Finite Group Databasebeta

← #30650213

Subgroups — #30650213

Order 16256.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861234C211yesmaximal, characteristic, complemented
2C2×D8×D127C211yesmaximal, characteristic, complemented
3#22550514C211yesmaximal, characteristic, complemented
4#22550510C211yesmaximal, characteristic, complemented
5#22550262C211yesmaximal, characteristic, complemented
6#22550127C211yesmaximal, characteristic, complemented
7#22550108C211yesmaximal, characteristic, complemented
8#22550094C211yesmaximal, characteristic, complemented
9#22550069C211yesmaximal, characteristic, complemented
10#22550025C211yesmaximal, characteristic, complemented
11#22550012C211yesmaximal, characteristic, complemented
12#22550011C211yesmaximal, characteristic, complemented
13#22549985C211yesmaximal, characteristic, complemented
14#22549950C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic, complemented
17C2×D4×D127—21no—
18C2×D4×D127C2211yescharacteristic
19C2×D4×D127C2211yescharacteristic, complemented
20#16861234—24no—
21C2×#14523235C2211yescharacteristic, complemented
22C2×#14523235C2211yescharacteristic, complemented
23C2×D1016C2211yescharacteristic
24C2×D1016C2211yescharacteristic, complemented
25C2×#14523233C2211yescharacteristic
26C2×#14523233C2211yescharacteristic, complemented
27C2×#14523233C2211yescharacteristic, complemented
28D8×D127—24no—
29C2×#14523232C2211yescharacteristic, complemented
30C2×C8×D127C2211yescharacteristic, complemented
31#16861205C2211yescharacteristic, complemented
32#16861198C2211yescharacteristic, complemented
33#16861192—21no—
34#16861185C2211yescharacteristic, complemented
35#16861184C2211yescharacteristic
36#16861183C2211yescharacteristic, complemented
37#16861182—21no—
38#16861181—21no—
39#16861179—21no—
40#16861177—21no—
41#16861177—21no—
42D4×Dic127C2211yescharacteristic, complemented
43#16861171C2211yescharacteristic, complemented
44C4×C22⋊D127C2211yescharacteristic
45C2×(C2×D127)⋊C4—21no—
46#16861140C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic, complemented
50#16861130C2211yescharacteristic, complemented
51#16861129C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861119C2211yescharacteristic, complemented
54C2×#41×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic, complemented
56C2×D8×C127C2211yescharacteristic, complemented
57#121×C127C2211yescharacteristic, complemented
58#103×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic, complemented
60C22⋊C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63C23×D127—21no—
64C2×D508—22no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508—41no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127D411yescharacteristic
77D4×D127—22no—
78D4×D127—22no—
79D4×D127—41no—
80D4×D127—41no—
81D4×D127—44no—
82C22×Dic127—21no—
83C2×C4×D127—21no—
84C2×C4×D127C2311yescharacteristic
85#14523235—22no—
86#14523235—22no—
87D1016—22no—
88D1016—41no—
89#14523233—22no—
90#14523233—22no—
91#14523233D414yescomplemented
92#14523232—22no—
93C2×C127⋊C8C2311yescharacteristic
94C2×C127⋊C8C2311yescharacteristic
95C8×D127—22no—
96C4⋊Dic127C2311yescharacteristic
97Dic127⋊C4—21no—
98Dic127⋊C4D411yescharacteristic
99C22⋊Dic127—21no—
100C22⋊Dic127D411yescharacteristic
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108C4×Dic127C2311yescharacteristic
109#41×C127—22no—
110#41×C127—41no—
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127—21no—
114C2×D4×C127C2311yescharacteristic
115C2×D4×C127C2311yescharacteristic
116C22×C4×C127—21no—
117C22×C4×C127C2311yescharacteristic
118SD16×C127—41no—
119D8×C127—22no—
120C2×C8×C127C2311yescharacteristic
121C2×C8×C127C2311yescharacteristic
122C4⋊C4×C127C2311yescharacteristic
123C22⋊C4×C127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—42no—
131C22×D127—42no—
132C22×D127—42no—
133C22×D127—81no—
134C22×D127—81no—
135C22×D127—81no—
136C22×D127C2×D411yescharacteristic
137D508—21no—
138D508—21no—
139D508—21no—
140D508—21no—
141D508—21no—
142D508—42no—
143D508—42no—
144D508C2×D412yes—
145C22⋊D127—22no—
146C22⋊D127—42no—
147C22⋊D127—42no—
148C22⋊D127—42no—
149C22⋊D127—81no—
150C22⋊D127—81no—
151C2×Dic127—21no—
152C2×Dic127—81no—
153C2×Dic127C2×D411yescharacteristic
154C2×Dic127C2×D411yescharacteristic
155C2×Dic127C2×D411yescharacteristic
156C4×D127—21no—
157C4×D127—21no—
158C4×D127—42no—
159C127⋊C8—21no—
160C127⋊C8C2×D412yescomplemented
161C23×C127—21no—
162C23×C127—21no—
163C23×C127C2×D411yescharacteristic
164Q8×C127—21no—
165Q8×C127—21no—
166D4×C127—21no—
167D4×C127—21no—
168D4×C127—21no—
169D4×C127—42no—
170D4×C127—42no—
171D4×C127C2×D412yescomplemented
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—42no—
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127—21no—
181C22×C127—21no—
182C22×C127—22no—
183C22×C127—21no—
184C22×C127—21no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127C22×D411yescharacteristic
191C2×D127—21no—
192C2×D127—21no—
193C2×D127—22no—
194C2×D127—41no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—41no—
198C2×D127—41no—
199C2×D127—41no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—84no—
203C2×D127—84no—
204C2×D127—84no—
205C2×D127AGL(1,ℤ/8)12yescomplemented
206C4×C127—21no—
207C4×C127—41no—
208C4×C127—41no—
209C4×C127C22×D411yescharacteristic
210C4×C127H(5,𝔽2)11yescharacteristic
211Dic127—21no—
212Dic127—21no—
213Dic127—21no—
214Dic127—41no—
215C2×C127—41no—
216C2×C127—42no—
217C2×C127—81no—
218C2×C127D4211yescharacteristic
219C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C127#57611yescharacteristic
221D127—22no—
222D127—42no—
223D127—81no—
224D127—81no—
225#3264—1271nomaximal, Sylow
226C127#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1271no—
228C22×D8—1271no—
229#546—1271no—
230D42—1271no—
231#496—1271no—
232#469—1271no—
233#465—1271no—
234#463—1271no—
235#459—1271no—
236#450—1271no—
237#449D12711yescharacteristic, complemented
238#447—1271no—
239#447—1271no—
240#442—1271no—
241#435—1271no—
242C2×#41C2×D12711yescharacteristic, complemented
243C22×D4—1271no—
244C22×D4—1271no—
245C22×D4—1271no—
246C22×D4—2541no—
247AGL(1,ℤ/8)—2544no—
248C2×SD16—1271no—
249C2×SD16—1271no—
250C2×SD16C2×D12711yescharacteristic, complemented
251C2×D8—1271no—
252C2×D8—1271no—
253C2×D8—1271no—
254C2×D8—1271no—
255C2×D8—1271no—
256C2×D8—2544no—
257C2×D8C2×D12711yescharacteristic, complemented
258C2×M16—1271no—
259C22×C8—1271no—
260#127—1271no—
261#124—1271no—
262#123—2541no—
263C22⋊Q8—1271no—
264#121—1271no—
265#121—1271no—
266#121—2541no—
267#121—2541no—
268#121C2×D12711yescharacteristic, complemented
269#120—2541no—
270#120—2541no—
271#120—2541no—
272C4×D4—1271no—
273C4×D4—1271no—
274C4×D4—1271no—
275C2×C22⋊C4—2541no—
276#106—1271no—
277C4⋊C8—1271no—
278#103C2×D12711yescharacteristic, complemented
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102—1271no—
283#102C2×D12711yescharacteristic, complemented
284C22⋊C8—1271no—
285C22⋊C8C2×D12711yescharacteristic, complemented
286#97—1271no—
287C24—2541no—
288C24—2541no—
289C24—2541no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D12711yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1271no—
296C2×D4—1271no—
297C2×D4—1271no—
298C2×D4—1271no—
299C2×D4—5081no—
300C2×D4—5084no—
301C2×D4—5081no—
302C2×D4—5081no—
303C2×D4—2542no—
304C2×D4—2541no—
305C2×D4—2541no—
306C2×D4—2541no—
307C2×D4—2541no—
308C2×D4—2541no—
309C2×D4—2542no—
310C2×D4—2541no—
311C2×D4—2542no—
312C2×D4—2541no—
313C2×D4C22×D12711yescharacteristic
314C2×D4C22×D12711yescharacteristic
315C22×C4—21no—
316C22×C4—1271no—
317C22×C4—2541no—
318C22×C4—2541no—
319C22×C4C22×D12711yescharacteristic
320SD16—41no—
321SD16—2542no—
322SD16—2542no—
323D8—22no—
324D8—1274no—
325D8—5081no—
326D8—2542no—
327D8—2542no—
328D8—2542no—
329M16—2542no—
330C2×C8—1271no—
331C2×C8—1271no—
332C2×C8—2542no—
333C2×C8C22×D12711yescharacteristic
334C2×C8C22×D12711yescharacteristic
335C4⋊C4—1271no—
336C4⋊C4—1271no—
337C4⋊C4—2541no—
338C4⋊C4C22×D12711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—1271no—
341C22⋊C4—1271no—
342C22⋊C4—2541no—
343C22⋊C4—2541no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C22⋊C4—2541no—
348C22⋊C4—2541no—
349C42—1271no—
350C23—21no—
351C23—21no—
352C23—1271no—
353C23—10161no—
354C23—10161no—
355C23—10161no—
356C23—5082no—
357C23—5082no—
358C23—5082no—
359C23—5082no—
360C23—5082no—
361C23—5082no—
362C23—2541no—
363C23—2541no—
364C23—2541no—
365C23D4×D12711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—42no—
373D4—1272no—
374D4—10161no—
375D4—10161no—
376D4—5082no—
377D4—5082no—
378D4—5082no—
379D4—5082no—
380D4—5082no—
381D4—2542no—
382D4—2541no—
383D4—2541no—
384D4—2541no—
385D4—2541no—
386D4—2541no—
387D4D4×D12712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—1271no—
394C2×C4—1271no—
395C2×C4—1271no—
396C2×C4—10161no—
397C2×C4—5082no—
398C2×C4—2541no—
399C2×C4—2541no—
400C2×C4—2541no—
401C2×C4D4×D12711yescharacteristic
402C2×C4C23×D12711yescharacteristic
403C8—21no—
404C8—21no—
405C8—1272no—
406C8—2541no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—1272no—
417C22—10164no—
418C22—10164no—
419C22—10164no—
420C22—5081no—
421C22—5081no—
422C22—5081no—
423C22—5081no—
424C22—5081no—
425C22—5081no—
426C22—5081no—
427C22—5081no—
428C22—2542no—
429C22—2541no—
430C22—2541no—
431C22C2×D4×D12711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—5081no—
436C4—2541no—
437C4—2541no—
438C4—2541no—
439C4C2×D4×D12711yescharacteristic
440C4#1686125411yescharacteristic
441C2—41no—
442C2—42no—
443C2—81no—
444C2—10161no—
445C2—10161no—
446C2—5082no—
447C2—2542no—
448C2#2255050811yescharacteristic
449C2AGL(1,ℤ/8)×D12711yescharacteristic
450C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×D8×D127C2semidirect
3. #22550514C2semidirect
4. #22550510C2semidirect
5. #22550262C2semidirect
6. #22550127C2semidirect
7. #22550108C2semidirect
8. #22550094C2semidirect
9. #22550069C2semidirect
10. #22550025C2semidirect
11. #22550012C2semidirect
12. #22550011C2semidirect
13. #22549985C2semidirect
14. #22549950C2semidirect
15. #449×C127C2semidirect
16. C22×D508C22semidirect
19. C2×D4×D127C22semidirect
21. C2×#14523235C22semidirect
22. C2×#14523235C22semidirect
24. C2×D1016C22semidirect
26. C2×#14523233C22semidirect
27. C2×#14523233C22semidirect
29. C2×#14523232C22semidirect
30. C2×C8×D127C22semidirect
31. #16861205C22semidirect
32. #16861198C22semidirect
34. #16861185C22semidirect
36. #16861183C22semidirect
42. D4×Dic127C22semidirect
43. #16861171C22semidirect
46. #16861140C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861131C22semidirect
49. #16861131C22semidirect
50. #16861130C22semidirect
51. #16861129C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861119C22semidirect
54. C2×#41×C127C22semidirect
55. C2×SD16×C127C22semidirect
56. C2×D8×C127C22semidirect
57. #121×C127C22semidirect
58. #103×C127C22semidirect
59. #102×C127C22semidirect
60. C22⋊C8×C127C22semidirect
91. #14523233D4semidirect
160. C127⋊C8C2×D4semidirect
171. D4×C127C2×D4semidirect
205. C2×D127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
237. #449D127semidirect
242. C2×#41C2×D127semidirect
250. C2×SD16C2×D127semidirect
257. C2×D8C2×D127semidirect
268. #121C2×D127semidirect
278. #103C2×D127semidirect
283. #102C2×D127semidirect
285. C22⋊C8C2×D127semidirect
387. D4D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×D8×D1272
#225505142
#225505102
#225502622
#225501272
#225501082
#225500942
#225500692
#225500252
#225500122
#225500112
#225499852
#225499502
#449×C1272
#3264127