Finite Group Databasebeta

← #30650337

Subgroups — #30650337

Order 16256.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861235C211yesmaximal, characteristic, complemented
2C2×#16861234C211yesmaximal, characteristic, complemented
3#22550522C211yesmaximal, characteristic, complemented
4#22550258C211yesmaximal, characteristic, complemented
5#22550124C211yesmaximal, characteristic, complemented
6#22550120C211yesmaximal, characteristic, complemented
7#22550091C211yesmaximal, characteristic, complemented
8#22550088C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550036C211yesmaximal, characteristic, complemented
11#22550010C211yesmaximal, characteristic, complemented
12#22550009C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549954C211yesmaximal, characteristic, complemented
15#546×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×D508C2211yescharacteristic, complemented
18#16861235—24no—
19#16861234—24no—
20C2×#14523238C2211yescharacteristic, complemented
21C2×#14523236C2211yescharacteristic, complemented
22C2×#14523236C2211yescharacteristic, complemented
23C2×#14523235C2211yescharacteristic, complemented
24C2×#14523235C2211yescharacteristic, complemented
25C2×#14523233C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523232C2211yescharacteristic, complemented
30C2×#14523232C2211yescharacteristic, complemented
31#16861197C2211yescharacteristic, complemented
32#16861194—21no—
33#16861188C2211yescharacteristic, complemented
34#16861183C2211yescharacteristic
35#16861182—21no—
36#16861177—21no—
37C4×D508C2211yescharacteristic, complemented
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, complemented
46#16861129C2211yescharacteristic, complemented
47C22⋊(C127⋊C8)C2211yescharacteristic, complemented
48C22⋊(C127⋊C8)C2211yescharacteristic, complemented
49C4×C127⋊C8C2211yescharacteristic, complemented
50C2×#41×C127C2211yescharacteristic
51C22×D4×C127C2211yescharacteristic, complemented
52#124×C127C2211yescharacteristic, complemented
53#123×C127—21no—
54C22⋊Q8×C127C2211yescharacteristic, complemented
55#121×C127—21no—
56#121×C127C2211yescharacteristic
57#120×C127—21no—
58C4×D4×C127C2211yescharacteristic
59C4×D4×C127C2211yescharacteristic, complemented
60C2×C22⋊C4×C127—21no—
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—41no—
75#14523236—22no—
76#14523236—22no—
77#14523235—22no—
78#14523235—22no—
79#14523233—22no—
80#14523233—22no—
81#14523233—41no—
82#14523233D414yescomplemented
83#14523232—22no—
84#14523232—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—
96C24×C127—21no—
97#41×C127—22no—
98#41×C127—41no—
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127—22no—
101C2×D4×C127—21no—
102C2×D4×C127—21no—
103C2×D4×C127—21no—
104C2×D4×C127—41no—
105C2×D4×C127D411yescharacteristic
106C2×D4×C127C2311yescharacteristic
107C2×D4×C127C2311yescharacteristic
108C22×C4×C127—21no—
109C22×C4×C127—21no—
110C22×C4×C127C2311yescharacteristic
111C22×C4×C127C2311yescharacteristic
112C4⋊C4×C127—21no—
113C4⋊C4×C127D411yescharacteristic
114C4⋊C4×C127C2311yescharacteristic
115C22⋊C4×C127—21no—
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C22⋊C4×C127D411yescharacteristic
121C22⋊C4×C127D411yescharacteristic
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×C127—42no—
148C23×C127—42no—
149C23×C127—81no—
150C23×C127C2×D411yescharacteristic
151C23×C127C2×D411yescharacteristic
152Q8×C127—21no—
153Q8×C127—21no—
154D4×C127—21no—
155D4×C127—21no—
156D4×C127—21no—
157D4×C127—41no—
158D4×C127—42no—
159D4×C127—42no—
160D4×C127—81no—
161D4×C127C2×D412yes—
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—81no—
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—22no—
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—21no—
181C22×C127—21no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—84no—
188C22×C127C22×D411yescharacteristic
189C2×D127—22no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—41no—
195C2×D127—84no—
196C4×C127—21no—
197C4×C127—21no—
198C4×C127—21no—
199C4×C127—21no—
200C4×C127—41no—
201C4×C127—41no—
202C4×C127C22×D411yescharacteristic
203C4×C127H(5,𝔽2)11yescharacteristic
204Dic127—41no—
205Dic127—41no—
206C2×C127—41no—
207C2×C127—41no—
208C2×C127—42no—
209C2×C127—81no—
210C2×C127D4211yescharacteristic
211C2×C127#57511yescharacteristic
212C2×C127#57611yescharacteristic
213D127—42no—
214D127—81no—
215#3273—1271nomaximal, Sylow
216C127#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—1271no—
218C2×AGL(1,ℤ/8)—1271no—
219#546—1271no—
220#546D12711yescharacteristic, complemented
221#495—1271no—
222#468—1271no—
223#468—1271no—
224#463—1271no—
225#463—1271no—
226#452—1271no—
227#452—1271no—
228#447—1271no—
229#447—1271no—
230C4×D8—1271no—
231#436—1271no—
232C2×#41—1271no—
233C2×#41C2×D12711yescharacteristic
234C22×D4—1271no—
235C22×D4C2×D12711yescharacteristic, complemented
236AGL(1,ℤ/8)—2544no—
237AGL(1,ℤ/8)—2544no—
238C2×Q16—1271no—
239C2×SD16—1271no—
240C2×SD16—1271no—
241C2×SD16—1271no—
242C2×SD16—1271no—
243C2×D8—1271no—
244C2×D8—1271no—
245C2×D8—1271no—
246C2×D8—1271no—
247C2×M16—1271no—
248C2×M16—1271no—
249#124—1271no—
250#124C2×D12711yescharacteristic, complemented
251#123—21no—
252#123—2541no—
253C22⋊Q8—1271no—
254C22⋊Q8C2×D12711yescharacteristic, complemented
255#121—21no—
256#121—1271no—
257#121—2541no—
258#121C2×D12711yescharacteristic
259#120—21no—
260#120—2541no—
261C4×D4—1271no—
262C4×D4C2×D12711yescharacteristic
263C4×D4C2×D12711yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—2541no—
266#107—1271no—
267C4⋊C8—1271no—
268#103—1271no—
269#103—1271no—
270#102—1271no—
271#102—1271no—
272#102—1271no—
273#102—1271no—
274C22⋊C8—1271no—
275C22⋊C8—1271no—
276C4×C8—1271no—
277C24—21no—
278C24—2541no—
279#41—22no—
280#41—41no—
281#41—5081no—
282#41—2542no—
283C2×Q8—1271no—
284C2×Q8C22×D12711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—1271no—
291C2×D4—1271no—
292C2×D4—5081no—
293C2×D4—2542no—
294C2×D4—2541no—
295C2×D4—2541no—
296C2×D4—2541no—
297C2×D4C22⋊D12711yescharacteristic
298C2×D4C22×D12711yescharacteristic
299C2×D4C22×D12711yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—2541no—
303C22×C4—2541no—
304C22×C4C22×D12711yescharacteristic
305C22×C4C22×D12711yescharacteristic
306Q16—5081no—
307SD16—2542no—
308SD16—2542no—
309SD16—2542no—
310SD16—2542no—
311D8—1274no—
312D8—5081no—
313D8—2542no—
314D8—2542no—
315M16—2542no—
316M16—2542no—
317C2×C8—1271no—
318C2×C8—1271no—
319C2×C8—1271no—
320C2×C8—1271no—
321C4⋊C4—21no—
322C4⋊C4—1271no—
323C4⋊C4—2541no—
324C4⋊C4C22⋊D12711yescharacteristic
325C4⋊C4C22×D12711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—2541no—
332C22⋊C4—2541no—
333C22⋊C4—2541no—
334C22⋊C4—2541no—
335C22⋊C4—2541no—
336C22⋊C4C22⋊D12711yescharacteristic
337C22⋊C4C22⋊D12711yescharacteristic
338C42C22×D12711yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—10161no—
345C23—5082no—
346C23—5082no—
347C23—2541no—
348C23—2541no—
349C23C2×C22⋊D12711yescharacteristic
350C23C2×C22⋊D12711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2541no—
354Q8—2541no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—1272no—
363D4—10161no—
364D4—5082no—
365D4—5082no—
366D4—2541no—
367D4—2541no—
368D4—2541no—
369D4D4×D12712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—10161no—
379C2×C4—5082no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4C2×C22⋊D12711yescharacteristic
383C2×C4C2×C22⋊D12711yescharacteristic
384C2×C4C2×C22⋊D12711yescharacteristic
385C2×C4C2×C22⋊D12711yescharacteristic
386C2×C4C23×D12711yescharacteristic
387C8—1272no—
388C8—2541no—
389C8—2541no—
390C8—2541no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—10164no—
405C22—5081no—
406C22—5081no—
407C22—5081no—
408C22—5081no—
409C22—5081no—
410C22—2542no—
411C22C22×C22⋊D12711yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—5081no—
419C4—5081no—
420C4C2×D4×D12711yescharacteristic
421C4#1686125411yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—10161no—
427C2—5082no—
428C2D4×C22⋊D12711yescharacteristic
429C2#2255067411yescharacteristic
430C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#546×C127C2
socleC22×C127C22×D4
hypercenter#546D127
hypocenterC127#3273
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. C2×#16861234C2semidirect
3. #22550522C2semidirect
4. #22550258C2semidirect
5. #22550124C2semidirect
6. #22550120C2semidirect
7. #22550091C2semidirect
8. #22550088C2semidirect
9. #22550038C2semidirect
10. #22550036C2semidirect
11. #22550010C2semidirect
12. #22550009C2semidirect
13. C4×#14523233C2semidirect
14. #22549954C2semidirect
15. #546×C127C2semidirect
17. C22×D508C22semidirect
20. C2×#14523238C22semidirect
21. C2×#14523236C22semidirect
22. C2×#14523236C22semidirect
23. C2×#14523235C22semidirect
24. C2×#14523235C22semidirect
29. C2×#14523232C22semidirect
30. C2×#14523232C22semidirect
31. #16861197C22semidirect
33. #16861188C22semidirect
37. C4×D508C22semidirect
39. #16861142C22semidirect
40. C4⋊(C127⋊C8)C22semidirect
41. #16861133C22semidirect
42. #16861132C22semidirect
43. #16861130C22semidirect
44. #16861130C22semidirect
45. #16861129C22semidirect
46. #16861129C22semidirect
47. C22⋊(C127⋊C8)C22semidirect
48. C22⋊(C127⋊C8)C22semidirect
49. C4×C127⋊C8C22semidirect
51. C22×D4×C127C22semidirect
52. #124×C127C22semidirect
54. C22⋊Q8×C127C22semidirect
59. C4×D4×C127C22semidirect
82. #14523233D4semidirect
216. C127#3273semidirect
220. #546D127semidirect
235. C22×D4C2×D127semidirect
250. #124C2×D127semidirect
254. C22⋊Q8C2×D127semidirect
263. C4×D4C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C2×#168612342
#225505222
#225502582
#225501242
#225501202
#225500912
#225500882
#225500382
#225500362
#225500102
#225500092
C4×#145232332
#225499542
#546×C1272
#3273127