Finite Group Databasebeta

← #30650211

Subgroups — #30650211

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
2C22×#14523233C211yesmaximal, characteristic, complemented
3D4×D508C211yesmaximal, characteristic, complemented
4#22550266C211yesmaximal, characteristic, complemented
5#22550127C211yesmaximal, characteristic, complemented
6#22550107C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550071C211yesmaximal, characteristic, complemented
9#22550026C211yesmaximal, characteristic, complemented
10#22550021C211yesmaximal, characteristic, complemented
11#22550010C211yesmaximal, characteristic, complemented
12#22550009C211yesmaximal, characteristic, complemented
13#22549986C211yesmaximal, characteristic, complemented
14#22549948C211yesmaximal, characteristic, complemented
15#546×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C22×D508C2211yescharacteristic, complemented
18C2×D4×D127—21no—
19#16861234—24no—
20C2×#14523235C2211yescharacteristic, complemented
21C2×#14523235C2211yescharacteristic, complemented
22C2×#14523235C2211yescharacteristic, complemented
23C2×#14523233—24no—
24C2×#14523233C2211yescharacteristic
25C2×#14523233C2211yescharacteristic
26C2×#14523233C2211yescharacteristic, complemented
27C2×#14523233C2211yescharacteristic, complemented
28C2×#14523233C2211yescharacteristic, complemented
29C2×#14523233C2211yescharacteristic, complemented
30C2×#14523232C2211yescharacteristic, complemented
31C22×C127⋊C8C2211yescharacteristic, complemented
32#16861206C2211yescharacteristic, complemented
33#16861184—21no—
34#16861183C2211yescharacteristic
35#16861183C2211yescharacteristic, complemented
36#16861179—21no—
37#16861179—21no—
38C4×D508C2211yescharacteristic, complemented
39#16861140C2211yescharacteristic, complemented
40C4⋊(C127⋊C8)C2211yescharacteristic, complemented
41#16861133C2211yescharacteristic, complemented
42#16861130C2211yescharacteristic, complemented
43#16861130C2211yescharacteristic, complemented
44#16861129C2211yescharacteristic, complemented
45#16861129C2211yescharacteristic, complemented
46#16861129C2211yescharacteristic, complemented
47C22⋊(C127⋊C8)C2211yescharacteristic, complemented
48C22⋊(C127⋊C8)C2211yescharacteristic, complemented
49#16861120C2211yescharacteristic, complemented
50C2×#41×C127C2211yescharacteristic, complemented
51C22×D4×C127C2211yescharacteristic, complemented
52#124×C127C2211yescharacteristic, complemented
53#123×C127—21no—
54C22⋊Q8×C127C2211yescharacteristic, complemented
55#121×C127—21no—
56#121×C127C2211yescharacteristic, complemented
57#120×C127—21no—
58C4×D4×C127C2211yescharacteristic
59C4×D4×C127C2211yescharacteristic, complemented
60C2×C22⋊C4×C127—21no—
61C23×D127—21no—
62C23×D127—21no—
63C2×D508—22no—
64C2×D508—22no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508—21no—
68C2×D508—21no—
69C2×D508—41no—
70C2×D508—41no—
71C2×D508C2311yescharacteristic
72C2×D508C2311yescharacteristic
73C2×D508C2311yescharacteristic
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76D4×D127—44no—
77C2×C4×D127—21no—
78#14523235—22no—
79#14523235—22no—
80#14523235—41no—
81#14523233—22no—
82#14523233—22no—
83#14523233—22no—
84#14523233—22no—
85#14523233—41no—
86#14523233D414yescomplemented
87#14523232—22no—
88C2×C127⋊C8—22no—
89C2×C127⋊C8C2311yescharacteristic
90C2×C127⋊C8C2311yescharacteristic
91C2×C127⋊C8C2311yescharacteristic
92C2×C127⋊C8C2311yescharacteristic
93C4⋊Dic127C2311yescharacteristic
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97C24×C127—21no—
98#41×C127—22no—
99#41×C127—41no—
100C2×Q8×C127C2311yescharacteristic
101C2×D4×C127—22no—
102C2×D4×C127—21no—
103C2×D4×C127—21no—
104C2×D4×C127—21no—
105C2×D4×C127—41no—
106C2×D4×C127D411yescharacteristic
107C2×D4×C127C2311yescharacteristic
108C2×D4×C127C2311yescharacteristic
109C22×C4×C127—21no—
110C22×C4×C127—21no—
111C22×C4×C127C2311yescharacteristic
112C22×C4×C127C2311yescharacteristic
113C4⋊C4×C127—21no—
114C4⋊C4×C127D411yescharacteristic
115C4⋊C4×C127C2311yescharacteristic
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C22⋊C4×C127—21no—
121C22⋊C4×C127D411yescharacteristic
122C22⋊C4×C127D411yescharacteristic
123C42×C127C2311yescharacteristic
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—42no—
131C22×D127—81no—
132C22×D127—81no—
133D508—21no—
134D508—21no—
135D508—21no—
136D508—21no—
137D508—21no—
138D508—42no—
139D508—42no—
140D508—42no—
141D508—81no—
142D508C2×D412yes—
143C22⋊D127—42no—
144C22⋊D127—42no—
145C2×Dic127—21no—
146C4×D127—42no—
147C127⋊C8—21no—
148C127⋊C8—21no—
149C127⋊C8—21no—
150C127⋊C8C2×D412yescomplemented
151C23×C127—21no—
152C23×C127—21no—
153C23×C127—42no—
154C23×C127—42no—
155C23×C127—81no—
156C23×C127C2×D411yescharacteristic
157C23×C127C2×D411yescharacteristic
158Q8×C127—21no—
159Q8×C127—21no—
160D4×C127—22no—
161D4×C127—21no—
162D4×C127—21no—
163D4×C127—21no—
164D4×C127—42no—
165D4×C127—42no—
166D4×C127—81no—
167D4×C127C2×D412yescomplemented
168C2×C4×C127—21no—
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—42no—
175C2×C4×C127—81no—
176C2×C4×C127C2×D411yescharacteristic
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2×D411yescharacteristic
179C2×C4×C127C2×D411yescharacteristic
180C2×C4×C127C2411yescharacteristic
181C22×C127—21no—
182C22×C127—21no—
183C22×C127—21no—
184C22×C127—22no—
185C22×C127—21no—
186C22×C127—21no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127—41no—
191C22×C127—41no—
192C22×C127—84no—
193C22×C127AGL(1,ℤ/8)12yescomplemented
194C22×C127C22×D411yescharacteristic
195C2×D127—22no—
196C2×D127—41no—
197C2×D127—41no—
198C2×D127—41no—
199C2×D127—41no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—41no—
203C2×D127—41no—
204C2×D127—84no—
205C2×D127—84no—
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—21no—
209C4×C127—21no—
210C4×C127—41no—
211C4×C127—41no—
212C4×C127C22×D411yescharacteristic
213C4×C127H(5,𝔽2)11yescharacteristic
214Dic127—41no—
215C2×C127—22no—
216C2×C127—41no—
217C2×C127—42no—
218C2×C127—81no—
219C2×C127D4211yescharacteristic
220C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C127#57611yescharacteristic
222D127—42no—
223D127—81no—
224D127—81no—
225#3264—1271nomaximal, Sylow
226C127#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1271no—
228C22×D8—1271no—
229#546D12711yescharacteristic, complemented
230D42—1271no—
231#496—1271no—
232#469—1271no—
233#465—1271no—
234#463—1271no—
235#459—1271no—
236#450—1271no—
237#449—1271no—
238#447—1271no—
239#447—1271no—
240#442—1271no—
241#435—1271no—
242C2×#41C2×D12711yescharacteristic, complemented
243C22×D4—1271no—
244C22×D4—1271no—
245C22×D4—2541no—
246C22×D4C2×D12711yescharacteristic, complemented
247AGL(1,ℤ/8)—2544no—
248C2×SD16—1271no—
249C2×SD16—1271no—
250C2×SD16—1271no—
251C2×D8—1271no—
252C2×D8—1271no—
253C2×D8—1271no—
254C2×D8—1271no—
255C2×D8—1271no—
256C2×D8—1271no—
257C2×D8—2544no—
258C2×M16—1271no—
259C22×C8—1271no—
260#127—1271no—
261#124C2×D12711yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D12711yescharacteristic, complemented
264#121—21no—
265#121—1271no—
266#121—1271no—
267#121—2541no—
268#121C2×D12711yescharacteristic, complemented
269#120—21no—
270#120—2541no—
271#120—2541no—
272C4×D4—1271no—
273C4×D4C2×D12711yescharacteristic
274C4×D4C2×D12711yescharacteristic, complemented
275C2×C22⋊C4—21no—
276#106—1271no—
277C4⋊C8—1271no—
278#103—1271no—
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102—1271no—
283#102—1271no—
284C22⋊C8—1271no—
285C22⋊C8—1271no—
286#97—1271no—
287C24—21no—
288C24—2541no—
289C24—2541no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D12711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—1271no—
299C2×D4—1271no—
300C2×D4—1271no—
301C2×D4—5081no—
302C2×D4—5081no—
303C2×D4—5084no—
304C2×D4—2542no—
305C2×D4—2542no—
306C2×D4—2541no—
307C2×D4—2541no—
308C2×D4—2541no—
309C2×D4—2541no—
310C2×D4—2541no—
311C2×D4—2541no—
312C2×D4C22⋊D12711yescharacteristic
313C2×D4C22×D12711yescharacteristic
314C2×D4C22×D12711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2541no—
318C22×C4C22×D12711yescharacteristic
319C22×C4C22×D12711yescharacteristic
320SD16—5081no—
321SD16—2542no—
322SD16—2542no—
323D8—1274no—
324D8—5081no—
325D8—2542no—
326D8—2542no—
327D8—2542no—
328D8—2542no—
329M16—2542no—
330C2×C8—1271no—
331C2×C8—1271no—
332C2×C8—1271no—
333C2×C8—1271no—
334C2×C8—2542no—
335C4⋊C4—21no—
336C4⋊C4—1271no—
337C4⋊C4C22⋊D12711yescharacteristic
338C4⋊C4C22×D12711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C22⋊C4C22⋊D12711yescharacteristic
348C22⋊C4C22⋊D12711yescharacteristic
349C42C22×D12711yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—10161no—
356C23—10161no—
357C23—5082no—
358C23—5082no—
359C23—5082no—
360C23—5082no—
361C23—2541no—
362C23—2541no—
363C23—2541no—
364C23C2×C22⋊D12711yescharacteristic
365C23C2×C22⋊D12711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—1272no—
376D4—10161no—
377D4—5082no—
378D4—5082no—
379D4—5082no—
380D4—5082no—
381D4—5082no—
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—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—5082no—
397C2×C4—2541no—
398C2×C4C2×C22⋊D12711yescharacteristic
399C2×C4C2×C22⋊D12711yescharacteristic
400C2×C4C2×C22⋊D12711yescharacteristic
401C2×C4C2×C22⋊D12711yescharacteristic
402C2×C4C23×D12711yescharacteristic
403C8—1272no—
404C8—2541no—
405C8—2541no—
406C8—2541no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—10164no—
420C22—10164no—
421C22—5081no—
422C22—5081no—
423C22—5081no—
424C22—5081no—
425C22—5081no—
426C22—5081no—
427C22—5081no—
428C22—5081no—
429C22—2542no—
430C22#1686123412yescomplemented
431C22C22×C22⋊D12711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—5081no—
439C4C2×D4×D12711yescharacteristic
440C4#1686125411yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—10161no—
446C2—10161no—
447C2—5082no—
448C2D4×C22⋊D12711yescharacteristic
449C2C2×#1686123411yescharacteristic
450C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#546×C127C2
socleC22×C127C22×D4
hypercenter#546D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C22×#14523233C2semidirect
3. D4×D508C2semidirect
4. #22550266C2semidirect
5. #22550127C2semidirect
6. #22550107C2semidirect
7. #22550088C2semidirect
8. #22550071C2semidirect
9. #22550026C2semidirect
10. #22550021C2semidirect
11. #22550010C2semidirect
12. #22550009C2semidirect
13. #22549986C2semidirect
14. #22549948C2semidirect
15. #546×C127C2semidirect
17. C22×D508C22semidirect
20. C2×#14523235C22semidirect
21. C2×#14523235C22semidirect
22. C2×#14523235C22semidirect
26. C2×#14523233C22semidirect
27. C2×#14523233C22semidirect
28. C2×#14523233C22semidirect
29. C2×#14523233C22semidirect
30. C2×#14523232C22semidirect
31. C22×C127⋊C8C22semidirect
32. #16861206C22semidirect
35. #16861183C22semidirect
38. C4×D508C22semidirect
39. #16861140C22semidirect
40. C4⋊(C127⋊C8)C22semidirect
41. #16861133C22semidirect
42. #16861130C22semidirect
43. #16861130C22semidirect
44. #16861129C22semidirect
45. #16861129C22semidirect
46. #16861129C22semidirect
47. C22⋊(C127⋊C8)C22semidirect
48. C22⋊(C127⋊C8)C22semidirect
49. #16861120C22semidirect
50. C2×#41×C127C22semidirect
51. C22×D4×C127C22semidirect
52. #124×C127C22semidirect
54. C22⋊Q8×C127C22semidirect
56. #121×C127C22semidirect
59. C4×D4×C127C22semidirect
86. #14523233D4semidirect
150. C127⋊C8C2×D4semidirect
167. D4×C127C2×D4semidirect
193. C22×C127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
229. #546D127semidirect
242. C2×#41C2×D127semidirect
246. C22×D4C2×D127semidirect
261. #124C2×D127semidirect
263. C22⋊Q8C2×D127semidirect
268. #121C2×D127semidirect
274. C4×D4C2×D127semidirect
387. D4D4×D127semidirect
430. C22#16861234semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C22×#145232332
D4×D5082
#225502662
#225501272
#225501072
#225500882
#225500712
#225500262
#225500212
#225500102
#225500092
#225499862
#225499482
#546×C1272
#3264127