Finite Group Databasebeta

← #30650221

Subgroups — #30650221

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×#16861235C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3#22550535C211yesmaximal, characteristic, complemented
4#22550263C211yesmaximal, characteristic, complemented
5#22550120C211yesmaximal, characteristic, complemented
6#22550110C211yesmaximal, characteristic, complemented
7#22550091C211yesmaximal, characteristic, complemented
8#22550073C211yesmaximal, characteristic, complemented
9#22550027C211yesmaximal, characteristic, complemented
10#22550019C211yesmaximal, characteristic, complemented
11#22550014C211yesmaximal, characteristic, complemented
12#22550010C211yesmaximal, characteristic, complemented
13#22549976C211yesmaximal, characteristic, complemented
14#22549948C211yesmaximal, characteristic, complemented
15D42×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic, complemented
17C22×Dic254C2211yescharacteristic, complemented
18#16861235—24no—
19C2×#14523236—24no—
20C2×#14523236C2211yescharacteristic
21C2×#14523236C2211yescharacteristic
22C2×#14523236C2211yescharacteristic, complemented
23C2×#14523236C2211yescharacteristic, complemented
24C2×#14523236C2211yescharacteristic, complemented
25C2×#14523236C2211yescharacteristic, complemented
26C2×#14523233C2211yescharacteristic
27C2×#14523233C2211yescharacteristic, complemented
28C2×#14523233C2211yescharacteristic, complemented
29C2×#14523232C2211yescharacteristic, complemented
30C22×C127⋊C8C2211yescharacteristic, complemented
31#16861197C2211yescharacteristic, complemented
32#16861195—21no—
33#16861188C2211yescharacteristic, complemented
34#16861187—21no—
35#16861183C2211yescharacteristic, complemented
36#16861181—21no—
37C4×Dic254C2211yescharacteristic, complemented
38(C22⋊D127)⋊C4—21no—
39#16861142C2211yescharacteristic, complemented
40C4⋊(C127⋊C8)C2211yescharacteristic, complemented
41#16861132C2211yescharacteristic, complemented
42#16861132C2211yescharacteristic, complemented
43#16861130C2211yescharacteristic
44#16861130C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic, 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
53#127×C127C2211yescharacteristic
54#121×C127—21no—
55#121×C127C2211yescharacteristic
56#121×C127C2211yescharacteristic
57#120×C127—21no—
58#120×C127—21no—
59C4×D4×C127C2211yescharacteristic
60C4×D4×C127C2211yescharacteristic
61#14523248—22no—
62#14523248—41no—
63C2×Dic254—22no—
64C2×Dic254—21no—
65C2×Dic254—41no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73#14523236—22no—
74#14523236—22no—
75#14523236—22no—
76#14523236—22no—
77#14523236—41no—
78#14523236D414yescomplemented
79#14523233—22no—
80#14523233—22no—
81#14523233—41no—
82#14523232—22no—
83C2×C127⋊C8—22no—
84C2×C127⋊C8C2311yescharacteristic
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C2×C127⋊C8C2311yescharacteristic
88C4⋊Dic127C2311yescharacteristic
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95C4×Dic127—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—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—42no—
129Dic254C2×D412yescomplemented
130D508—21no—
131D508—21no—
132C22⋊D127—42no—
133C22⋊D127—42no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—42no—
138C4×D127—42no—
139C127⋊C8—21no—
140C127⋊C8—21no—
141C127⋊C8—21no—
142C127⋊C8C2×D412yescomplemented
143C23×C127—21no—
144C23×C127—21no—
145C23×C127—21no—
146C23×C127—42no—
147C23×C127—42no—
148C23×C127—42no—
149C23×C127—42no—
150C23×C127—81no—
151C23×C127—81no—
152C23×C127C2×D411yescharacteristic
153C23×C127C2×D411yescharacteristic
154D4×C127—22no—
155D4×C127—21no—
156D4×C127—21no—
157D4×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—42no—
161D4×C127—42no—
162D4×C127—42no—
163D4×C127—42no—
164D4×C127—42no—
165D4×C127—81no—
166D4×C127C2×D412yes—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
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×C127C2×D411yescharacteristic
176C2×C4×C127C2×D411yescharacteristic
177C2×C4×C127C2411yescharacteristic
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—21no—
181C22×C127—22no—
182C22×C127—21no—
183C22×C127—21no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127—41no—
191C22×C127—41no—
192C22×C127—84no—
193C22×C127—84no—
194C22×C127AGL(1,ℤ/8)12yescomplemented
195C22×C127C22×D411yescharacteristic
196C2×D127—41no—
197C2×D127—41no—
198C2×D127—41no—
199C4×C127—21no—
200C4×C127—21no—
201C4×C127—21no—
202C4×C127—21no—
203C4×C127—41no—
204C4×C127C22×D411yescharacteristic
205C4×C127H(5,𝔽2)11yescharacteristic
206Dic127—22no—
207Dic127—41no—
208Dic127—41no—
209C2×C127—22no—
210C2×C127—41no—
211C2×C127—42no—
212C2×C127—81no—
213C2×C127—81no—
214C2×C127D4211yescharacteristic
215C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C127#57711yescharacteristic
217D127—81no—
218#3265—1271nomaximal, Sylow
219C127#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1271no—
221C22×SD16—1271no—
222#548—1271no—
223D42D12711yescharacteristic, complemented
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#440—1271no—
234#435—1271no—
235C2×#41—1271no—
236C22×Q8—1271no—
237C22×D4—21no—
238C22×D4C2×D12711yescharacteristic
239C22×D4C2×D12711yescharacteristic
240AGL(1,ℤ/8)—2544no—
241C2×SD16—1271no—
242C2×SD16—1271no—
243C2×SD16—1271no—
244C2×SD16—1271no—
245C2×SD16—1271no—
246C2×SD16—1271no—
247C2×SD16—2544no—
248C2×D8—1271no—
249C2×D8—1271no—
250C2×D8—1271no—
251C2×M16—1271no—
252C22×C8—1271no—
253#127C2×D12711yescharacteristic
254#124—1271no—
255#124—2541no—
256C22⋊Q8—1271no—
257C22⋊Q8—2541no—
258#121—21no—
259#121—1271no—
260#121—2541no—
261#121C2×D12711yescharacteristic
262#121C2×D12711yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—1271no—
266C4×D4—2541no—
267C4×D4C2×D12711yescharacteristic
268C4×D4C2×D12711yescharacteristic
269#107—1271no—
270C4⋊C8—1271no—
271#103—1271no—
272#103—1271no—
273#102—1271no—
274#102—1271no—
275#102—1271no—
276#102—1271no—
277C22⋊C8—1271no—
278C22⋊C8—1271no—
279#97—1271no—
280C24—21no—
281C24—21no—
282#41—5081no—
283#41—2542no—
284C2×Q8—1271no—
285C2×Q8—1271no—
286C2×Q8—5081no—
287C2×Q8—2542no—
288C2×Q8—2541no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—1271no—
301C2×D4—2541no—
302C2×D4—2541no—
303C2×D4C22⋊D12711yescharacteristic
304C2×D4C22×D12711yescharacteristic
305C2×D4C22×D12711yescharacteristic
306C2×D4C22×D12711yescharacteristic
307C22×C4—21no—
308C22×C4—2541no—
309C22×C4—2541no—
310C22×C4C22×D12711yescharacteristic
311C22×C4C22×D12711yescharacteristic
312SD16—1274no—
313SD16—5081no—
314SD16—2542no—
315SD16—2542no—
316SD16—2542no—
317SD16—2542no—
318D8—5081no—
319D8—2542no—
320D8—2542no—
321M16—2542no—
322C2×C8—1271no—
323C2×C8—1271no—
324C2×C8—1271no—
325C2×C8—1271no—
326C2×C8—2542no—
327C4⋊C4—1271no—
328C4⋊C4—2541no—
329C4⋊C4—2541no—
330C4⋊C4C22⋊D12711yescharacteristic
331C4⋊C4C22×D12711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—2541no—
336C22⋊C4—2541no—
337C22⋊C4—2541no—
338C22⋊C4—2541no—
339C22⋊C4C22⋊D12711yescharacteristic
340C22⋊C4C22⋊D12711yescharacteristic
341C42—2541no—
342C42C22×D12711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—2541no—
353C23C2×C22⋊D12711yescharacteristic
354C23C2×C22⋊D12711yescharacteristic
355Q8—1272no—
356Q8—5082no—
357Q8—2541no—
358Q8—2541no—
359Q8—2541no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—5082no—
373D4—5082no—
374D4—2541no—
375D4—2541no—
376D4D4×D12712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—5082no—
384C2×C4—5082no—
385C2×C4—2541no—
386C2×C4—2541no—
387C2×C4—2541no—
388C2×C4C2×C22⋊D12711yescharacteristic
389C2×C4C2×C22⋊D12711yescharacteristic
390C2×C4C2×C22⋊D12711yescharacteristic
391C2×C4C2×C22⋊D12711yescharacteristic
392C2×C4C23×D12711yescharacteristic
393C8—1272no—
394C8—2541no—
395C8—2541no—
396C8—2541no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—5081no—
414C22—5081no—
415C22—5081no—
416C22#1686123512yescomplemented
417C22C22×C22⋊D12711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—5081no—
424C4—5081no—
425C4—2542no—
426C4C2×D4×D12711yescharacteristic
427C4#1686125411yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—10161no—
434C2D4×C22⋊D12711yescharacteristic
435C2C2×#1686123511yescharacteristic
436C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD42×C127C2
socleC22×C127C22×D4
hypercenterD42D127
hypocenterC127#3265
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×#14523236C2semidirect
3. #22550535C2semidirect
4. #22550263C2semidirect
5. #22550120C2semidirect
6. #22550110C2semidirect
7. #22550091C2semidirect
8. #22550073C2semidirect
9. #22550027C2semidirect
10. #22550019C2semidirect
11. #22550014C2semidirect
12. #22550010C2semidirect
13. #22549976C2semidirect
14. #22549948C2semidirect
15. D42×C127C2semidirect
16. C2×#14523248C22semidirect
17. C22×Dic254C22semidirect
22. C2×#14523236C22semidirect
23. C2×#14523236C22semidirect
24. C2×#14523236C22semidirect
25. C2×#14523236C22semidirect
27. C2×#14523233C22semidirect
28. C2×#14523233C22semidirect
29. C2×#14523232C22semidirect
30. C22×C127⋊C8C22semidirect
31. #16861197C22semidirect
33. #16861188C22semidirect
35. #16861183C22semidirect
37. C4×Dic254C22semidirect
39. #16861142C22semidirect
40. C4⋊(C127⋊C8)C22semidirect
41. #16861132C22semidirect
42. #16861132C22semidirect
44. #16861130C22semidirect
45. #16861130C22semidirect
46. #16861129C22semidirect
47. C22⋊(C127⋊C8)C22semidirect
48. C22⋊(C127⋊C8)C22semidirect
49. #16861120C22semidirect
78. #14523236D4semidirect
129. Dic254C2×D4semidirect
142. C127⋊C8C2×D4semidirect
194. C22×C127AGL(1,ℤ/8)semidirect
219. C127#3265semidirect
223. D42D127semidirect
416. 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×#145232362
#225505352
#225502632
#225501202
#225501102
#225500912
#225500732
#225500272
#225500192
#225500142
#225500102
#225499762
#225499482
D42×C1272
#3265127