Finite Group Databasebeta

← #25753

Subgroups — #25753

Order 256.

999 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 999 conjugacy classes of subgroups — too many to draw. Only its 167 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×D42C211yesmaximal, characteristic
2C2×#492C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4C2×#450C214yesmaximal, complemented
5#2542C211yesmaximal, characteristic, complemented
6#2000C214yesmaximal, complemented
7#1982C211yesmaximal, characteristic, complemented
8#1954C211yesmaximal, characteristic, complemented
9C4×#102C211yesmaximal, characteristic, complemented
10C23×D4—21no—
11C23×D4C2212yes—
12C22×SD16C2212yes—
13C22×SD16C2214yescomplemented
14D42—24no—
15D42—24no—
16C2×C4⋊Q8C2211yescharacteristic, complemented
17C2×#127C2211yescharacteristic
18C2×C22⋊Q8C2214yescomplemented
19C2×#121—21no—
20C2×#121C2212yes—
21C2×#120—22no—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24#492—24no—
25#461—24no—
26#450—216no—
27C2×C4⋊C8C2211yescharacteristic, complemented
28C2×#103C2212yescomplemented
29C2×#102C2212yes—
30C2×#102C2214yescomplemented
31C2×C22⋊C8C2214yescomplemented
32C2×C4×C8C2211yescharacteristic, complemented
33#399—22no—
34#390—21no—
35#384—21no—
36C4×C4⋊C4C2211yescharacteristic, complemented
37#340C2212yescomplemented
38C25—22no—
39C22×Q8C2312yes—
40C22×D4—28no—
41C22×D4—22no—
42C22×D4—24no—
43C22×D4—24no—
44C22×D4—28no—
45C22×D4—22no—
46C22×D4—21no—
47C22×D4—21no—
48C22×D4—44no—
49C22×D4—44no—
50C22×D4—44no—
51C22×D4D411yescharacteristic
52C22×D4C2312yes—
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4C2312yes—
56C2×SD16—28no—
57C2×SD16—24no—
58C2×SD16—28no—
59C2×SD16—28no—
60C2×SD16—44no—
61C22×C8C2312yes—
62C22×C8C2312yes—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65#127—22no—
66#127D414yes—
67C22⋊Q8—28no—
68C22⋊Q8—44no—
69#121—28no—
70#121—24no—
71#121—24no—
72#120—216no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4D414yes—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4D412yes—
87C2×C42—21no—
88C2×C42C2311yescharacteristic
89C4⋊C8—22no—
90#103—24no—
91#102—28no—
92#102D418yescomplemented
93C22⋊C8—28no—
94C4×C8—22no—
95#95—21no—
96C24—21no—
97C24—28no—
98C24—22no—
99C24—48no—
100C24—44no—
101C24—44no—
102C24—84no—
103C24C2×D412yes—
104C2×Q8—22no—
105C2×Q8—24no—
106C2×Q8—22no—
107C2×Q8—24no—
108C2×D4—28no—
109C2×D4—28no—
110C2×D4—22no—
111C2×D4—24no—
112C2×D4—24no—
113C2×D4—22no—
114C2×D4—21no—
115C2×D4—24no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—24no—
119C2×D4—48no—
120C2×D4—416no—
121C2×D4—48no—
122C2×D4—432no—
123C2×D4—44no—
124C2×D4—44no—
125C2×D4—48no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—82no—
129C2×D4SD1614yescomplemented
130C2×D4C2×D414yes—
131C2×D4C2×D414yes—
132C2×D4C2×D412yes—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4C2×D412yes—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147SD16—416no—
148SD16—48no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—24no—
152C2×C8—22no—
153C2×C8C2×D414yescomplemented
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—48no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D414yescomplemented
162C22⋊C4—28no—
163C22⋊C4—24no—
164C22⋊C4—24no—
165C22⋊C4—48no—
166C42—42no—
167C42C2×D412yes—
168C42C2×D412yes—
169C23—22no—
170C23—22no—
171C23—24no—
172C23—24no—
173C23—22no—
174C23—24no—
175C23—24no—
176C23—416no—
177C23—416no—
178C23—41no—
179C23—41no—
180C23—42no—
181C23—44no—
182C23—42no—
183C23—42no—
184C23—44no—
185C23—816no—
186C23—88no—
187C23—816no—
188C23C2×SD1614yes—
189C23C22×D411yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—44no—
193D4—22no—
194D4—24no—
195D4—28no—
196D4—28no—
197D4—416no—
198D4—416no—
199D4—42no—
200D4—42no—
201D4—48no—
202D4—84no—
203D4#12714yes—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—48no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4#12012yes—
220C2×C4#12012yes—
221C2×C4C2×SD1612yes—
222C2×C4AGL(1,ℤ/8)12yes—
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D412yes—
225C2×C4C22×D411yescharacteristic
226C2×C4H(5,𝔽2)12yes—
227C8—24no—
228C8—42no—
229C22—24no—
230C22—24no—
231C22—28no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—44no—
237C22—44no—
238C22—44no—
239C22—44no—
240C22—44no—
241C22—44no—
242C22—832no—
243C22—84no—
244C22—84no—
245C22C2×#12011yescharacteristic
246C22D4212yes—
247C22C22×SD1611yescharacteristic
248C22C2×AGL(1,ℤ/8)11yescharacteristic
249C22#57712yes—
250C4—22no—
251C4—42no—
252C4—42no—
253C4—44no—
254C4—82no—
255C4#45014yescomplemented
256C4C2×#12711yescharacteristic
257C4#53511yescharacteristic
258C4D4212yes—
259C2—44no—
260C2—44no—
261C2—84no—
262C2#243311yescharacteristic
263C2C2×#45011yescharacteristic
264C2#300911yescharacteristic
265C2#327212yes—
266C2D4×SD1612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×D42C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#492C2semidirect
3. C2×#461C2semidirect
4. C2×#450C2semidirect
5. #2542C2semidirect
6. #2000C2semidirect
7. #1982C2semidirect
8. #1954C2semidirect
9. C4×#102C2semidirect
13. C22×SD16C22semidirect
16. C2×C4⋊Q8C22semidirect
18. C2×C22⋊Q8C22semidirect
27. C2×C4⋊C8C22semidirect
28. C2×#103C22semidirect
30. C2×#102C22semidirect
31. C2×C22⋊C8C22semidirect
32. C2×C4×C8C22semidirect
36. C4×C4⋊C4C22semidirect
37. #340C22semidirect
92. #102D4semidirect
129. C2×D4SD16semidirect
153. C2×C8C2×D4semidirect
161. C4⋊C4C2×D4semidirect
255. C4#450semidirect

Maximal subgroups

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

subgroupindex
C2×D422
C2×#4922
C2×#4612
C2×#4502
#25422
#20002
#19822
#19542
C4×#1022