Finite Group Databasebeta

← #30650271

Subgroups — #30650271

Order 16256.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861234C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550522C211yesmaximal, characteristic, complemented
4#22550263C211yesmaximal, characteristic, complemented
5#22550120C211yesmaximal, characteristic, complemented
6#22550110C211yesmaximal, characteristic, complemented
7#22550098C211yesmaximal, characteristic, complemented
8#22550068C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550021C211yesmaximal, characteristic, complemented
11#22550019C211yesmaximal, characteristic, complemented
12#22550009C211yesmaximal, characteristic, complemented
13#22549974C211yesmaximal, characteristic, complemented
14#22549948C211yesmaximal, characteristic, complemented
15#550×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×D508C2211yescharacteristic
18#16861234—24no—
19#16861229—24no—
20C2×#14523238C2211yescharacteristic, complemented
21C2×#14523236C2211yescharacteristic, complemented
22C2×#14523236C2211yescharacteristic, complemented
23C2×#14523235C2211yescharacteristic
24C2×#14523235C2211yescharacteristic
25C2×#14523233C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523232C2211yescharacteristic
30C22×C127⋊C8C2211yescharacteristic, complemented
31#16861197C2211yescharacteristic
32#16861194—21no—
33#16861188C2211yescharacteristic
34#16861183C2211yescharacteristic
35#16861182—21no—
36#16861177—21no—
37C4×D508C2211yescharacteristic
38C22⋊C4×D127—21no—
39#16861142C2211yescharacteristic, complemented
40C4⋊(C127⋊C8)C2211yescharacteristic, complemented
41#16861133C2211yescharacteristic, complemented
42#16861132C2211yescharacteristic, complemented
43#16861130C2211yescharacteristic, complemented
44#16861130C2211yescharacteristic, complemented
45#16861129C2211yescharacteristic
46#16861129C2211yescharacteristic
47C22⋊(C127⋊C8)C2211yescharacteristic
48C22⋊(C127⋊C8)C2211yescharacteristic, complemented
49#16861120C2211yescharacteristic, complemented
50C2×#41×C127C2211yescharacteristic
51C2×#41×C127C2211yescharacteristic, complemented
52#127×C127—21no—
53#127×C127C2211yescharacteristic, complemented
54#121×C127—21no—
55#121×C127—21no—
56#121×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic, complemented
58C4×Q8×C127C2211yescharacteristic, complemented
59C4×D4×C127—21no—
60C4×D4×C127C2211yescharacteristic
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#14523238—22no—
75#14523236—22no—
76#14523236—41no—
77#14523235—22no—
78#14523235D414yescomplemented
79#14523233—22no—
80#14523233—22no—
81#14523233—22no—
82#14523233—41no—
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—
96#41×C127—22no—
97#41×C127—22no—
98#41×C127—41no—
99#41×C127—41no—
100C2×Q8×C127C2311yescharacteristic
101C2×D4×C127—21no—
102C2×D4×C127—21no—
103C2×D4×C127—21no—
104C2×D4×C127—21no—
105C2×D4×C127—21no—
106C2×D4×C127—21no—
107C2×D4×C127D411yescharacteristic
108C2×D4×C127C2311yescharacteristic
109C2×D4×C127C2311yescharacteristic
110C22×C4×C127—21no—
111C22×C4×C127—21no—
112C22×C4×C127C2311yescharacteristic
113C22×C4×C127C2311yescharacteristic
114C4⋊C4×C127—21no—
115C4⋊C4×C127D411yescharacteristic
116C4⋊C4×C127C2311yescharacteristic
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C22⋊C4×C127D411yescharacteristic
120C22⋊C4×C127D411yescharacteristic
121C42×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—81no—
128Dic254—21no—
129Dic254—21no—
130D508—21no—
131D508—21no—
132D508—21no—
133D508—42no—
134D508C2×D412yes—
135C22⋊D127—42no—
136C22⋊D127—81no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C4×D127—42no—
140C4×D127—81no—
141C127⋊C8—21no—
142C127⋊C8—21no—
143C127⋊C8—21no—
144C127⋊C8C2×D412yes—
145C23×C127—21no—
146C23×C127—21no—
147C23×C127C2×D411yescharacteristic
148C23×C127C2×D411yescharacteristic
149Q8×C127—21no—
150Q8×C127C2×D412yes—
151D4×C127—21no—
152D4×C127—21no—
153D4×C127—21no—
154D4×C127—21no—
155D4×C127—41no—
156D4×C127—42no—
157D4×C127—42no—
158D4×C127—42no—
159D4×C127—81no—
160D4×C127—81no—
161D4×C127—81no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—42no—
169C2×C4×C127—42no—
170C2×C4×C127C2×D411yescharacteristic
171C2×C4×C127C2×D411yescharacteristic
172C2×C4×C127C2×D411yescharacteristic
173C2×C4×C127C2×D411yescharacteristic
174C2×C4×C127C2411yescharacteristic
175C22×C127—21no—
176C22×C127—21no—
177C22×C127—21no—
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—21no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127C22×D411yescharacteristic
188C2×D127—22no—
189C2×D127—41no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—84no—
195C4×C127—21no—
196C4×C127—21no—
197C4×C127—22no—
198C4×C127—21no—
199C4×C127—21no—
200C4×C127—41no—
201C4×C127C22×D411yescharacteristic
202C4×C127H(5,𝔽2)11yescharacteristic
203Dic127—41no—
204Dic127—41no—
205C2×C127—41no—
206C2×C127—41no—
207C2×C127—81no—
208C2×C127—81no—
209C2×C127D4211yescharacteristic
210C2×C127#57511yescharacteristic
211C2×C127#57611yescharacteristic
212D127—42no—
213D127—81no—
214#3268—1271nomaximal, Sylow
215C127#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1271no—
217C2×#135—1271no—
218#550D12711yescharacteristic, complemented
219#546—1271no—
220#496—1271no—
221#468—1271no—
222#466—1271no—
223#464—1271no—
224#459—1271no—
225#452—1271no—
226#449—1271no—
227#449—1271no—
228#447—1271no—
229#440—1271no—
230#435—1271no—
231C2×#41—1271no—
232C2×#41C2×D12711yescharacteristic
233C2×#41C2×D12711yescharacteristic, complemented
234C22×D4—1271no—
235AGL(1,ℤ/8)—2544no—
236#135—2544no—
237C2×Q16—1271no—
238C2×SD16—1271no—
239C2×SD16—1271no—
240C2×SD16—1271no—
241C2×SD16—1271no—
242C2×D8—1271no—
243C2×D8—1271no—
244C2×D8—1271no—
245C2×D8—1271no—
246C2×M16—1271no—
247C22×C8—1271no—
248#127—21no—
249#127C2×D12711yescharacteristic, complemented
250#124—1271no—
251#123—2541no—
252C22⋊Q8—1271no—
253#121—21no—
254#121—21no—
255#121—1271no—
256#121—2541no—
257#121C2×D12711yescharacteristic
258#121C2×D12711yescharacteristic, complemented
259#120—2541no—
260C4×Q8C2×D12711yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—1271no—
263C4×D4C2×D12711yescharacteristic
264C2×C22⋊C4—2541no—
265#107—1271no—
266C4⋊C8—1271no—
267#103—1271no—
268#103—1271no—
269#102—1271no—
270#102—1271no—
271#102—1271no—
272#102—1271no—
273C22⋊C8—1271no—
274C22⋊C8—1271no—
275#97—1271no—
276C24—2541no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—5081no—
282#41—2542no—
283C2×Q8—1271no—
284C2×Q8C22×D12711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—1271no—
292C2×D4—1271no—
293C2×D4—5081no—
294C2×D4—2542no—
295C2×D4—2541no—
296C2×D4—2541no—
297C2×D4—2541no—
298C2×D4C22⋊D12711yescharacteristic
299C2×D4C22×D12711yescharacteristic
300C2×D4C22×D12711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—2541no—
304C22×C4—2541no—
305C22×C4C22×D12711yescharacteristic
306C22×C4C22×D12711yescharacteristic
307Q16—2542no—
308SD16—1274no—
309SD16—5081no—
310SD16—2542no—
311SD16—2542no—
312D8—5081no—
313D8—2542no—
314D8—2542no—
315D8—2542no—
316M16—2542no—
317C2×C8—1271no—
318C2×C8—1271no—
319C2×C8—1271no—
320C2×C8—1271no—
321C2×C8—2542no—
322C4⋊C4—21no—
323C4⋊C4—1271no—
324C4⋊C4—2541no—
325C4⋊C4C22⋊D12711yescharacteristic
326C4⋊C4C22×D12711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—2541no—
330C22⋊C4—2541no—
331C22⋊C4—2541no—
332C22⋊C4—2541no—
333C22⋊C4—2541no—
334C22⋊C4C22⋊D12711yescharacteristic
335C22⋊C4C22⋊D12711yescharacteristic
336C42—21no—
337C42C22×D12711yescharacteristic
338C23—21no—
339C23—21no—
340C23—10161no—
341C23—5082no—
342C23—5082no—
343C23—2541no—
344C23—2541no—
345C23C2×C22⋊D12711yescharacteristic
346C23C2×C22⋊D12711yescharacteristic
347Q8—21no—
348Q8—2541no—
349Q8—2541no—
350Q8D4×D12712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—1272no—
363D4—10161no—
364D4—5082no—
365D4—5082no—
366D4—2541no—
367D4—2541no—
368D4—2541no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—10161no—
378C2×C4—5082no—
379C2×C4—2541no—
380C2×C4—2541no—
381C2×C4C2×C22⋊D12711yescharacteristic
382C2×C4C2×C22⋊D12711yescharacteristic
383C2×C4C2×C22⋊D12711yescharacteristic
384C2×C4C2×C22⋊D12711yescharacteristic
385C2×C4C23×D12711yescharacteristic
386C8—1272no—
387C8—2541no—
388C8—2541no—
389C8—2541no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—10164no—
403C22—5081no—
404C22—5081no—
405C22—5081no—
406C22—5081no—
407C22—5081no—
408C22—2542no—
409C22C22×C22⋊D12711yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—5081no—
417C4—5081no—
418C4C2×D4×D12711yescharacteristic
419C4#1686125411yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—10161no—
425C2—5082no—
426C2D4×C22⋊D12711yescharacteristic
427C2#2255067411yescharacteristic
428C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#550×C127C2
socleC22×C127C22×D4
hypercenter#550D127
hypocenterC127#3268
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×#16861229C2semidirect
3. #22550522C2semidirect
4. #22550263C2semidirect
5. #22550120C2semidirect
6. #22550110C2semidirect
7. #22550098C2semidirect
8. #22550068C2semidirect
9. #22550038C2semidirect
10. #22550021C2semidirect
11. #22550019C2semidirect
12. #22550009C2semidirect
13. #22549974C2semidirect
14. #22549948C2semidirect
15. #550×C127C2semidirect
20. C2×#14523238C22semidirect
21. C2×#14523236C22semidirect
22. C2×#14523236C22semidirect
30. C22×C127⋊C8C22semidirect
39. #16861142C22semidirect
40. C4⋊(C127⋊C8)C22semidirect
41. #16861133C22semidirect
42. #16861132C22semidirect
43. #16861130C22semidirect
44. #16861130C22semidirect
48. C22⋊(C127⋊C8)C22semidirect
49. #16861120C22semidirect
51. C2×#41×C127C22semidirect
53. #127×C127C22semidirect
57. #121×C127C22semidirect
58. C4×Q8×C127C22semidirect
78. #14523235D4semidirect
215. C127#3268semidirect
218. #550D127semidirect
233. C2×#41C2×D127semidirect
249. #127C2×D127semidirect
258. #121C2×D127semidirect
260. C4×Q8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168612292
#225505222
#225502632
#225501202
#225501102
#225500982
#225500682
#225500382
#225500212
#225500192
#225500092
#225499742
#225499482
#550×C1272
#3268127