Finite Group Databasebeta

← #30650227

Subgroups — #30650227

Order 16256.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2C2×SD16×D127C211yesmaximal, characteristic, complemented
3#22550538C211yesmaximal, characteristic, complemented
4D4×D508C211yesmaximal, characteristic, complemented
5#22550265C211yesmaximal, characteristic, complemented
6#22550123C211yesmaximal, characteristic, complemented
7#22550113C211yesmaximal, characteristic, complemented
8#22550091C211yesmaximal, characteristic, complemented
9#22550076C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22550024C211yesmaximal, characteristic, complemented
12#22550016C211yesmaximal, characteristic, complemented
13#22550011C211yesmaximal, characteristic, complemented
14#22549949C211yesmaximal, characteristic, complemented
15#440×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic, complemented
17C2×Q8×D127C2211yescharacteristic, complemented
18C22×D508—21no—
19C2×D4×D127C2211yescharacteristic
20C2×D4×D127C2211yescharacteristic
21#16861236—24no—
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic, complemented
24C2×#14523236C2211yescharacteristic
25C2×#14523235C2211yescharacteristic, complemented
26C2×#14523235C2211yescharacteristic, complemented
27SD16×D127—24no—
28C2×D1016C2211yescharacteristic, complemented
29C2×D1016C2211yescharacteristic, complemented
30C2×#14523233C2211yescharacteristic
31C2×#14523231C2211yescharacteristic, complemented
32C2×C8×D127C2211yescharacteristic, complemented
33#16861206C2211yescharacteristic
34#16861197—21no—
35#16861197C2211yescharacteristic, complemented
36#16861189—21no—
37#16861189C2211yescharacteristic, complemented
38#16861184—21no—
39#16861184C2211yescharacteristic
40#16861184C2211yescharacteristic
41#16861184C2211yescharacteristic, complemented
42#16861183—21no—
43#16861179—21no—
44#16861179—21no—
45C4×D508—21no—
46C4×D508C2211yescharacteristic
47C4⋊(C127⋊C8)C2211yescharacteristic, complemented
48#16861132C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic, complemented
51#16861129C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53(C2×D127)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic
55C4×Q8×C127C2211yescharacteristic, complemented
56C4×D4×C127C2211yescharacteristic
57#107×C127C2211yescharacteristic, complemented
58#103×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic
60#97×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63#14523249—22no—
64#14523249—41no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67Q8×D127—22no—
68Q8×D127—41no—
69C2×D508—21no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508—21no—
73C2×D508—21no—
74C2×D508—21no—
75C2×D508—44no—
76C2×D508D411yescharacteristic
77C2×D508C2311yescharacteristic
78C2×D508C2311yescharacteristic
79C2×D508C2311yescharacteristic
80C2×C22⋊D127—21no—
81C2×C22⋊D127—21no—
82D4×D127—22no—
83D4×D127—22no—
84D4×D127—41no—
85D4×D127—41no—
86C2×C4×D127—21no—
87C2×C4×D127—21no—
88C2×C4×D127C2311yescharacteristic
89C2×C4×D127C2311yescharacteristic
90#14523237—22no—
91#14523237—41no—
92#14523236—22no—
93#14523235—22no—
94#14523235—22no—
95D1016—22no—
96D1016—41no—
97#14523233—22no—
98#14523231—22no—
99C2×C127⋊C8C2311yescharacteristic
100C2×C127⋊C8C2311yescharacteristic
101C8×D127—22no—
102C4⋊Dic127—21no—
103C4⋊Dic127D411yescharacteristic
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4—21no—
109(C2×D127)⋊C4—21no—
110(C2×D127)⋊C4D411yescharacteristic
111(C2×D127)⋊C4D411yescharacteristic
112C2×Q8×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115SD16×C127D414yescomplemented
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C4⋊C4×C127—21no—
119C4⋊C4×C127C2311yescharacteristic
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C42×C127—21no—
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—
133C22×D127C2×D411yescharacteristic
134C22×D127C2×D411yescharacteristic
135Dic254—21no—
136Dic254—21no—
137Dic254—42no—
138D508—22no—
139D508—21no—
140D508—21no—
141D508—21no—
142D508—21no—
143D508—21no—
144D508—21no—
145D508—42no—
146D508—42no—
147D508—42no—
148D508—42no—
149D508—42no—
150D508—81no—
151C22⋊D127—42no—
152C22⋊D127—42no—
153C2×Dic127—21no—
154C2×Dic127C2×D411yescharacteristic
155C2×Dic127C2×D411yescharacteristic
156C4×D127—21no—
157C4×D127—21no—
158C4×D127—21no—
159C4×D127—21no—
160C4×D127—42no—
161C4×D127—42no—
162C127⋊C8—21no—
163C127⋊C8—21no—
164C23×C127—21no—
165Q8×C127—21no—
166Q8×C127C2×D412yescomplemented
167D4×C127—21no—
168D4×C127C2×D412yes—
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—42no—
173C2×C4×C127C2×D411yescharacteristic
174C2×C4×C127C2×D411yescharacteristic
175C2×C4×C127C2411yescharacteristic
176C8×C127—21no—
177C8×C127C2×D412yescomplemented
178C22×C127—22no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127C22×D411yescharacteristic
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C2×D127—21no—
186C2×D127—21no—
187C2×D127—41no—
188C2×D127—41no—
189C2×D127—41no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—41no—
195C2×D127—41no—
196C2×D127—84no—
197C2×D127—84no—
198C2×D127AGL(1,ℤ/8)12yescomplemented
199C4×C127—22no—
200C4×C127—21no—
201C4×C127—21no—
202C4×C127—41no—
203C4×C127—41no—
204C4×C127C22×D411yescharacteristic
205C4×C127H(5,𝔽2)11yescharacteristic
206Dic127—21no—
207Dic127—21no—
208Dic127—41no—
209C2×C127—42no—
210C2×C127D4211yescharacteristic
211C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
212C2×C127#57711yescharacteristic
213D127—22no—
214D127—41no—
215D127—81no—
216D127—81no—
217D127—81no—
218#3265—1271nomaximal, Sylow
219C127#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1271no—
221C22×SD16—1271no—
222#548—1271no—
223D42—1271no—
224#496—1271no—
225#468—1271no—
226#466—1271no—
227#463—1271no—
228#460—1271no—
229#450—1271no—
230#449—1271no—
231#448—1271no—
232#447—1271no—
233#440D12711yescharacteristic, complemented
234#435—1271no—
235C2×#41—1271no—
236C22×Q8—1271no—
237C22×D4—1271no—
238C22×D4—1271no—
239C22×D4—2541no—
240AGL(1,ℤ/8)—2544no—
241C2×SD16—1271no—
242C2×SD16—1271no—
243C2×SD16—1271no—
244C2×SD16—1271no—
245C2×SD16—1271no—
246C2×SD16—2544no—
247C2×SD16C2×D12711yescharacteristic
248C2×D8—1271no—
249C2×D8—1271no—
250C2×D8—1271no—
251C2×M16—1271no—
252C22×C8—1271no—
253#127—1271no—
254#124—1271no—
255#124—2541no—
256C22⋊Q8—1271no—
257C22⋊Q8—2541no—
258#121—1271no—
259#121—1271no—
260#121—1271no—
261#121—2541no—
262#121—2541no—
263#120—2541no—
264#120—2541no—
265C4×Q8C2×D12711yescharacteristic, complemented
266C4×D4—1271no—
267C4×D4—2541no—
268C4×D4C2×D12711yescharacteristic
269#107C2×D12711yescharacteristic, complemented
270C4⋊C8—1271no—
271#103—1271no—
272#103C2×D12711yescharacteristic, complemented
273#102—1271no—
274#102—1271no—
275#102—1271no—
276#102C2×D12711yescharacteristic
277C22⋊C8—1271no—
278C22⋊C8—1271no—
279#97C2×D12711yescharacteristic, complemented
280C24—2541no—
281C24—2541no—
282#41—5081no—
283#41—2542no—
284C2×Q8—1271no—
285C2×Q8—5081no—
286C2×Q8—2541no—
287C2×Q8—2542no—
288C2×Q8C22×D12711yescharacteristic
289C2×D4—1271no—
290C2×D4—1271no—
291C2×D4—1271no—
292C2×D4—1271no—
293C2×D4—5084no—
294C2×D4—5081no—
295C2×D4—5081no—
296C2×D4—2541no—
297C2×D4—2541no—
298C2×D4—2541no—
299C2×D4—2541no—
300C2×D4—2541no—
301C2×D4—2541no—
302C2×D4—2542no—
303C2×D4—2542no—
304C2×D4—2541no—
305C2×D4—2541no—
306C2×D4C22×D12711yescharacteristic
307C22×C4—21no—
308C22×C4—1271no—
309C22×C4—1271no—
310C22×C4—2541no—
311C22×C4—2541no—
312SD16—5081no—
313SD16—2542no—
314SD16—2542no—
315SD16—2542no—
316SD16—2542no—
317SD16D50814yescomplemented
318D8—5081no—
319D8—2542no—
320D8—2542no—
321M16—2542no—
322C2×C8—1271no—
323C2×C8—1271no—
324C2×C8—2542no—
325C2×C8C22×D12711yescharacteristic
326C2×C8C22×D12711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—1271no—
329C4⋊C4—2541no—
330C4⋊C4C22×D12711yescharacteristic
331C4⋊C4C22×D12711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—1271no—
334C22⋊C4—1271no—
335C22⋊C4—2541no—
336C22⋊C4—2541no—
337C22⋊C4—2541no—
338C22⋊C4—2541no—
339C22⋊C4—2541no—
340C22⋊C4—2541no—
341C42—21no—
342C42C22×D12711yescharacteristic
343C23—21no—
344C23—1271no—
345C23—1271no—
346C23—10161no—
347C23—10161no—
348C23—5082no—
349C23—5082no—
350C23—5082no—
351C23—5082no—
352C23—2541no—
353C23—2541no—
354C23—2541no—
355Q8—21no—
356Q8—5082no—
357Q8—2541no—
358Q8—2541no—
359Q8C2×D50812yescomplemented
360D4—21no—
361D4—10161no—
362D4—5082no—
363D4—5082no—
364D4—5082no—
365D4—5082no—
366D4—5082no—
367D4—5082no—
368D4—5082no—
369D4—2542no—
370D4—2541no—
371D4—2541no—
372D4—2541no—
373D4—2541no—
374D4—2541no—
375D4—2541no—
376D4C2×D50812yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—1271no—
382C2×C4—1271no—
383C2×C4—5082no—
384C2×C4—5082no—
385C2×C4—2541no—
386C2×C4—2541no—
387C2×C4—2541no—
388C2×C4—2541no—
389C2×C4—2541no—
390C2×C4D4×D12711yescharacteristic
391C2×C4D4×D12711yescharacteristic
392C2×C4C23×D12711yescharacteristic
393C8—21no—
394C8—2541no—
395C8—2541no—
396C8C2×D50812yescomplemented
397C22—22no—
398C22—41no—
399C22—41no—
400C22—1272no—
401C22—10164no—
402C22—10164no—
403C22—5081no—
404C22—5081no—
405C22—5081no—
406C22—5081no—
407C22—5081no—
408C22—5081no—
409C22—5081no—
410C22—5081no—
411C22—5081no—
412C22—2541no—
413C22—2541no—
414C22—2541no—
415C22—2541no—
416C22—2541no—
417C22C2×D4×D12711yescharacteristic
418C4—21no—
419C4—21no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—5081no—
424C4—2541no—
425C4—2541no—
426C4C22×D50811yescharacteristic
427C4#1686125511yescharacteristic
428C2—42no—
429C2—10161no—
430C2—10161no—
431C2—10161no—
432C2—5081no—
433C2—2542no—
434C2D4×D50811yescharacteristic
435C2AGL(1,ℤ/8)×D12711yescharacteristic
436C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#440×C127C2
socleC22×C127C22×D4
hypercenter#440D127
hypocenterC127#3265
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×SD16×D127C2semidirect
3. #22550538C2semidirect
4. D4×D508C2semidirect
5. #22550265C2semidirect
6. #22550123C2semidirect
7. #22550113C2semidirect
8. #22550091C2semidirect
9. #22550076C2semidirect
10. #22550028C2semidirect
11. #22550024C2semidirect
12. #22550016C2semidirect
13. #22550011C2semidirect
14. #22549949C2semidirect
15. #440×C127C2semidirect
16. C2×#14523249C22semidirect
17. C2×Q8×D127C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523237C22semidirect
25. C2×#14523235C22semidirect
26. C2×#14523235C22semidirect
28. C2×D1016C22semidirect
29. C2×D1016C22semidirect
31. C2×#14523231C22semidirect
32. C2×C8×D127C22semidirect
35. #16861197C22semidirect
37. #16861189C22semidirect
41. #16861184C22semidirect
47. C4⋊(C127⋊C8)C22semidirect
48. #16861132C22semidirect
49. #16861129C22semidirect
50. #16861129C22semidirect
51. #16861129C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. (C2×D127)⋊C8C22semidirect
55. C4×Q8×C127C22semidirect
57. #107×C127C22semidirect
58. #103×C127C22semidirect
60. #97×C127C22semidirect
115. SD16×C127D4semidirect
166. Q8×C127C2×D4semidirect
177. C8×C127C2×D4semidirect
198. C2×D127AGL(1,ℤ/8)semidirect
219. C127#3265semidirect
233. #440D127semidirect
265. C4×Q8C2×D127semidirect
269. #107C2×D127semidirect
272. #103C2×D127semidirect
279. #97C2×D127semidirect
317. SD16D508semidirect
359. Q8C2×D508semidirect
396. C8C2×D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×SD16×D1272
#225505382
D4×D5082
#225502652
#225501232
#225501132
#225500912
#225500762
#225500282
#225500242
#225500162
#225500112
#225499492
#440×C1272
#3265127