Finite Group Databasebeta

← #25774

Subgroups — #25774

Order 256.

880 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C23×SD16C211yesmaximal, characteristic, complemented
2C2×#492C212yesmaximal, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4C2×#448C211yesmaximal, characteristic, complemented
5C2×#450C211yesmaximal, characteristic, complemented
6#2434C211yesmaximal, characteristic, complemented
7#2433C211yesmaximal, characteristic
8#2014C212yesmaximal, complemented
9#2013C212yesmaximal, complemented
10#1968C212yesmaximal, complemented
11C8×C22⋊C4C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C22×SD16—28no—
15C22×SD16—24no—
16C22×SD16C2212yes—
17C22×SD16C2212yes—
18C22×SD16C2212yes—
19C22×SD16C2211yescharacteristic, complemented
20C22×SD16C2212yescomplemented
21C23×C8C2211yescharacteristic
22C2×C4⋊Q8C2212yescomplemented
23C2×#127C2212yes—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#120—22no—
27#492—28no—
28#465—24no—
29#450—24no—
30#448—24no—
31C2×#106C2211yescharacteristic, complemented
32C2×#103C2212yescomplemented
33C2×#103C2211yescharacteristic, complemented
34C2×#102C2212yes—
35C2×#102C2211yescharacteristic, complemented
36C2×C22⋊C8C2211yescharacteristic
37C2×C4×C8C2212yescomplemented
38#395—22no—
39#392—22no—
40#391C2212yescomplemented
41#390C2212yes—
42#385—22no—
43C4×C22⋊C4C2211yescharacteristic
44#336C2212yescomplemented
45C25—21no—
46C22×Q8—21no—
47C22×Q8—24no—
48C22×Q8—22no—
49C22×Q8—22no—
50C22×Q8—42no—
51C22×Q8C2311yescharacteristic
52C22×Q8C2312yes—
53C22×D4—24no—
54C22×D4—21no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—21no—
58C22×D4—24no—
59C22×D4—42no—
60C22×D4C2311yescharacteristic
61C22×D4C2312yes—
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C2×SD16—232no—
65C2×SD16—24no—
66C2×SD16—24no—
67C2×SD16—24no—
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—24no—
71C2×SD16—24no—
72C2×SD16—24no—
73C2×SD16—24no—
74C2×SD16—22no—
75C2×SD16—24no—
76C2×SD16—42no—
77C2×SD16—42no—
78C2×SD16D418yescomplemented
79C22×C8—24no—
80C22×C8D414yescomplemented
81C22×C8C2311yescharacteristic
82C22×C8C2312yes—
83C22×C8C2311yescharacteristic
84C4⋊Q8—24no—
85C4⋊Q8—42no—
86#127—24no—
87#127—42no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90#121—22no—
91#121—22no—
92#120—48no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4D412yes—
101C2×C42C2312yes—
102#106—22no—
103#103—24no—
104#103—22no—
105#102—24no—
106#102—22no—
107C22⋊C8D414yescomplemented
108C4×C8—24no—
109#95—24no—
110#95D412yes—
111C24—22no—
112C24—21no—
113C24—44no—
114C24—42no—
115C24—44no—
116C24—82no—
117C24C2×D411yescharacteristic
118C2×Q8—22no—
119C2×Q8—28no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—44no—
129C2×Q8—48no—
130C2×Q8—44no—
131C2×Q8—42no—
132C2×Q8C2×D414yes—
133C2×D4—28no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—48no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—48no—
147C2×D4—42no—
148C2×D4—44no—
149C2×D4—42no—
150C2×D4—48no—
151C2×D4—84no—
152C2×D4C2×D414yes—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D412yes—
163C22×C4C2×D412yes—
164C22×C4C2×D414yes—
165C22×C4C2411yescharacteristic
166SD16—216no—
167SD16—216no—
168SD16—48no—
169SD16—48no—
170SD16—44no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—28no—
175C2×C8C2×D414yes—
176C2×C8C2×D412yes—
177C2×C8C2×D414yes—
178C2×C8C2×D412yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—48no—
184C4⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—48no—
188C22⋊C4SD1618yescomplemented
189C42—22no—
190C42—22no—
191C23—22no—
192C23—22no—
193C23—24no—
194C23—48no—
195C23—42no—
196C23—42no—
197C23—44no—
198C23—44no—
199C23—41no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—816no—
204C23—84no—
205C23—84no—
206C23#12712yes—
207C23C2×SD1614yes—
208C23C22×D411yescharacteristic
209Q8—24no—
210Q8—24no—
211Q8—48no—
212Q8—42no—
213Q8—42no—
214Q8—41no—
215Q8—41no—
216Q8—42no—
217D4—24no—
218D4—24no—
219D4—48no—
220D4—41no—
221D4—42no—
222D4—41no—
223D4—42no—
224D4—42no—
225D4—816no—
226D4—84no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—48no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4#12712yes—
244C2×C4C2×SD1618yescomplemented
245C2×C4C22×D411yescharacteristic
246C2×C4C22×D412yes—
247C2×C4C22×D411yescharacteristic
248C2×C4H(5,𝔽2)12yes—
249C8—22no—
250C8—24no—
251C8#12014yescomplemented
252C22—22no—
253C22—22no—
254C22—28no—
255C22—44no—
256C22—48no—
257C22—816no—
258C22—82no—
259C22—82no—
260C22—84no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—84no—
265C22#49214yescomplemented
266C22C2×#12711yescharacteristic
267C22D4212yes—
268C22C22×SD1612yes—
269C22#57712yes—
270C4—22no—
271C4—28no—
272C4—44no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4C2×#12011yescharacteristic
277C4#53511yescharacteristic
278C4D4212yes—
279C2—24no—
280C2—84no—
281C2—161no—
282C2#243311yescharacteristic
283C2C2×#49211yescharacteristic
284C2#315411yescharacteristic
285C2D4×SD1614yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2433C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×SD16C2semidirect
2. C2×#492C2semidirect
3. C2×#465C2semidirect
4. C2×#448C2semidirect
5. C2×#450C2semidirect
6. #2434C2semidirect
8. #2014C2semidirect
9. #2013C2semidirect
10. #1968C2semidirect
11. C8×C22⋊C4C2semidirect
19. C22×SD16C22semidirect
20. C22×SD16C22semidirect
22. C2×C4⋊Q8C22semidirect
31. C2×#106C22semidirect
32. C2×#103C22semidirect
33. C2×#103C22semidirect
35. C2×#102C22semidirect
37. C2×C4×C8C22semidirect
40. #391C22semidirect
44. #336C22semidirect
78. C2×SD16D4semidirect
80. C22×C8D4semidirect
107. C22⋊C8D4semidirect
188. C22⋊C4SD16semidirect
244. C2×C4C2×SD16semidirect
251. C8#120semidirect
265. C22#492semidirect

Maximal subgroups

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

subgroupindex
C23×SD162
C2×#4922
C2×#4652
C2×#4482
C2×#4502
#24342
#24332
#20142
#20132
#19682
C8×C22⋊C42