Finite Group Databasebeta

← #30648001

Subgroups — #30648001

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×#16861234C211yesmaximal, characteristic, complemented
3C22×D1016C211yesmaximal, characteristic, complemented
4#22550445C211yesmaximal, characteristic, complemented
5#22550024C214yesmaximal, complemented
6#22550011C214yesmaximal, complemented
7#22549880C211yesmaximal, characteristic, complemented
8#22549842C211yesmaximal, characteristic, complemented
9#416×C127C211yesmaximal, characteristic, complemented
10#16861255—24no—
11#16861255—24no—
12C2×#14523249C2212yescomplemented
13C2×#14523248—21no—
14C2×#14523248C2211yescharacteristic
15C22×D508—21no—
16C22×D508C2211yescharacteristic
17C22×D508C2211yescharacteristic, complemented
18C2×D4×D127—22no—
19C2×D4×D127C2212yescomplemented
20#16861234—24no—
21C2×#14523235C2212yescomplemented
22C2×D1016—24no—
23C2×D1016C2214yescomplemented
24C2×#14523233C2212yescomplemented
25C2×#14523232C2211yescharacteristic, complemented
26#16861206—21no—
27#16861197—21no—
28#16861184C2214yescomplemented
29#16861179—22no—
30#16861129C2214yescomplemented
31(C2×D127)⋊C8C2214yescomplemented
32C2×#41×C127C2211yescharacteristic
33C22×C8×C127C2211yescharacteristic, complemented
34#117×C127C2211yescharacteristic, complemented
35#103×C127C2212yescomplemented
36#102×C127C2212yescomplemented
37C23×D127—22no—
38C23×D127—21no—
39C23×D127—21no—
40#14523249—24no—
41#14523249—24no—
42#14523248—28no—
43#14523248—22no—
44#14523248—22no—
45C2×Dic254D411yescharacteristic
46C2×D508—28no—
47C2×D508—22no—
48C2×D508—24no—
49C2×D508—21no—
50C2×D508—22no—
51C2×D508—22no—
52C2×D508—22no—
53C2×D508—21no—
54C2×D508D411yescharacteristic
55C2×D508D414yescomplemented
56C2×D508C2312yes—
57C2×D508C2312yes—
58C2×C22⋊D127—22no—
59C2×C22⋊D127D412yes—
60D4×D127—216no—
61D4×D127—24no—
62D4×D127—24no—
63C2×C4×D127—22no—
64C2×C4×D127C2312yes—
65#14523235—24no—
66D1016—28no—
67#14523233—24no—
68#14523232—22no—
69C2×C127⋊C8C2312yes—
70(C2×D127)⋊C4—24no—
71#41×C127—22no—
72#41×C127D414yescomplemented
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127—21no—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78C2×C8×C127—22no—
79C2×C8×C127C2312yes—
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C42×C127—21no—
83C22×D127—22no—
84C22×D127—28no—
85C22×D127—24no—
86C22×D127—22no—
87C22×D127—42no—
88C22×D127—84no—
89C22×D127—82no—
90C22×D127—82no—
91C22×D127C2×D412yes—
92Dic254—22no—
93D508—28no—
94D508—22no—
95D508—24no—
96D508—22no—
97D508—22no—
98D508—22no—
99D508—42no—
100D508—84no—
101D508C2×D414yes—
102C22⋊D127—28no—
103C22⋊D127—24no—
104C2×Dic127C2×D412yes—
105C4×D127—28no—
106C4×D127—22no—
107C4×D127—22no—
108C127⋊C8—22no—
109C23×C127—21no—
110C23×C127C2×D411yescharacteristic
111Q8×C127—21no—
112Q8×C127C2×D412yes—
113D4×C127—24no—
114D4×C127—21no—
115D4×C127C2×D412yes—
116C2×C4×C127—21no—
117C2×C4×C127—24no—
118C2×C4×C127—22no—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2411yescharacteristic
123C8×C127—22no—
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—42no—
127C22×C127#12012yes—
128C22×C127C22×D411yescharacteristic
129C2×D127—22no—
130C2×D127—24no—
131C2×D127—24no—
132C2×D127—44no—
133C2×D127—42no—
134C2×D127—44no—
135C2×D127—84no—
136C2×D127—88no—
137C2×D127—84no—
138C4×C127—22no—
139C4×C127—42no—
140C4×C127#12012yes—
141C4×C127C22×D411yescharacteristic
142C4×C127C22×D411yescharacteristic
143Dic127—22no—
144C2×C127—22no—
145C2×C127—42no—
146C2×C127C2×#12011yescharacteristic
147C2×C127#57612yes—
148D127—42no—
149D127—44no—
150D127—82no—
151#3008—1271nomaximal, Sylow
152C127#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—1271no—
154C2×AGL(1,ℤ/8)—1271no—
155C22×D8—1271no—
156#535—1271no—
157#449—1274no—
158#447—1274no—
159#418—1271no—
160#416D12711yescharacteristic, complemented
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—1272no—
169C22×D4—1271no—
170C22×D4—1271no—
171C22×D4—2541no—
172C22×D4—2542no—
173AGL(1,ℤ/8)—2544no—
174C2×SD16—1272no—
175C2×D8—1272no—
176C2×D8—1274no—
177C2×D8—2544no—
178C2×M16—1271no—
179C22×C8C2×D12711yescharacteristic, complemented
180#127—2541no—
181#124—2541no—
182#121—1274no—
183#120—2542no—
184#117C2×D12711yescharacteristic, complemented
185#103C2×D12712yescomplemented
186#102—1274no—
187#102C2×D12712yescomplemented
188C22⋊C8—1274no—
189C24—2542no—
190C24—2541no—
191C24—2541no—
192#41—22no—
193#41—2548no—
194#41—2544no—
195#41—2544no—
196#41—2542no—
197#41—2542no—
198#41D50814yescomplemented
199C2×Q8—1271no—
200C2×Q8C22×D12711yescharacteristic
201C2×D4—21no—
202C2×D4—1272no—
203C2×D4—1274no—
204C2×D4—1271no—
205C2×D4—1272no—
206C2×D4—1272no—
207C2×D4—2544no—
208C2×D4—2542no—
209C2×D4—2548no—
210C2×D4—2542no—
211C2×D4—2544no—
212C2×D4—25416no—
213C2×D4—2544no—
214C2×D4—2541no—
215C2×D4—2542no—
216C2×D4—2541no—
217C2×D4—2542no—
218C2×D4—2542no—
219C2×D4C22×D12711yescharacteristic
220C22×C4—21no—
221C22×C4—1272no—
222C22×C4—2542no—
223C22×C4C22×D12711yescharacteristic
224SD16—2544no—
225D8—2548no—
226D8—2544no—
227M16—2542no—
228C2×C8—22no—
229C2×C8—1272no—
230C2×C8C22×D12712yes—
231C4⋊C4C22×D12712yes—
232C22⋊C4—21no—
233C22⋊C4—2544no—
234C42—21no—
235C23—21no—
236C23—1272no—
237C23—10162no—
238C23—10162no—
239C23—10164no—
240C23—5082no—
241C23—2548no—
242C23—2544no—
243C23—2542no—
244C23—2542no—
245C23D4×D12711yescharacteristic
246Q8—21no—
247Q8—2542no—
248Q8C2×D50812yes—
249D4—24no—
250D4—21no—
251D4—1274no—
252D4—10164no—
253D4—5082no—
254D4—2548no—
255D4—2548no—
256D4—2544no—
257D4—2542no—
258D4—2542no—
259D4—2542no—
260D4—2544no—
261D4—2542no—
262D4C2×D50812yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—1272no—
267C2×C4—2548no—
268C2×C4—2542no—
269C2×C4—2542no—
270C2×C4D4×D12712yes—
271C2×C4D4×D12711yescharacteristic
272C2×C4C2×D50812yes—
273C2×C4C23×D12711yescharacteristic
274C8—22no—
275C8—2542no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—10164no—
280C22—10164no—
281C22—10168no—
282C22—5082no—
283C22—5084no—
284C22—5084no—
285C22—2544no—
286C22—2542no—
287C22—2544no—
288C22#1686117912yes—
289C22C2×D4×D12711yescharacteristic
290C4—22no—
291C4—42no—
292C4—2542no—
293C4#1686117912yes—
294C4C2×D4×D12711yescharacteristic
295C4C22×D50811yescharacteristic
296C2—22no—
297C2—42no—
298C2—10162no—
299C2—5084no—
300C2—5082no—
301C2C2×#1686117911yescharacteristic
302C2#2255067812yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
hypercenter#416D127
hypocenterC127#3008
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2semidirect
2. C2×#16861234C2semidirect
3. C22×D1016C2semidirect
4. #22550445C2semidirect
5. #22550024C2semidirect
6. #22550011C2semidirect
7. #22549880C2semidirect
8. #22549842C2semidirect
9. #416×C127C2semidirect
12. C2×#14523249C22semidirect
17. C22×D508C22semidirect
19. C2×D4×D127C22semidirect
21. C2×#14523235C22semidirect
23. C2×D1016C22semidirect
24. C2×#14523233C22semidirect
25. C2×#14523232C22semidirect
28. #16861184C22semidirect
30. #16861129C22semidirect
31. (C2×D127)⋊C8C22semidirect
33. C22×C8×C127C22semidirect
34. #117×C127C22semidirect
35. #103×C127C22semidirect
36. #102×C127C22semidirect
55. C2×D508D4semidirect
72. #41×C127D4semidirect
152. C127#3008semidirect
160. #416D127semidirect
179. C22×C8C2×D127semidirect
184. #117C2×D127semidirect
185. #103C2×D127semidirect
187. #102C2×D127semidirect
198. #41D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612342
C22×D10162
#225504452
#225500242
#225500112
#225498802
#225498422
#416×C1272
#3008127