Finite Group Databasebeta

← #30647958

Subgroups — #30647958

Order 16256.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×D4×D127C211yesmaximal, characteristic, complemented
2C2×#16861236C212yesmaximal, complemented
3#22550389C211yesmaximal, characteristic, complemented
4#22550026C211yesmaximal, characteristic, complemented
5#22550025C212yesmaximal, complemented
6#22550012C211yesmaximal, characteristic, complemented
7#22550011C212yesmaximal, complemented
8#22550010C211yesmaximal, characteristic, complemented
9#22549879C212yesmaximal, complemented
10#22549838C211yesmaximal, characteristic, complemented
11#450×C127C211yesmaximal, characteristic, complemented
12C24×D127—21no—
13C2×#14523249C2211yescharacteristic, complemented
14C22×D508C2211yescharacteristic
15C22×C22⋊D127—21no—
16C2×D4×D127—28no—
17C2×D4×D127—21no—
18C2×D4×D127—21no—
19C2×D4×D127C2212yes—
20C2×D4×D127C2212yes—
21C22×C4×D127C2211yescharacteristic
22#16861236—28no—
23C2×#14523235C2212yescomplemented
24C2×D1016C2212yescomplemented
25C2×#14523233C2212yescomplemented
26C2×#14523231C2212yescomplemented
27#16861194—21no—
28#16861185C2211yescharacteristic, complemented
29#16861184C2211yescharacteristic, complemented
30#16861183C2211yescharacteristic, complemented
31#16861179—21no—
32#16861171—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167C2211yescharacteristic, complemented
35#16861131C2212yescomplemented
36#16861130C2212yescomplemented
37#16861129C2212yescomplemented
38C22⋊(C127⋊C8)C2211yescharacteristic, complemented
39(C2×D127)⋊C8C2212yescomplemented
40C22×D4×C127C2211yescharacteristic
41C2×SD16×C127C2212yescomplemented
42C22⋊Q8×C127C2211yescharacteristic, complemented
43#102×C127C2212yescomplemented
44C22⋊C8×C127C2211yescharacteristic, complemented
45C23×D127—28no—
46C23×D127—22no—
47C23×D127—22no—
48C23×D127—21no—
49C23×D127—44no—
50C23×D127—44no—
51C23×D127D411yescharacteristic
52#14523249—24no—
53C2×D508—24no—
54C2×D508—21no—
55C2×D508—21no—
56C2×D508—21no—
57C2×D508C2312yes—
58C2×D508C2311yescharacteristic, complemented
59C2×C22⋊D127—28no—
60C2×C22⋊D127—22no—
61C2×C22⋊D127—22no—
62C2×C22⋊D127—21no—
63D4×D127—28no—
64D4×D127—28no—
65D4×D127—24no—
66D4×D127—24no—
67D4×D127—24no—
68D4×D127D418yescomplemented
69C22×Dic127D411yescharacteristic
70C2×C4×D127—24no—
71C2×C4×D127—21no—
72C2×C4×D127D411yescharacteristic
73C2×C4×D127D411yescharacteristic
74C2×C4×D127C2311yescharacteristic
75C2×C4×D127C2311yescharacteristic
76#14523235—24no—
77D1016—24no—
78#14523233—24no—
79#14523231—24no—
80C2×C127⋊C8C2312yescomplemented
81C4⋊Dic127C2311yescharacteristic, complemented
82Dic127⋊C4—21no—
83(C2×D127)⋊C4—21no—
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87C4×Dic127—21no—
88C24×C127—21no—
89C2×Q8×C127C2311yescharacteristic, complemented
90C2×D4×C127—24no—
91C2×D4×C127—21no—
92C2×D4×C127C2312yes—
93C22×C4×C127C2311yescharacteristic
94SD16×C127—24no—
95C2×C8×C127C2312yescomplemented
96C4⋊C4×C127—21no—
97C4⋊C4×C127C2311yescharacteristic, complemented
98C22⋊C4×C127—21no—
99C22×D127—22no—
100C22×D127—22no—
101C22×D127—22no—
102C22×D127—28no—
103C22×D127—24no—
104C22×D127—22no—
105C22×D127—28no—
106C22×D127—21no—
107C22×D127—432no—
108C22×D127—44no—
109C22×D127—44no—
110C22×D127—44no—
111C22×D127—42no—
112C22×D127—42no—
113C22×D127—44no—
114C22×D127C2×D411yescharacteristic
115C22×D127C2×D411yescharacteristic
116D508—24no—
117D508—22no—
118D508—22no—
119D508—42no—
120D508—42no—
121D508C2×D414yes—
122C22⋊D127—28no—
123C22⋊D127—28no—
124C22⋊D127—42no—
125C2×Dic127—22no—
126C2×Dic127—21no—
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—22no—
130C4×D127—22no—
131C4×D127—22no—
132C4×D127—42no—
133C4×D127C2×D414yes—
134C127⋊C8—22no—
135C23×C127—24no—
136C23×C127—22no—
137C23×C127—42no—
138C23×C127—42no—
139C23×C127C2×D411yescharacteristic
140Q8×C127—22no—
141D4×C127—24no—
142D4×C127—22no—
143D4×C127C2×D414yes—
144C2×C4×C127—22no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2411yescharacteristic
149C8×C127—22no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127—24no—
153C22×C127—48no—
154C22×C127—42no—
155C22×C127—42no—
156C22×C127AGL(1,ℤ/8)12yescomplemented
157C22×C127C22×D411yescharacteristic
158C2×D127—28no—
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—21no—
162C2×D127—21no—
163C2×D127—24no—
164C2×D127—416no—
165C2×D127—416no—
166C2×D127—48no—
167C2×D127—42no—
168C2×D127—42no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127#12012yes—
173C2×D127AGL(1,ℤ/8)12yescomplemented
174C4×C127—21no—
175C4×C127—41no—
176C4×C127—41no—
177C4×C127C22×D412yes—
178Dic127—21no—
179Dic127—41no—
180Dic127#12012yes—
181C2×C127—22no—
182C2×C127—44no—
183C2×C127C2×#12011yescharacteristic
184C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
186D127—22no—
187D127—22no—
188D127—44no—
189D127—81no—
190#3003—1271nomaximal, Sylow
191C127#300311yescharacteristic, complemented, Sylow
192C23×D4—1271no—
193C2×AGL(1,ℤ/8)—1272no—
194#525—1271no—
195#450—1272no—
196#450—1271no—
197#450D12711yescharacteristic, complemented
198#447—1271no—
199#447—1272no—
200#447—1271no—
201#418—1272no—
202#407—1271no—
203C25—2541no—
204C2×#41—1271no—
205C22×D4—1272no—
206C22×D4—1272no—
207C22×D4—1271no—
208C22×D4—2541no—
209C22×D4—2541no—
210C22×D4—2541no—
211C22×D4—2548no—
212C22×D4C2×D12711yescharacteristic
213C23×C4—1271no—
214AGL(1,ℤ/8)—2548no—
215C2×SD16—1272no—
216C2×SD16C2×D12712yescomplemented
217C2×D8—1272no—
218C2×D8—1272no—
219C2×M16—1272no—
220#123—2541no—
221C22⋊Q8—1271no—
222C22⋊Q8C2×D12711yescharacteristic, complemented
223#121—1271no—
224#121—1271no—
225#120—2541no—
226C4×D4—2541no—
227C4×D4—2541no—
228#117—1271no—
229#102—1272no—
230#102—1272no—
231#102—1272no—
232#102C2×D12712yescomplemented
233C22⋊C8—1271no—
234C22⋊C8—1272no—
235C22⋊C8C2×D12711yescharacteristic, complemented
236C24—21no—
237C24—1271no—
238C24—5084no—
239C24—5084no—
240C24—2542no—
241C24—2542no—
242C24—2548no—
243C24—2541no—
244#41—2544no—
245C2×Q8C22×D12711yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—1278no—
249C2×D4—1272no—
250C2×D4—1271no—
251C2×D4—2544no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4—2544no—
255C2×D4—2542no—
256C2×D4—2542no—
257C2×D4—2544no—
258C2×D4—2548no—
259C2×D4—2548no—
260C2×D4—2541no—
261C2×D4—2548no—
262C2×D4—2544no—
263C2×D4—2541no—
264C2×D4C22×D12712yes—
265C22×C4—1271no—
266C22×C4—1271no—
267C22×C4—1271no—
268C22×C4—1271no—
269C22×C4—1271no—
270C22×C4—2541no—
271C22×C4—2544no—
272C22×C4C22×D12711yescharacteristic
273SD16—24no—
274SD16—2544no—
275D8—2544no—
276D8—2544no—
277M16—2544no—
278C2×C8—1272no—
279C2×C8C22×D12712yescomplemented
280C4⋊C4—21no—
281C4⋊C4—1271no—
282C4⋊C4—2541no—
283C4⋊C4C22×D12711yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C42—2541no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—1271no—
295C23—1271no—
296C23—5082no—
297C23—5084no—
298C23—50832no—
299C23—5084no—
300C23—5084no—
301C23—5084no—
302C23—5082no—
303C23—2548no—
304C23—2542no—
305C23—2542no—
306C23—2544no—
307C23—2548no—
308C23—2541no—
309C23—2542no—
310C23—2542no—
311C23D4×D12711yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—1274no—
316D4—5082no—
317D4—5082no—
318D4—5082no—
319D4—2544no—
320D4—2548no—
321D4—2548no—
322D4—2542no—
323D4—2542no—
324D4D4×D12714yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—1271no—
329C2×C4—1274no—
330C2×C4—1271no—
331C2×C4—5082no—
332C2×C4—2542no—
333C2×C4—2542no—
334C2×C4—2542no—
335C2×C4—2541no—
336C2×C4—2542no—
337C2×C4D4×D12711yescharacteristic
338C2×C4C23×D12711yescharacteristic
339C8—22no—
340C8—2542no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—1272no—
348C22—1272no—
349C22—5088no—
350C22—50816no—
351C22—50816no—
352C22—5081no—
353C22—5081no—
354C22—5081no—
355C22—5082no—
356C22—5082no—
357C22—2548no—
358C22—2544no—
359C22—2541no—
360C22—2541no—
361C22—2541no—
362C22—2541no—
363C22#1686123612yescomplemented
364C22C2×D4×D12711yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—1272no—
369C4—5081no—
370C4—2541no—
371C4C2×D4×D12712yes—
372C2—22no—
373C2—44no—
374C2—10161no—
375C2—5084no—
376C2—2542no—
377C2—2542no—
378C2#120×D12711yescharacteristic
379C2AGL(1,ℤ/8)×D12711yescharacteristic
380C2C2×#1686123611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
hypercenter#450D127
hypocenterC127#3003
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D127C2semidirect
2. C2×#16861236C2semidirect
3. #22550389C2semidirect
4. #22550026C2semidirect
5. #22550025C2semidirect
6. #22550012C2semidirect
7. #22550011C2semidirect
8. #22550010C2semidirect
9. #22549879C2semidirect
10. #22549838C2semidirect
11. #450×C127C2semidirect
13. C2×#14523249C22semidirect
23. C2×#14523235C22semidirect
24. C2×D1016C22semidirect
25. C2×#14523233C22semidirect
26. C2×#14523231C22semidirect
28. #16861185C22semidirect
29. #16861184C22semidirect
30. #16861183C22semidirect
34. #16861167C22semidirect
35. #16861131C22semidirect
36. #16861130C22semidirect
37. #16861129C22semidirect
38. C22⋊(C127⋊C8)C22semidirect
39. (C2×D127)⋊C8C22semidirect
41. C2×SD16×C127C22semidirect
42. C22⋊Q8×C127C22semidirect
43. #102×C127C22semidirect
44. C22⋊C8×C127C22semidirect
58. C2×D508C23semidirect
68. D4×D127D4semidirect
80. C2×C127⋊C8C23semidirect
81. C4⋊Dic127C23semidirect
89. C2×Q8×C127C23semidirect
95. C2×C8×C127C23semidirect
97. C4⋊C4×C127C23semidirect
156. C22×C127AGL(1,ℤ/8)semidirect
173. C2×D127AGL(1,ℤ/8)semidirect
191. C127#3003semidirect
197. #450D127semidirect
216. C2×SD16C2×D127semidirect
222. C22⋊Q8C2×D127semidirect
232. #102C2×D127semidirect
235. C22⋊C8C2×D127semidirect
245. C2×Q8C22×D127semidirect
279. C2×C8C22×D127semidirect
283. C4⋊C4C22×D127semidirect
363. C22#16861236semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C2×#168612362
#225503892
#225500262
#225500252
#225500122
#225500112
#225500102
#225498792
#225498382
#450×C1272
#3003127