×#100C2212yes—
31#407C2212yescomplemented
32#391C2212yescomplemented
33#390C2212yes—
34#386—22no—
35#380—22no—
36#339C2214yescomplemented
37H(5,𝔽2)—24no—
38H(5,𝔽2)—42no—
39C2×#41—22no—
40C2×#41—24no—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—42no—
44C2×#41C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4—22no—
47C22×D4—22no—
48C22×D4D412yes—
49C22×D4C2311yescharacteristic
50C23×C4—21no—
51C23×C4D412yes—
52C23×C4C2311yescharacteristic
53C2×M16C2314yes—
54C4⋊Q8—22no—
55#124—22no—
56#123—24no—
57#123—24no—
58#123—24no—
59#123—22no—
60#123—24no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8D412yes—
64C22⋊Q8D412yes—
65#121—24no—
66#121—24no—
67#121D412yes—
68#121D412yes—
69#120—24no—
70#120—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75#117—42no—
76#117C2312yes—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4C2312yes—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—24no—
89C2×C42C2312yes—
90#107—24no—
91#106—24no—
92#104—44no—
93#103—24no—
94#102—24no—
95#101—42no—
96#100—24no—
97C22⋊C8—22no—
98C22⋊C8—22no—
99#95—24no—
100C24—22no—
101C24C2×D411yescharacteristic
102C24C2×D412yes—
103#41—24no—
104#41—24no—
105#41—48no—
106#41—48no—
107#41—48no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8C2×D412yes—
112C2×D4—24no—
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—42no—
121C2×D4—48no—
122C2×D4—44no—
123C2×D4—44no—
124C2×D4C2×D412yes—
125C2×D4#4114yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—82no—
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D412yes—
143C22×C4C2×D412yes—
144C22×C4C2411yescharacteristic
145M16—24no—
146M16—24no—
147C2×C8—24no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—24no—
153C4⋊C4—42no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D412yes—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—42no—
167C22⋊C4—84no—
168C42—22no—
169C42—24no—
170C23—22no—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—42no—
178C23—42no—
179C23—44no—
180C23—82no—
181C23—82no—
182C23#12012yes—
183C23#12314yes—
184C23C22×D411yescharacteristic
185Q8—24no—
186Q8—44no—
187D4—28no—
188D4—24no—
189D4—48no—
190D4—48no—
191D4—42no—
192D4—42no—
193D4—82no—
194D4—82no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—84no—
217C2×C4#12011yescharacteristic
218C2×C4#12011yescharacteristic
219C2×C4C22×D412yes—
220C2×C4C2×#4112yes—
221C2×C4H(5,𝔽2)12yes—
222C8—44no—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—24no—
227C22—22no—
228C22—22no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—82no—
238C22—164no—
239C22C2×#12011yescharacteristic
240C22C2×#12311yescharacteristic
241C22#53411yescharacteristic
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—82no—
249C4—82no—
250C4#54614yes—
251C2—21no—
252C2—21no—
253C2—42no—
254C2—44no—
255C2—82no—
256C2#240511yescharacteristic
257C2#302111yescharacteristic
258C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3485C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3092C2semidirect
3. #2311C2semidirect
4. #2038C2semidirect
5. #1897C2semidirect
6. #1856C2semidirect
7. #1792C2semidirect
8. #1785C2semidirect
20. C2×#117C22semidirect
21. C22×C4⋊C4C22semidirect
26. #428C22semidirect
29. C2×#101C22semidirect
31. #407C22semidirect
32. #391C22semidirect
36. #339C22semidirect

Maximal subgroups

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

subgroupindex
#34852
#30922
#23112
#20382
#18972
#18562
#17922
#17852