Finite Group Databasebeta

← #30648032

Subgroups — #30648032

Order 16256.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861255C211yesmaximal, characteristic, complemented
2C2×#16861233C211yesmaximal, characteristic, complemented
3C2×#16861230C211yesmaximal, characteristic, complemented
4#22550455C211yesmaximal, characteristic, complemented
5#22550041C211yesmaximal, characteristic, complemented
6#22550038C211yesmaximal, characteristic, complemented
7#22550029C211yesmaximal, characteristic, complemented
8#22550028C211yesmaximal, characteristic, complemented
9#22550020C211yesmaximal, characteristic, complemented
10#22550009C211yesmaximal, characteristic, complemented
11#22550008C211yesmaximal, characteristic, complemented
12#22549876C211yesmaximal, characteristic, complemented
13#102×D127C211yesmaximal, characteristic, complemented
14#22549841C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16#16861255—24no—
17#16861255—24no—
18C2×#14523249C2211yescharacteristic
19C2×#14523249C2211yescharacteristic
20C2×#14523248—21no—
21C2×#14523248C2211yescharacteristic
22C2×#14523247C2211yescharacteristic, complemented
23C22×D508—21no—
24C22×D508C2211yescharacteristic
25C2×D4×D127—21no—
26C2×D4×D127—21no—
27C2×D4×D127C2211yescharacteristic
28C2×D4×D127C2211yescharacteristic
29#16861233—24no—
30#16861230—24no—
31C2×#14523238C2211yescharacteristic, complemented
32C2×#14523237C2211yescharacteristic, complemented
33C2×#14523237C2211yescharacteristic, complemented
34C2×#14523236C2211yescharacteristic, complemented
35C2×#14523233C2211yescharacteristic, complemented
36C2×#14523233C2211yescharacteristic, complemented
37C2×#14523231C2211yescharacteristic, complemented
38C2×C8×D127C2211yescharacteristic, complemented
39#16861189C2211yescharacteristic, complemented
40#16861188C2211yescharacteristic, complemented
41#16861186—21no—
42#16861182—21no—
43#16861181—21no—
44#16861180—21no—
45#16861180C2211yescharacteristic, complemented
46C4⋊C4×D127C2211yescharacteristic, complemented
47#16861133C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic, complemented
50#16861130C2211yescharacteristic, complemented
51#16861129C2211yescharacteristic, complemented
52#16861129C2211yescharacteristic, complemented
53C22⋊(C127⋊C8)C2211yescharacteristic, complemented
54(C2×D127)⋊C8C2211yescharacteristic, complemented
55(C2×D127)⋊C8C2211yescharacteristic, complemented
56C2×#41×C127C2211yescharacteristic
57C2×SD16×C127C2211yescharacteristic, complemented
58C2×D8×C127C2211yescharacteristic, complemented
59#121×C127C2211yescharacteristic, complemented
60#103×C127C2211yescharacteristic, complemented
61#102×C127C2211yescharacteristic, complemented
62C22⋊C8×C127C2211yescharacteristic, complemented
63C23×D127—21no—
64C23×D127—21no—
65C23×D127—21no—
66#14523249—22no—
67#14523249—22no—
68#14523249—22no—
69#14523249D414yescomplemented
70#14523248—28no—
71#14523248—22no—
72#14523248—22no—
73#14523247—22no—
74#14523247—22no—
75C2×Dic254D411yescharacteristic
76C2×Dic254C2311yescharacteristic
77C2×D508—28no—
78C2×D508—21no—
79C2×D508—21no—
80C2×D508—22no—
81C2×D508—22no—
82C2×D508—21no—
83C2×D508D411yescharacteristic
84C2×D508C2311yescharacteristic
85C2×D508C2311yescharacteristic
86C2×C22⋊D127—21no—
87C2×C22⋊D127—21no—
88C2×C22⋊D127—21no—
89C2×C22⋊D127—21no—
90C2×C22⋊D127D411yescharacteristic
91C2×C22⋊D127D411yescharacteristic
92D4×D127—28no—
93D4×D127—28no—
94D4×D127—22no—
95D4×D127—22no—
96D4×D127—22no—
97D4×D127D414yescomplemented
98C22×Dic127—21no—
99C2×C4×D127—21no—
100C2×C4×D127—21no—
101C2×C4×D127—21no—
102C2×C4×D127C2311yescharacteristic
103C2×C4×D127C2311yescharacteristic
104#14523238—22no—
105#14523237—22no—
106#14523237—22no—
107#14523236—22no—
108#14523233—22no—
109#14523233—22no—
110#14523231—22no—
111C2×C127⋊C8C2311yescharacteristic
112C2×C127⋊C8C2311yescharacteristic
113C8×D127—22no—
114C4⋊Dic127—21no—
115C4⋊Dic127C2311yescharacteristic
116Dic127⋊C4—21no—
117Dic127⋊C4—21no—
118C22⋊Dic127—21no—
119C22⋊Dic127—21no—
120(C2×D127)⋊C4—21no—
121#41×C127—22no—
122#41×C127—22no—
123C2×Q8×C127C2311yescharacteristic
124C2×D4×C127—21no—
125C2×D4×C127—21no—
126C2×D4×C127C2311yescharacteristic
127C2×D4×C127C2311yescharacteristic
128C22×C4×C127—21no—
129C22×C4×C127C2311yescharacteristic
130SD16×C127—22no—
131D8×C127—22no—
132C2×C8×C127C2311yescharacteristic
133C2×C8×C127C2311yescharacteristic
134C4⋊C4×C127C2311yescharacteristic
135C22⋊C4×C127—21no—
136C22×D127—21no—
137C22×D127—21no—
138C22×D127—24no—
139C22×D127—24no—
140C22×D127—24no—
141C22×D127—81no—
142C22×D127—81no—
143C22×D127—81no—
144C22×D127—81no—
145C22×D127—81no—
146C22×D127—81no—
147C22×D127C2×D411yescharacteristic
148C22×D127C2×D411yescharacteristic
149Dic254—22no—
150Dic254—21no—
151Dic254—21no—
152D508—24no—
153D508—24no—
154D508—24no—
155D508—22no—
156D508—21no—
157D508—21no—
158D508C2×D412yes—
159D508C2×D412yes—
160C22⋊D127—24no—
161C22⋊D127—22no—
162C22⋊D127—22no—
163C22⋊D127—24no—
164C22⋊D127—42no—
165C22⋊D127—81no—
166C2×Dic127—21no—
167C2×Dic127—21no—
168C2×Dic127—42no—
169C2×Dic127C2×D411yescharacteristic
170C2×Dic127C2×D411yescharacteristic
171C4×D127—24no—
172C4×D127—21no—
173C4×D127—21no—
174C4×D127—24no—
175C4×D127—81no—
176C4×D127C2×D412yes—
177C4×D127C2×D412yes—
178C127⋊C8—21no—
179C127⋊C8—21no—
180C23×C127—21no—
181C23×C127—21no—
182C23×C127C2×D411yescharacteristic
183Q8×C127—21no—
184Q8×C127C2×D412yes—
185D4×C127—24no—
186D4×C127—21no—
187D4×C127—21no—
188D4×C127—21no—
189D4×C127—81no—
190D4×C127C2×D412yes—
191C2×C4×C127—24no—
192C2×C4×C127—21no—
193C2×C4×C127—21no—
194C2×C4×C127—21no—
195C2×C4×C127—21no—
196C2×C4×C127C2×D411yescharacteristic
197C2×C4×C127C2411yescharacteristic
198C8×C127—21no—
199C8×C127—21no—
200C22×C127—21no—
201C22×C127—22no—
202C22×C127—21no—
203C22×C127—21no—
204C22×C127—41no—
205C22×C127—41no—
206C22×C127—41no—
207C22×C127—41no—
208C22×C127—41no—
209C22×C127C22×D411yescharacteristic
210C2×D127—21no—
211C2×D127—21no—
212C2×D127—21no—
213C2×D127—22no—
214C2×D127—22no—
215C2×D127—21no—
216C2×D127—21no—
217C2×D127—41no—
218C2×D127—41no—
219C2×D127—41no—
220C2×D127—41no—
221C2×D127—84no—
222C2×D127—84no—
223C2×D127—84no—
224C2×D127#12012yes—
225C4×C127—21no—
226C4×C127—22no—
227C4×C127—41no—
228C4×C127C22×D411yescharacteristic
229C4×C127C22×D411yescharacteristic
230Dic127—21no—
231Dic127—41no—
232Dic127—41no—
233Dic127#12012yes—
234C2×C127—41no—
235C2×C127—42no—
236C2×C127—81no—
237C2×C127C2×#12011yescharacteristic
238C2×C127#57611yescharacteristic
239C2×C127#57711yescharacteristic
240D127—22no—
241D127—41no—
242D127—42no—
243D127—42no—
244#3012—1271nomaximal, Sylow
245C127#301211yescharacteristic, complemented, Sylow
246C2×H(5,𝔽2)—1271no—
247C2×AGL(1,ℤ/8)—1271no—
248C2×#135—1271no—
249#537—1271no—
250#452—1271no—
251#452—1271no—
252#450—1271no—
253#450—1271no—
254#449—1271no—
255#449D12711yescharacteristic, complemented
256#447—1271no—
257#447—1271no—
258#417—1271no—
259C2×#102—1271no—
260#408—1271no—
261H(5,𝔽2)—2544no—
262H(5,𝔽2)—2544no—
263C2×#41—1271no—
264C2×#41—1271no—
265C2×#41—1271no—
266C2×#41—1271no—
267C2×#41—2541no—
268C2×#41C2×D12711yescharacteristic
269C22×D4—1271no—
270C22×D4—1271no—
271C22×D4—1271no—
272C22×D4—2541no—
273C22×D4—2541no—
274C22×D4—2541no—
275AGL(1,ℤ/8)—2544no—
276#135—2544no—
277C2×Q16—1271no—
278C2×SD16—1271no—
279C2×SD16—1271no—
280C2×SD16—1271no—
281C2×SD16C2×D12711yescharacteristic, complemented
282C2×D8—1271no—
283C2×D8—1271no—
284C2×D8C2×D12711yescharacteristic, complemented
285C2×M16—1271no—
286C22×C8—1271no—
287C22⋊Q8—1271no—
288C22⋊Q8—1271no—
289C22⋊Q8—2541no—
290#121—1271no—
291#121—2541no—
292#121—2541no—
293#121—2541no—
294#121C2×D12711yescharacteristic, complemented
295C2×C4⋊C4—1271no—
296#103—1271no—
297#103C2×D12711yescharacteristic, complemented
298#102—1271no—
299#102—1271no—
300#102—1271no—
301#102—1271no—
302#102—1271no—
303#102C2×D12711yescharacteristic, complemented
304C22⋊C8—1271no—
305C22⋊C8—1271no—
306C22⋊C8—1271no—
307C22⋊C8C2×D12711yescharacteristic, complemented
308C24—2541no—
309C24—2541no—
310C24—2541no—
311#41—22no—
312#41—22no—
313#41—1274no—
314#41—2542no—
315#41—2542no—
316#41—2548no—
317#41—2542no—
318#41—2542no—
319#41—2542no—
320#41—2542no—
321#41—2542no—
322C2×Q8—1271no—
323C2×Q8—1271no—
324C2×Q8C22×D12711yescharacteristic
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—1274no—
328C2×D4—1271no—
329C2×D4—1271no—
330C2×D4—1271no—
331C2×D4—1271no—
332C2×D4—1271no—
333C2×D4—2542no—
334C2×D4—2542no—
335C2×D4—2541no—
336C2×D4—2541no—
337C2×D4—2548no—
338C2×D4—2542no—
339C2×D4—2541no—
340C2×D4—2541no—
341C2×D4—2542no—
342C2×D4—2548no—
343C2×D4—2548no—
344C2×D4—2541no—
345C2×D4—2541no—
346C2×D4—2542no—
347C2×D4—2541no—
348C2×D4C22×D12711yescharacteristic
349C2×D4C22×D12711yescharacteristic
350C22×C4—21no—
351C22×C4—1271no—
352C22×C4—1271no—
353C22×C4—2541no—
354C22×C4—2541no—
355C22×C4—2541no—
356C22×C4—2541no—
357C22×C4C22×D12711yescharacteristic
358Q16—2542no—
359SD16—22no—
360SD16—2542no—
361SD16—2542no—
362SD16—2542no—
363D8—22no—
364D8—2542no—
365D8—2542no—
366M16—2542no—
367C2×C8—1271no—
368C2×C8—1271no—
369C2×C8—2542no—
370C2×C8C22×D12711yescharacteristic
371C2×C8C22×D12711yescharacteristic
372C4⋊C4—1271no—
373C4⋊C4—2541no—
374C4⋊C4—2541no—
375C4⋊C4—2541no—
376C4⋊C4C22×D12711yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—2541no—
379C22⋊C4—2541no—
380C22⋊C4—2541no—
381C23—21no—
382C23—21no—
383C23—1271no—
384C23—1271no—
385C23—10161no—
386C23—10161no—
387C23—10161no—
388C23—10161no—
389C23—10161no—
390C23—10161no—
391C23—2544no—
392C23—2544no—
393C23—2544no—
394C23—2541no—
395C23—2541no—
396C23D4×D12711yescharacteristic
397Q8—21no—
398Q8—2542no—
399Q8—2541no—
400Q8—2541no—
401Q8D4×D12712yes—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—21no—
406D4—81no—
407D4—1272no—
408D4—1272no—
409D4—10161no—
410D4—5082no—
411D4—2542no—
412D4—2544no—
413D4—2544no—
414D4—2544no—
415D4—2542no—
416D4—2544no—
417D4—2542no—
418D4—2544no—
419D4—2541no—
420D4—2541no—
421D4D4×D12712yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—1271no—
428C2×C4—1272no—
429C2×C4—1272no—
430C2×C4—1271no—
431C2×C4—10161no—
432C2×C4—5082no—
433C2×C4—2544no—
434C2×C4—2544no—
435C2×C4—2541no—
436C2×C4—2541no—
437C2×C4—2541no—
438C2×C4—2541no—
439C2×C4D4×D12711yescharacteristic
440C2×C4C23×D12711yescharacteristic
441C8—21no—
442C8—21no—
443C8—2541no—
444C8—2541no—
445C22—21no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—1272no—
455C22—10164no—
456C22—10164no—
457C22—10164no—
458C22—5081no—
459C22—5081no—
460C22—5081no—
461C22—5081no—
462C22—2542no—
463C22—2542no—
464C22—2541no—
465C22—2541no—
466C22—2541no—
467C22—2541no—
468C22—2541no—
469C22C2×D4×D12711yescharacteristic
470C4—21no—
471C4—22no—
472C4—41no—
473C4—1272no—
474C4—5081no—
475C4—5081no—
476C4—2541no—
477C4C2×D4×D12711yescharacteristic
478C4C2×D4×D12711yescharacteristic
479C2—41no—
480C2—42no—
481C2—81no—
482C2—5082no—
483C2—5082no—
484C2—5081no—
485C2—2542no—
486C2#120×D12711yescharacteristic
487C2#2255068011yescharacteristic
488C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3012
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2semidirect
2. C2×#16861233C2semidirect
3. C2×#16861230C2semidirect
4. #22550455C2semidirect
5. #22550041C2semidirect
6. #22550038C2semidirect
7. #22550029C2semidirect
8. #22550028C2semidirect
9. #22550020C2semidirect
10. #22550009C2semidirect
11. #22550008C2semidirect
12. #22549876C2semidirect
13. #102×D127C2semidirect
14. #22549841C2semidirect
15. #449×C127C2semidirect
22. C2×#14523247C22semidirect
31. C2×#14523238C22semidirect
32. C2×#14523237C22semidirect
33. C2×#14523237C22semidirect
34. C2×#14523236C22semidirect
35. C2×#14523233C22semidirect
36. C2×#14523233C22semidirect
37. C2×#14523231C22semidirect
38. C2×C8×D127C22semidirect
39. #16861189C22semidirect
40. #16861188C22semidirect
45. #16861180C22semidirect
46. C4⋊C4×D127C22semidirect
47. #16861133C22semidirect
48. #16861131C22semidirect
49. #16861131C22semidirect
50. #16861130C22semidirect
51. #16861129C22semidirect
52. #16861129C22semidirect
53. C22⋊(C127⋊C8)C22semidirect
54. (C2×D127)⋊C8C22semidirect
55. (C2×D127)⋊C8C22semidirect
57. C2×SD16×C127C22semidirect
58. C2×D8×C127C22semidirect
59. #121×C127C22semidirect
60. #103×C127C22semidirect
61. #102×C127C22semidirect
62. C22⋊C8×C127C22semidirect
69. #14523249D4semidirect
97. D4×D127D4semidirect
245. C127#3012semidirect
255. #449D127semidirect
281. C2×SD16C2×D127semidirect
284. C2×D8C2×D127semidirect
294. #121C2×D127semidirect
297. #103C2×D127semidirect
303. #102C2×D127semidirect
307. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612332
C2×#168612302
#225504552
#225500412
#225500382
#225500292
#225500282
#225500202
#225500092
#225500082
#225498762
#102×D1272
#225498412
#449×C1272
#3012127