Finite Group Databasebeta

← #30650215

Subgroups — #30650215

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×#16861237C211yesmaximal, characteristic, complemented
2C22×D1016C211yesmaximal, characteristic, complemented
3#22550522C211yesmaximal, characteristic, complemented
4D4×D508C211yesmaximal, characteristic, complemented
5#22550265C211yesmaximal, characteristic, complemented
6#22550129C211yesmaximal, characteristic, complemented
7#22550105C211yesmaximal, characteristic, complemented
8#22550092C211yesmaximal, characteristic, complemented
9#22550072C211yesmaximal, characteristic, complemented
10#22550029C211yesmaximal, characteristic, complemented
11#22550023C211yesmaximal, characteristic, complemented
12#22550012C211yesmaximal, characteristic, complemented
13#22550012C211yesmaximal, characteristic, complemented
14#22549989C211yesmaximal, characteristic, complemented
15#435×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic, complemented
17C22×D508C2211yescharacteristic
18C22×D508C2211yescharacteristic, complemented
19C22×D508C2211yescharacteristic, complemented
20C2×D4×D127—21no—
21#16861237—24no—
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic, complemented
24C2×#14523237C2211yescharacteristic, complemented
25C2×D1016—24no—
26C2×D1016C2211yescharacteristic
27C2×D1016C2211yescharacteristic
28C2×D1016C2211yescharacteristic, complemented
29C2×D1016C2211yescharacteristic, complemented
30C2×D1016C2211yescharacteristic, complemented
31C2×D1016C2211yescharacteristic, complemented
32#16861206C2211yescharacteristic, complemented
33#16861197C2211yescharacteristic, complemented
34#16861194—21no—
35#16861188C2211yescharacteristic, complemented
36#16861184—21no—
37#16861183C2211yescharacteristic
38#16861183C2211yescharacteristic, complemented
39#16861183C2211yescharacteristic, complemented
40#16861182—21no—
41#16861179—21no—
42#16861179—21no—
43#16861177—21no—
44C4×D508C2211yescharacteristic, complemented
45C4×D508C2211yescharacteristic, complemented
46C22⋊C4×D127—21no—
47#16861141C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic, complemented
51#16861131C2211yescharacteristic, complemented
52#16861131C2211yescharacteristic, complemented
53#16861131C2211yescharacteristic, complemented
54C2×M16×C127C2211yescharacteristic, complemented
55C22×C8×C127C2211yescharacteristic, complemented
56C4×D4×C127C2211yescharacteristic
57C4⋊C8×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic, complemented
59C22⋊C8×C127C2211yescharacteristic, complemented
60#97×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63C23×D127—21no—
64#14523248—22no—
65#14523248—41no—
66C2×Dic254C2311yescharacteristic
67C2×D508—22no—
68C2×D508—22no—
69C2×D508—22no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508—21no—
73C2×D508—21no—
74C2×D508—21no—
75C2×D508—41no—
76C2×D508—41no—
77C2×D508—41no—
78C2×D508C2311yescharacteristic
79C2×D508C2311yescharacteristic
80C2×D508C2311yescharacteristic
81C2×D508C2311yescharacteristic
82C2×D508C2311yescharacteristic
83C2×C22⋊D127—21no—
84C2×C22⋊D127—21no—
85C2×C22⋊D127—21no—
86C2×C22⋊D127—21no—
87D4×D127—44no—
88C2×C4×D127—21no—
89C2×C4×D127—21no—
90C2×C4×D127—21no—
91#14523237—22no—
92#14523237—22no—
93#14523237—41no—
94D1016—22no—
95D1016—22no—
96D1016—22no—
97D1016—22no—
98D1016—41no—
99D1016D414yescomplemented
100C4⋊Dic127C2311yescharacteristic
101C4⋊Dic127C2311yescharacteristic
102Dic127⋊C4—21no—
103C22⋊Dic127—21no—
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)⋊C4—21no—
111C2×D4×C127D411yescharacteristic
112C22×C4×C127C2311yescharacteristic
113C22×C4×C127C2311yescharacteristic
114M16×C127—22no—
115C2×C8×C127—22no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127D411yescharacteristic
121C22⋊C4×C127D411yescharacteristic
122C22⋊C4×C127D411yescharacteristic
123C42×C127C2311yescharacteristic
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—21no—
128C22×D127—21no—
129C22×D127—42no—
130C22×D127—42no—
131C22×D127—42no—
132C22×D127—42no—
133C22×D127—42no—
134C22×D127—42no—
135C22×D127—81no—
136C22×D127—81no—
137C22×D127—81no—
138Dic254—21no—
139Dic254—21no—
140D508—21no—
141D508—21no—
142D508—21no—
143D508—21no—
144D508—21no—
145D508—21no—
146D508—21no—
147D508—21no—
148D508—42no—
149D508—42no—
150D508—42no—
151D508—42no—
152D508—81no—
153D508C2×D412yes—
154D508C2×D412yescomplemented
155C22⋊D127—42no—
156C22⋊D127—42no—
157C22⋊D127—42no—
158C22⋊D127—81no—
159C2×Dic127—21no—
160C2×Dic127—21no—
161C2×Dic127—21no—
162C4×D127—42no—
163C4×D127—42no—
164C4×D127—81no—
165C23×C127C2×D411yescharacteristic
166C23×C127C2×D411yescharacteristic
167D4×C127—22no—
168C2×C4×C127—21no—
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127C2×D411yescharacteristic
173C2×C4×C127C2×D411yescharacteristic
174C2×C4×C127C2×D411yescharacteristic
175C2×C4×C127C2×D411yescharacteristic
176C2×C4×C127C2411yescharacteristic
177C8×C127—21no—
178C8×C127—21no—
179C8×C127—21no—
180C8×C127C2×D412yescomplemented
181C22×C127—21no—
182C22×C127—21no—
183C22×C127—21no—
184C22×C127—21no—
185C22×C127—21no—
186C22×C127AGL(1,ℤ/8)12yescomplemented
187C22×C127C22×D411yescharacteristic
188C2×D127—22no—
189C2×D127—22no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
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—41no—
203C2×D127—84no—
204C2×D127—84no—
205C2×D127—84no—
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—21no—
209C4×C127—21no—
210C4×C127C22×D411yescharacteristic
211C4×C127H(5,𝔽2)11yescharacteristic
212Dic127—41no—
213Dic127—41no—
214Dic127—41no—
215C2×C127—22no—
216C2×C127—41no—
217C2×C127D4211yescharacteristic
218C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C127#57611yescharacteristic
220D127—42no—
221D127—42no—
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#442—1271no—
241#435D12711yescharacteristic, complemented
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—1271no—
257C2×D8—2544no—
258C2×M16C2×D12711yescharacteristic, complemented
259C22×C8C2×D12711yescharacteristic, complemented
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×D4—1271no—
274C4×D4C2×D12711yescharacteristic
275C2×C22⋊C4—2541no—
276#106—1271no—
277C4⋊C8C2×D12711yescharacteristic, complemented
278#103—1271no—
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102—1271no—
283#102—1271no—
284C22⋊C8C2×D12711yescharacteristic, complemented
285C22⋊C8C2×D12711yescharacteristic, complemented
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—1271no—
298C2×D4—5081no—
299C2×D4—5081no—
300C2×D4—5081no—
301C2×D4—5084no—
302C2×D4—2542no—
303C2×D4—2542no—
304C2×D4—2542no—
305C2×D4—2541no—
306C2×D4—2541no—
307C2×D4—2541no—
308C2×D4—2541no—
309C2×D4—2541no—
310C2×D4—2541no—
311C2×D4—2541no—
312C2×D4—2541no—
313C2×D4—2541no—
314C2×D4D50811yescharacteristic
315C22×C4—2541no—
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—22no—
330C2×C8—22no—
331C2×C8C22×D12711yescharacteristic
332C2×C8C22×D12711yescharacteristic
333C2×C8C22×D12711yescharacteristic
334C2×C8C22×D12711yescharacteristic
335C4⋊C4—1271no—
336C4⋊C4—1271no—
337C4⋊C4—2541no—
338C4⋊C4D50811yescharacteristic
339C22⋊C4—2541no—
340C22⋊C4—2541no—
341C22⋊C4—2541no—
342C22⋊C4—2541no—
343C22⋊C4—2541no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C22⋊C4D50811yescharacteristic
348C22⋊C4D50811yescharacteristic
349C42C22×D12711yescharacteristic
350C23—10161no—
351C23—10161no—
352C23—10161no—
353C23—5082no—
354C23—5082no—
355C23—5082no—
356C23—5082no—
357C23—5082no—
358C23—5082no—
359C23—2541no—
360C23—2541no—
361C23—2541no—
362C23—2541no—
363C23—2541no—
364C23C2×D50811yescharacteristic
365C23C2×D50811yescharacteristic
366Q8—2541no—
367Q8—2541no—
368D4—22no—
369D4—1272no—
370D4—1272no—
371D4—10161no—
372D4—10161no—
373D4—5082no—
374D4—5082no—
375D4—5082no—
376D4—5082no—
377D4—5082no—
378D4—5082no—
379D4—5082no—
380D4—2541no—
381D4—2541no—
382D4—2541no—
383D4—2541no—
384D4—2541no—
385D4—2541no—
386D4—2541no—
387D4—2541no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—10161no—
393C2×C4—5082no—
394C2×C4—5082no—
395C2×C4—2541no—
396C2×C4—2541no—
397C2×C4—2541no—
398C2×C4C2×D50811yescharacteristic
399C2×C4C2×D50811yescharacteristic
400C2×C4C2×D50811yescharacteristic
401C2×C4C2×D50811yescharacteristic
402C2×C4C23×D12711yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D12712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—10164no—
413C22—10164no—
414C22—10164no—
415C22—5081no—
416C22—5081no—
417C22—5081no—
418C22—5081no—
419C22—5081no—
420C22—5081no—
421C22—5081no—
422C22—5081no—
423C22—5081no—
424C22—5081no—
425C22—5081no—
426C22—5081no—
427C22—5081no—
428C22—2542no—
429C22—2542no—
430C22#1686123712yescomplemented
431C22C22×D50811yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—5081no—
437C4—5081no—
438C4—5081no—
439C4C2×D4×D12711yescharacteristic
440C4#1686125511yescharacteristic
441C2—22no—
442C2—41no—
443C2—10161no—
444C2—10161no—
445C2—10161no—
446C2—5082no—
447C2—5082no—
448C2D4×D50811yescharacteristic
449C2C2×#1686123711yescharacteristic
450C2#2255068111yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#435×C127C2
socleC22×C127C22×D4
hypercenter#435D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. C22×D1016C2semidirect
3. #22550522C2semidirect
4. D4×D508C2semidirect
5. #22550265C2semidirect
6. #22550129C2semidirect
7. #22550105C2semidirect
8. #22550092C2semidirect
9. #22550072C2semidirect
10. #22550029C2semidirect
11. #22550023C2semidirect
12. #22550012C2semidirect
13. #22550012C2semidirect
14. #22549989C2semidirect
15. #435×C127C2semidirect
16. C2×#14523248C22semidirect
18. C22×D508C22semidirect
19. C22×D508C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523237C22semidirect
24. C2×#14523237C22semidirect
28. C2×D1016C22semidirect
29. C2×D1016C22semidirect
30. C2×D1016C22semidirect
31. C2×D1016C22semidirect
32. #16861206C22semidirect
33. #16861197C22semidirect
35. #16861188C22semidirect
38. #16861183C22semidirect
39. #16861183C22semidirect
44. C4×D508C22semidirect
45. C4×D508C22semidirect
47. #16861141C22semidirect
48. #16861134C22semidirect
49. #16861131C22semidirect
50. #16861131C22semidirect
51. #16861131C22semidirect
52. #16861131C22semidirect
53. #16861131C22semidirect
54. C2×M16×C127C22semidirect
55. C22×C8×C127C22semidirect
57. C4⋊C8×C127C22semidirect
58. C22⋊C8×C127C22semidirect
59. C22⋊C8×C127C22semidirect
60. #97×C127C22semidirect
99. D1016D4semidirect
154. D508C2×D4semidirect
180. C8×C127C2×D4semidirect
186. C22×C127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
241. #435D127semidirect
258. C2×M16C2×D127semidirect
259. C22×C8C2×D127semidirect
277. C4⋊C8C2×D127semidirect
284. C22⋊C8C2×D127semidirect
285. C22⋊C8C2×D127semidirect
286. #97C2×D127semidirect
406. C8D4×D127semidirect
430. C22#16861237semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C22×D10162
#225505222
D4×D5082
#225502652
#225501292
#225501052
#225500922
#225500722
#225500292
#225500232
#225500122
#225499892
#435×C1272
#3264127