Finite Group Databasebeta

← #30647996

Subgroups — #30647996

Order 16256.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861255C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3C22×#14523233C211yesmaximal, characteristic, complemented
4#22550019C214yesmaximal, complemented
5#22550009C214yesmaximal, complemented
6#22549880C211yesmaximal, characteristic, complemented
7#22549870C211yesmaximal, characteristic, complemented
8#22549843C211yesmaximal, characteristic, complemented
9#535×C127C211yesmaximal, characteristic, complemented
10#16861255—24no—
11#16861255—24no—
12C2×#14523249—21no—
13C2×#14523248C2211yescharacteristic
14C2×#14523248C2212yescomplemented
15C22×D508C2211yescharacteristic
16C22×D508C2212yescomplemented
17C2×D4×D127—22no—
18C2×D4×D127—21no—
19#16861235—24no—
20C2×#14523236C2212yescomplemented
21C2×#14523233—24no—
22C2×#14523233C2214yescomplemented
23C2×#14523233C2212yescomplemented
24C2×#14523232C2211yescharacteristic, complemented
25C22×C127⋊C8C2211yescharacteristic, complemented
26#16861132C2212yescomplemented
27#16861129C2214yescomplemented
28#16861129C2212yescomplemented
29C22⋊(C127⋊C8)C2214yescomplemented
30C2×#41×C127C2211yescharacteristic
31C22×D4×C127C2211yescharacteristic, complemented
32#127×C127—21no—
33#124×C127—21no—
34#121×C127C2214yescomplemented
35#120×C127—22no—
36#117×C127C2211yescharacteristic, complemented
37C23×D127—21no—
38C23×D127—22no—
39#14523249—28no—
40#14523248—24no—
41#14523248—24no—
42#14523248—22no—
43#14523248D414yescomplemented
44C2×Dic254C2311yescharacteristic
45C2×D508—24no—
46C2×D508—24no—
47C2×D508—21no—
48C2×D508—22no—
49C2×D508—22no—
50C2×D508D414yescomplemented
51C2×D508C2312yes—
52C2×D508C2311yescharacteristic
53C2×C22⋊D127—21no—
54C2×C22⋊D127—22no—
55D4×D127—216no—
56D4×D127—28no—
57C2×C4×D127—21no—
58C2×C4×D127—22no—
59#14523236—24no—
60#14523233—28no—
61#14523233—24no—
62#14523232—22no—
63C2×C127⋊C8—22no—
64C2×C127⋊C8C2312yes—
65C2×C127⋊C8C2312yes—
66C24×C127—21no—
67#41×C127—22no—
68#41×C127—22no—
69C2×Q8×C127D411yescharacteristic
70C2×D4×C127—24no—
71C2×D4×C127—22no—
72C2×D4×C127—22no—
73C2×D4×C127—21no—
74C2×D4×C127D412yes—
75C2×D4×C127D411yescharacteristic
76C2×D4×C127C2312yes—
77C22×C4×C127C2312yes—
78C22×C4×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81C22⋊C4×C127—24no—
82C42×C127—21no—
83C22×D127—21no—
84C22×D127—22no—
85C22×D127—28no—
86C22×D127—24no—
87C22×D127—82no—
88C22×D127—84no—
89Dic254—21no—
90Dic254C2×D412yes—
91D508—28no—
92D508—24no—
93D508—22no—
94D508—21no—
95D508C2×D412yes—
96D508C2×D414yes—
97C22⋊D127—28no—
98C22⋊D127—24no—
99C2×Dic127—21no—
100C4×D127—28no—
101C4×D127—24no—
102C127⋊C8—22no—
103C127⋊C8—22no—
104C23×C127—22no—
105C23×C127—42no—
106C23×C127—82no—
107C23×C127C2×D412yes—
108C23×C127C2×D411yescharacteristic
109Q8×C127—22no—
110D4×C127—24no—
111D4×C127—22no—
112D4×C127—22no—
113D4×C127—22no—
114D4×C127—42no—
115D4×C127—84no—
116C2×C4×C127—22no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D411yescharacteristic
123C2×C4×C127C2411yescharacteristic
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—24no—
127C22×C127—42no—
128C22×C127—44no—
129C22×C127—84no—
130C22×C127#12012yes—
131C22×C127C22×D411yescharacteristic
132C2×D127—24no—
133C2×D127—22no—
134C2×D127—44no—
135C2×D127—42no—
136C2×D127—84no—
137C2×D127—88no—
138C4×C127—22no—
139C4×C127—42no—
140C4×C127#12012yes—
141C4×C127C22×D411yescharacteristic
142C4×C127C22×D411yescharacteristic
143Dic127—22no—
144C2×C127—22no—
145C2×C127—42no—
146C2×C127—82no—
147C2×C127C2×#12011yescharacteristic
148C2×C127#57612yes—
149D127—44no—
150D127—42no—
151#3008—1271nomaximal, Sylow
152C127#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—1271no—
154C2×AGL(1,ℤ/8)—1271no—
155C22×D8—1271no—
156#535D12711yescharacteristic, complemented
157#449—1274no—
158#447—1274no—
159#418—1271no—
160#416—1271no—
161#408—1271no—
162H(5,𝔽2)—2544no—
163H(5,𝔽2)—2544no—
164C2×#41—1271no—
165C2×#41—1272no—
166C2×#41—2541no—
167C2×#41C2×D12711yescharacteristic
168C22×D4—1271no—
169C22×D4—1272no—
170C22×D4—2542no—
171C22×D4—2541no—
172C22×D4C2×D12711yescharacteristic, complemented
173AGL(1,ℤ/8)—2544no—
174C2×SD16—1272no—
175C2×D8—1274no—
176C2×D8—1272no—
177C2×D8—2544no—
178C2×M16—1271no—
179C22×C8—1271no—
180#127—21no—
181#124—21no—
182#121C2×D12714yescomplemented
183#120—22no—
184#117C2×D12711yescharacteristic, complemented
185#103—1272no—
186#102—1272no—
187#102—1274no—
188C22⋊C8—1274no—
189C24—21no—
190C24—2542no—
191C24—2541no—
192#41—22no—
193#41—22no—
194#41—1274no—
195#41—2544no—
196#41—2544no—
197#41—2548no—
198#41—2542no—
199C2×Q8—1271no—
200C2×Q8C22⋊D12711yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—1274no—
206C2×D4—1272no—
207C2×D4—1271no—
208C2×D4—2544no—
209C2×D4—2544no—
210C2×D4—25416no—
211C2×D4—2548no—
212C2×D4—2542no—
213C2×D4—2541no—
214C2×D4—2542no—
215C2×D4—2542no—
216C2×D4—2541no—
217C2×D4C22⋊D12712yes—
218C2×D4C22⋊D12711yescharacteristic
219C2×D4C22×D12712yes—
220C22×C4—2542no—
221C22×C4—2541no—
222C22×C4C22×D12712yes—
223C22×C4C22×D12711yescharacteristic
224SD16—2544no—
225D8—2548no—
226D8—2544no—
227M16—2542no—
228C2×C8—1272no—
229C2×C8—1272no—
230C2×C8—2542no—
231C4⋊C4C22×D12712yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—10162no—
239C23—10164no—
240C23—2548no—
241C23—2544no—
242C23—2542no—
243C23—2541no—
244C23C2×C22⋊D12712yes—
245C23C2×C22⋊D12711yescharacteristic
246Q8—22no—
247Q8—1272no—
248Q8—2541no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—1274no—
256D4—1272no—
257D4—2548no—
258D4—2548no—
259D4—2544no—
260D4—2541no—
261D4—2542no—
262D4—2544no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—2548no—
267C2×C4—2541no—
268C2×C4—2544no—
269C2×C4D4×D12712yes—
270C2×C4D4×D12712yes—
271C2×C4C2×C22⋊D12712yes—
272C2×C4C2×C22⋊D12711yescharacteristic
273C2×C4C23×D12711yescharacteristic
274C8—2542no—
275C8—2542no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—10164no—
283C22—10168no—
284C22—5084no—
285C22—5082no—
286C22—2544no—
287C22—2542no—
288C22#1686117712yes—
289C22C22×C22⋊D12711yescharacteristic
290C4—22no—
291C4—42no—
292C4—2542no—
293C4#1686117712yes—
294C4C2×D4×D12711yescharacteristic
295C4C2×D4×D12711yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—5084no—
300C2—5082no—
301C2C2×#1686117711yescharacteristic
302C2#2255067912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#535×C127C2
socleC22×C127C22×D4
hypercenter#535D127
hypocenterC127#3008
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2semidirect
2. C2×#16861235C2semidirect
3. C22×#14523233C2semidirect
4. #22550019C2semidirect
5. #22550009C2semidirect
6. #22549880C2semidirect
7. #22549870C2semidirect
8. #22549843C2semidirect
9. #535×C127C2semidirect
14. C2×#14523248C22semidirect
16. C22×D508C22semidirect
20. C2×#14523236C22semidirect
22. C2×#14523233C22semidirect
23. C2×#14523233C22semidirect
24. C2×#14523232C22semidirect
25. C22×C127⋊C8C22semidirect
26. #16861132C22semidirect
27. #16861129C22semidirect
28. #16861129C22semidirect
29. C22⋊(C127⋊C8)C22semidirect
31. C22×D4×C127C22semidirect
34. #121×C127C22semidirect
36. #117×C127C22semidirect
43. #14523248D4semidirect
50. C2×D508D4semidirect
152. C127#3008semidirect
156. #535D127semidirect
172. C22×D4C2×D127semidirect
182. #121C2×D127semidirect
184. #117C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612352
C22×#145232332
#225500192
#225500092
#225498802
#225498702
#225498432
#535×C1272
#3008127