Finite Group Databasebeta

← #30650206

Subgroups — #30650206

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×#16861235C211yesmaximal, characteristic, complemented
2C22×#14523233C211yesmaximal, characteristic, complemented
3#22550522C211yesmaximal, characteristic, complemented
4#22550263C211yesmaximal, characteristic, complemented
5#22550127C211yesmaximal, characteristic, complemented
6#22550103C211yesmaximal, characteristic, complemented
7#22550091C211yesmaximal, characteristic, complemented
8#22550068C211yesmaximal, characteristic, complemented
9#22550027C211yesmaximal, characteristic, complemented
10#22550019C211yesmaximal, characteristic, complemented
11#22550010C211yesmaximal, characteristic, complemented
12#22550009C211yesmaximal, characteristic, complemented
13#22549986C211yesmaximal, characteristic, complemented
14#22549948C211yesmaximal, characteristic, complemented
15D42×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic, complemented
17C22×D508C2211yescharacteristic, complemented
18#16861235—24no—
19C2×#14523236C2211yescharacteristic, complemented
20C2×#14523236C2211yescharacteristic, complemented
21C2×#14523236C2211yescharacteristic, complemented
22C2×#14523233—24no—
23C2×#14523233C2211yescharacteristic
24C2×#14523233C2211yescharacteristic
25C2×#14523233C2211yescharacteristic, complemented
26C2×#14523233C2211yescharacteristic, complemented
27C2×#14523233C2211yescharacteristic, complemented
28C2×#14523233C2211yescharacteristic, complemented
29C2×#14523232C2211yescharacteristic, complemented
30C22×C127⋊C8C2211yescharacteristic, complemented
31#16861197C2211yescharacteristic, complemented
32#16861194—21no—
33#16861188C2211yescharacteristic, complemented
34#16861183C2211yescharacteristic, complemented
35#16861182—21no—
36#16861177—21no—
37C4×D508C2211yescharacteristic, complemented
38C22⋊C4×D127—21no—
39#16861140C2211yescharacteristic, complemented
40C4⋊(C127⋊C8)C2211yescharacteristic, complemented
41#16861132C2211yescharacteristic, complemented
42#16861130C2211yescharacteristic, complemented
43#16861130C2211yescharacteristic, complemented
44#16861130C2211yescharacteristic, complemented
45#16861129C2211yescharacteristic, complemented
46#16861129C2211yescharacteristic, complemented
47C22⋊(C127⋊C8)C2211yescharacteristic, complemented
48C22⋊(C127⋊C8)C2211yescharacteristic, complemented
49#16861120C2211yescharacteristic, complemented
50C22×D4×C127—21no—
51C22×D4×C127C2211yescharacteristic
52C22×D4×C127C2211yescharacteristic, complemented
53#127×C127C2211yescharacteristic, complemented
54#121×C127—21no—
55#121×C127C2211yescharacteristic
56#121×C127C2211yescharacteristic, complemented
57#120×C127—21no—
58#120×C127—21no—
59C4×D4×C127C2211yescharacteristic
60C4×D4×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523248—22no—
63#14523248—41no—
64C2×Dic254C2311yescharacteristic
65C2×D508—22no—
66C2×D508—21no—
67C2×D508—41no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74#14523236—22no—
75#14523236—22no—
76#14523236—41no—
77#14523233—22no—
78#14523233—22no—
79#14523233—22no—
80#14523233—22no—
81#14523233—41no—
82#14523233D414yescomplemented
83#14523232—22no—
84C2×C127⋊C8—22no—
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C2×C127⋊C8C2311yescharacteristic
88C2×C127⋊C8C2311yescharacteristic
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96C24×C127—21no—
97C24×C127—21no—
98C2×D4×C127—22no—
99C2×D4×C127—22no—
100C2×D4×C127—21no—
101C2×D4×C127—21no—
102C2×D4×C127—21no—
103C2×D4×C127—21no—
104C2×D4×C127—21no—
105C2×D4×C127—21no—
106C2×D4×C127—41no—
107C2×D4×C127—41no—
108C2×D4×C127—44no—
109C2×D4×C127D411yescharacteristic
110C2×D4×C127C2311yescharacteristic
111C2×D4×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127—21no—
114C22×C4×C127C2311yescharacteristic
115C22×C4×C127C2311yescharacteristic
116C4⋊C4×C127D411yescharacteristic
117C4⋊C4×C127C2311yescharacteristic
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—42no—
127C22×D127—42no—
128C22×D127—81no—
129Dic254—21no—
130Dic254—21no—
131D508—21no—
132D508—21no—
133D508—21no—
134D508—42no—
135D508C2×D412yescomplemented
136C22⋊D127—42no—
137C22⋊D127—81no—
138C2×Dic127—21no—
139C2×Dic127—21no—
140C4×D127—42no—
141C4×D127—81no—
142C127⋊C8—21no—
143C127⋊C8—21no—
144C127⋊C8—21no—
145C127⋊C8C2×D412yescomplemented
146C23×C127—21no—
147C23×C127—21no—
148C23×C127—21no—
149C23×C127—42no—
150C23×C127—42no—
151C23×C127—42no—
152C23×C127—42no—
153C23×C127—81no—
154C23×C127—81no—
155C23×C127C2×D411yescharacteristic
156C23×C127C2×D411yescharacteristic
157D4×C127—22no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127—21no—
163D4×C127—42no—
164D4×C127—42no—
165D4×C127—42no—
166D4×C127—42no—
167D4×C127—42no—
168D4×C127—81no—
169D4×C127C2×D412yes—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—42no—
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—41no—
193C22×C127—41no—
194C22×C127—41no—
195C22×C127—84no—
196C22×C127—84no—
197C22×C127AGL(1,ℤ/8)12yescomplemented
198C22×C127C22×D411yescharacteristic
199C2×D127—22no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—41no—
203C2×D127—41no—
204C2×D127—41no—
205C2×D127—84no—
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—21no—
209C4×C127—21no—
210C4×C127—41no—
211C4×C127C22×D411yescharacteristic
212C4×C127H(5,𝔽2)11yescharacteristic
213Dic127—41no—
214Dic127—41no—
215C2×C127—22no—
216C2×C127—41no—
217C2×C127—42no—
218C2×C127—81no—
219C2×C127—81no—
220C2×C127D4211yescharacteristic
221C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C127#57611yescharacteristic
223D127—42no—
224D127—81no—
225#3264—1271nomaximal, Sylow
226C127#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1271no—
228C22×D8—1271no—
229#546—1271no—
230D42D12711yescharacteristic, complemented
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×#41—1271no—
243C22×D4—21no—
244C22×D4—1271no—
245C22×D4C2×D12711yescharacteristic
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#127C2×D12711yescharacteristic, complemented
261#124—1271no—
262#123—2541no—
263C22⋊Q8—1271no—
264#121—21no—
265#121—1271no—
266#121—2541no—
267#121C2×D12711yescharacteristic
268#121C2×D12711yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—2541no—
272C4×D4—1271no—
273C4×D4C2×D12711yescharacteristic
274C4×D4C2×D12711yescharacteristic, complemented
275C2×C22⋊C4—2541no—
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—21no—
289C24—2541no—
290#41—5081no—
291#41—2542no—
292C2×Q8—1271no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—1271no—
305C2×D4—1271no—
306C2×D4—5081no—
307C2×D4—2542no—
308C2×D4—2541no—
309C2×D4—2541no—
310C2×D4—2541no—
311C2×D4C22⋊D12711yescharacteristic
312C2×D4C22×D12711yescharacteristic
313C2×D4C22×D12711yescharacteristic
314C2×D4C22×D12711yescharacteristic
315C22×C4—21no—
316C22×C4—2541no—
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—1271no—
336C4⋊C4—2541no—
337C4⋊C4C22⋊D12711yescharacteristic
338C4⋊C4C22×D12711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—2541no—
343C22⋊C4—2541no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C22⋊C4C22⋊D12711yescharacteristic
348C22⋊C4C22⋊D12711yescharacteristic
349C42C22×D12711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—10161no—
360C23—5082no—
361C23—5082no—
362C23—2541no—
363C23—2541no—
364C23C2×C22⋊D12711yescharacteristic
365C23C2×C22⋊D12711yescharacteristic
366Q8—2541no—
367Q8—2541no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—1272no—
381D4—10161no—
382D4—5082no—
383D4—5082no—
384D4—2541no—
385D4—2541no—
386D4—2541no—
387D4D4×D12712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—10161no—
395C2×C4—5082no—
396C2×C4—2541no—
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—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—10164no—
424C22—5081no—
425C22—5081no—
426C22—5081no—
427C22—5081no—
428C22—5081no—
429C22—2542no—
430C22#1686123512yescomplemented
431C22C22×C22⋊D12711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—5081no—
438C4—5081no—
439C4C2×D4×D12711yescharacteristic
440C4#1686125411yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—10161no—
447C2—5082no—
448C2D4×C22⋊D12711yescharacteristic
449C2C2×#1686123511yescharacteristic
450C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD42×C127C2
socleC22×C127C22×D4
hypercenterD42D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. C22×#14523233C2semidirect
3. #22550522C2semidirect
4. #22550263C2semidirect
5. #22550127C2semidirect
6. #22550103C2semidirect
7. #22550091C2semidirect
8. #22550068C2semidirect
9. #22550027C2semidirect
10. #22550019C2semidirect
11. #22550010C2semidirect
12. #22550009C2semidirect
13. #22549986C2semidirect
14. #22549948C2semidirect
15. D42×C127C2semidirect
16. C2×#14523248C22semidirect
17. C22×D508C22semidirect
19. C2×#14523236C22semidirect
20. C2×#14523236C22semidirect
21. C2×#14523236C22semidirect
25. C2×#14523233C22semidirect
26. C2×#14523233C22semidirect
27. C2×#14523233C22semidirect
28. C2×#14523233C22semidirect
29. C2×#14523232C22semidirect
30. C22×C127⋊C8C22semidirect
31. #16861197C22semidirect
33. #16861188C22semidirect
34. #16861183C22semidirect
37. C4×D508C22semidirect
39. #16861140C22semidirect
40. C4⋊(C127⋊C8)C22semidirect
41. #16861132C22semidirect
42. #16861130C22semidirect
43. #16861130C22semidirect
44. #16861130C22semidirect
45. #16861129C22semidirect
46. #16861129C22semidirect
47. C22⋊(C127⋊C8)C22semidirect
48. C22⋊(C127⋊C8)C22semidirect
49. #16861120C22semidirect
52. C22×D4×C127C22semidirect
53. #127×C127C22semidirect
56. #121×C127C22semidirect
60. C4×D4×C127C22semidirect
82. #14523233D4semidirect
135. D508C2×D4semidirect
145. C127⋊C8C2×D4semidirect
197. C22×C127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
230. D42D127semidirect
246. C22×D4C2×D127semidirect
260. #127C2×D127semidirect
268. #121C2×D127semidirect
274. C4×D4C2×D127semidirect
430. C22#16861235semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C22×#145232332
#225505222
#225502632
#225501272
#225501032
#225500912
#225500682
#225500272
#225500192
#225500102
#225500092
#225499862
#225499482
D42×C1272
#3264127