Finite Group Databasebeta

← #30650212

Subgroups — #30650212

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×#16861233C211yesmaximal, characteristic, complemented
2C2×D8×D127C211yesmaximal, characteristic, complemented
3#22550521C211yesmaximal, characteristic, complemented
4D4×D508C211yesmaximal, characteristic, complemented
5#22550265C211yesmaximal, characteristic, complemented
6#22550130C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550091C211yesmaximal, characteristic, complemented
9#22550071C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22550022C211yesmaximal, characteristic, complemented
12#22550011C211yesmaximal, characteristic, complemented
13#22550011C211yesmaximal, characteristic, complemented
14#22549949C211yesmaximal, characteristic, complemented
15#442×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic, complemented
17C22×D508—21no—
18C2×D4×D127C2211yescharacteristic
19C2×D4×D127C2211yescharacteristic, complemented
20C2×D4×D127C2211yescharacteristic, complemented
21#16861233—24no—
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic, complemented
24C2×#14523236C2211yescharacteristic, complemented
25C2×D1016C2211yescharacteristic, complemented
26C2×D1016C2211yescharacteristic, complemented
27C2×#14523233C2211yescharacteristic
28C2×#14523233C2211yescharacteristic, complemented
29C2×#14523233C2211yescharacteristic, complemented
30D8×D127—24no—
31C2×#14523231C2211yescharacteristic, complemented
32C2×C8×D127C2211yescharacteristic, complemented
33#16861206C2211yescharacteristic, complemented
34#16861197C2211yescharacteristic, complemented
35#16861193—21no—
36#16861189C2211yescharacteristic, complemented
37#16861184C2211yescharacteristic
38#16861184C2211yescharacteristic, complemented
39#16861184C2211yescharacteristic, complemented
40#16861183—21no—
41#16861183—21no—
42#16861179—21no—
43#16861179—21no—
44#16861179—21no—
45C4×D508C2211yescharacteristic
46C2×(C2×D127)⋊C4—21no—
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×D8×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic, complemented
56C4×D4×C127C2211yescharacteristic, complemented
57#106×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic, complemented
60#97×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63C23×D127—21no—
64#14523247—22no—
65#14523247—41no—
66C2×Dic254C2311yescharacteristic
67C2×D508—21no—
68C2×D508—21no—
69C2×D508—21no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508—44no—
73C2×D508D411yescharacteristic
74C2×D508C2311yescharacteristic
75C2×D508C2311yescharacteristic
76C2×D508C2311yescharacteristic
77C2×C22⋊D127—21no—
78C2×C22⋊D127—21no—
79C2×C22⋊D127—21no—
80C2×C22⋊D127—21no—
81D4×D127—22no—
82D4×D127—22no—
83D4×D127—22no—
84D4×D127—41no—
85D4×D127—41no—
86D4×D127—41no—
87C22×Dic127—21no—
88C2×C4×D127C2311yescharacteristic
89C2×C4×D127C2311yescharacteristic
90#14523237—22no—
91#14523237—41no—
92#14523236—22no—
93D1016—22no—
94D1016—41no—
95#14523233—22no—
96#14523233—22no—
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×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127—21no—
116D8×C127D414yescomplemented
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C4⋊C4×C127C2311yescharacteristic
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C22⋊C4×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—42no—
132C22×D127—42no—
133C22×D127—81no—
134C22×D127—81no—
135C22×D127—81no—
136C22×D127C2×D411yescharacteristic
137C22×D127C2×D411yescharacteristic
138Dic254—21no—
139Dic254—21no—
140D508—22no—
141D508—21no—
142D508—21no—
143D508—21no—
144D508—21no—
145D508—21no—
146D508—21no—
147D508—42no—
148D508—42no—
149D508—42no—
150D508—81no—
151D508—81no—
152C22⋊D127—42no—
153C22⋊D127—42no—
154C22⋊D127—42no—
155C22⋊D127—42no—
156C2×Dic127—21no—
157C2×Dic127—42no—
158C2×Dic127C2×D411yescharacteristic
159C2×Dic127C2×D411yescharacteristic
160C4×D127—21no—
161C4×D127—21no—
162C4×D127—21no—
163C4×D127—21no—
164C127⋊C8—21no—
165C127⋊C8—21no—
166C23×C127—21no—
167C23×C127—21no—
168D4×C127—21no—
169D4×C127—21no—
170D4×C127C2×D412yes—
171D4×C127C2×D412yescomplemented
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×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127C2×D412yescomplemented
181C22×C127—22no—
182C22×C127—22no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127C22×D411yescharacteristic
188C2×D127—21no—
189C2×D127—21no—
190C2×D127—21no—
191C2×D127—21no—
192C2×D127—21no—
193C2×D127—41no—
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—21no—
208C4×C127—41no—
209C4×C127—41no—
210C4×C127C22×D411yescharacteristic
211C4×C127H(5,𝔽2)11yescharacteristic
212Dic127—21no—
213Dic127—21no—
214Dic127—41no—
215C2×C127—42no—
216C2×C127—42no—
217C2×C127D4211yescharacteristic
218C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C127#57611yescharacteristic
220D127—22no—
221D127—41no—
222D127—81no—
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#449—1271no—
238#447—1271no—
239#447—1271no—
240#442D12711yescharacteristic, complemented
241#435—1271no—
242C2×#41—1271no—
243C22×D4—1271no—
244C22×D4—1271no—
245C22×D4—1271no—
246C22×D4—2541no—
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—2544no—
257C2×D8C2×D12711yescharacteristic
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—1271no—
267#121—2541no—
268#121—2541no—
269#120—2541no—
270#120—2541no—
271#120—2541no—
272C4×D4—1271no—
273C4×D4C2×D12711yescharacteristic, complemented
274C4×D4C2×D12711yescharacteristic, complemented
275C2×C22⋊C4—2541no—
276#106C2×D12711yescharacteristic, complemented
277C4⋊C8—1271no—
278#103—1271no—
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102C2×D12711yescharacteristic, complemented
283#102C2×D12711yescharacteristic, complemented
284C22⋊C8—1271no—
285C22⋊C8—1271no—
286#97C2×D12711yescharacteristic, complemented
287C24—2541no—
288C24—2541no—
289C24—2541no—
290#41—5081no—
291#41—2542no—
292C2×Q8—1271no—
293C2×D4—1271no—
294C2×D4—1271no—
295C2×D4—1271no—
296C2×D4—1271no—
297C2×D4—5084no—
298C2×D4—5081no—
299C2×D4—5081no—
300C2×D4—5081no—
301C2×D4—2541no—
302C2×D4—2541no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4—2541no—
306C2×D4—2541no—
307C2×D4—2541no—
308C2×D4—2542no—
309C2×D4—2542no—
310C2×D4—2542no—
311C2×D4—2541no—
312C2×D4—2541no—
313C2×D4C22×D12711yescharacteristic
314C2×D4C22×D12711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—1271no—
318C22×C4—1271no—
319C22×C4—2541no—
320SD16—5081no—
321SD16—2542no—
322SD16—2542no—
323D8—5081no—
324D8—2542no—
325D8—2542no—
326D8—2542no—
327D8—2542no—
328D8D50814yescomplemented
329M16—2542no—
330C2×C8—1271no—
331C2×C8—1271no—
332C2×C8—2542no—
333C2×C8C22×D12711yescharacteristic
334C2×C8C22×D12711yescharacteristic
335C4⋊C4—1271no—
336C4⋊C4—2541no—
337C4⋊C4C22×D12711yescharacteristic
338C4⋊C4C22×D12711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—1271no—
342C22⋊C4—1271no—
343C22⋊C4—2541no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C22⋊C4—2541no—
348C22⋊C4—2541no—
349C42C22×D12711yescharacteristic
350C23—21no—
351C23—21no—
352C23—1271no—
353C23—1271no—
354C23—10161no—
355C23—10161no—
356C23—10161no—
357C23—5082no—
358C23—5082no—
359C23—5082no—
360C23—5082no—
361C23—5082no—
362C23—5082no—
363C23—2541no—
364C23—2541no—
365C23—2541no—
366Q8—2541no—
367Q8—2541no—
368D4—21no—
369D4—21no—
370D4—10161no—
371D4—10161no—
372D4—5082no—
373D4—5082no—
374D4—5082no—
375D4—5082no—
376D4—5082no—
377D4—5082no—
378D4—5082no—
379D4—2542no—
380D4—2541no—
381D4—2541no—
382D4—2541no—
383D4—2541no—
384D4—2541no—
385D4—2541no—
386D4C2×D50812yes—
387D4C2×D50812yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—1271no—
393C2×C4—1271no—
394C2×C4—5082no—
395C2×C4—2541no—
396C2×C4—2541no—
397C2×C4—2541no—
398C2×C4—2541no—
399C2×C4—2541no—
400C2×C4D4×D12711yescharacteristic
401C2×C4D4×D12711yescharacteristic
402C2×C4C23×D12711yescharacteristic
403C8—21no—
404C8—2541no—
405C8—2541no—
406C8C2×D50812yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—1272no—
414C22—10164no—
415C22—10164no—
416C22—10164no—
417C22—5081no—
418C22—5081no—
419C22—5081no—
420C22—5081no—
421C22—5081no—
422C22—5081no—
423C22—5081no—
424C22—5081no—
425C22—5081no—
426C22—2541no—
427C22—2541no—
428C22—2541no—
429C22—2541no—
430C22—2541no—
431C22C2×D4×D12711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—5081no—
437C4—2541no—
438C4—2541no—
439C4C22×D50811yescharacteristic
440C4#1686125511yescharacteristic
441C2—42no—
442C2—42no—
443C2—10161no—
444C2—10161no—
445C2—10161no—
446C2—5081no—
447C2—2542no—
448C2D4×D50811yescharacteristic
449C2AGL(1,ℤ/8)×D12711yescharacteristic
450C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#442×C127C2
socleC22×C127C22×D4
hypercenter#442D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. C2×D8×D127C2semidirect
3. #22550521C2semidirect
4. D4×D508C2semidirect
5. #22550265C2semidirect
6. #22550130C2semidirect
7. #22550106C2semidirect
8. #22550091C2semidirect
9. #22550071C2semidirect
10. #22550028C2semidirect
11. #22550022C2semidirect
12. #22550011C2semidirect
13. #22550011C2semidirect
14. #22549949C2semidirect
15. #442×C127C2semidirect
16. C2×#14523247C22semidirect
19. C2×D4×D127C22semidirect
20. C2×D4×D127C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523237C22semidirect
24. C2×#14523236C22semidirect
25. C2×D1016C22semidirect
26. C2×D1016C22semidirect
28. C2×#14523233C22semidirect
29. C2×#14523233C22semidirect
31. C2×#14523231C22semidirect
32. C2×C8×D127C22semidirect
33. #16861206C22semidirect
34. #16861197C22semidirect
36. #16861189C22semidirect
38. #16861184C22semidirect
39. #16861184C22semidirect
47. C4⋊(C127⋊C8)C22semidirect
48. #16861132C22semidirect
49. #16861129C22semidirect
50. #16861129C22semidirect
51. #16861129C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. (C2×D127)⋊C8C22semidirect
55. C4×D4×C127C22semidirect
56. C4×D4×C127C22semidirect
57. #106×C127C22semidirect
58. #102×C127C22semidirect
59. #102×C127C22semidirect
60. #97×C127C22semidirect
116. D8×C127D4semidirect
171. D4×C127C2×D4semidirect
180. C8×C127C2×D4semidirect
205. C2×D127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
240. #442D127semidirect
273. C4×D4C2×D127semidirect
274. C4×D4C2×D127semidirect
276. #106C2×D127semidirect
282. #102C2×D127semidirect
283. #102C2×D127semidirect
286. #97C2×D127semidirect
328. D8D508semidirect
387. D4C2×D508semidirect
406. C8C2×D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
C2×D8×D1272
#225505212
D4×D5082
#225502652
#225501302
#225501062
#225500912
#225500712
#225500282
#225500222
#225500112
#225499492
#442×C1272
#3264127