Finite Group Databasebeta

← #30650344

Subgroups — #30650344

Order 16256.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2#22550545C211yesmaximal, characteristic, complemented
3D4×C22⋊D127C211yesmaximal, characteristic, complemented
4#22550251C211yesmaximal, characteristic, complemented
5#22550129C211yesmaximal, characteristic, complemented
6#22550126C211yesmaximal, characteristic, complemented
7#22550074C211yesmaximal, characteristic, complemented
8#22550069C211yesmaximal, characteristic, complemented
9#22550021C211yesmaximal, characteristic, complemented
10#22550020C211yesmaximal, characteristic, complemented
11#22550010C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13SD16×Dic127C211yesmaximal, characteristic, complemented
14#22549952C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic, complemented
17C22×C22⋊D127—21no—
18C2×D4×D127C2211yescharacteristic
19#16861236—24no—
20C2×#14523235C2211yescharacteristic, complemented
21C2×D1016C2211yescharacteristic, complemented
22C2×#14523233C2211yescharacteristic
23C2×#14523233C2211yescharacteristic, complemented
24C2×#14523233C2211yescharacteristic, complemented
25C2×#14523231C2211yescharacteristic, complemented
26#16861205—21no—
27#16861205C2211yescharacteristic
28#16861205C2211yescharacteristic, complemented
29#16861183—21no—
30#16861183C2211yescharacteristic
31#16861183C2211yescharacteristic, complemented
32#16861181—21no—
33#16861180—21no—
34#16861180C2211yescharacteristic
35#16861180C2211yescharacteristic, complemented
36#16861178—21no—
37#16861177—21no—
38Q8×Dic127C2211yescharacteristic, complemented
39D4×Dic127C2211yescharacteristic
40C4×C22⋊D127—21no—
41C4×C22⋊D127C2211yescharacteristic
42#16861141C2211yescharacteristic, complemented
43Dic127⋊C8C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50C22⋊(C127⋊C8)C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52C8×Dic127C2211yescharacteristic, complemented
53C2×#41×C127C2211yescharacteristic, complemented
54C22×D4×C127C2211yescharacteristic
55AGL(1,ℤ/8)×C127—24no—
56C2×SD16×C127C2211yescharacteristic
57C2×SD16×C127C2211yescharacteristic, complemented
58C2×D8×C127C2211yescharacteristic
59C2×D8×C127C2211yescharacteristic, complemented
60C2×M16×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—41no—
64C2×D508—21no—
65C2×D508C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—44no—
77C2×C22⋊D127D411yescharacteristic
78D4×D127—22no—
79D4×D127—41no—
80C22×Dic127—21no—
81C2×C4×D127—21no—
82C2×C4×D127C2311yescharacteristic
83#14523235—22no—
84D1016—22no—
85#14523233—22no—
86#14523233—41no—
87#14523233—41no—
88#14523231—22no—
89C2×C127⋊C8C2311yescharacteristic
90C2×C127⋊C8C2311yescharacteristic
91C4⋊Dic127—21no—
92C4⋊Dic127C2311yescharacteristic
93C4⋊Dic127C2311yescharacteristic
94Dic127⋊C4D411yescharacteristic
95C22⋊Dic127—21no—
96C22⋊Dic127—21no—
97C22⋊Dic127—21no—
98C22⋊Dic127D411yescharacteristic
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4D411yescharacteristic
102C4×Dic127—21no—
103C4×Dic127C2311yescharacteristic
104C24×C127—21no—
105#41×C127—22no—
106#41×C127—41no—
107C2×Q8×C127C2311yescharacteristic
108C2×D4×C127—22no—
109C2×D4×C127—21no—
110C2×D4×C127—21no—
111C2×D4×C127—41no—
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127—21no—
116C22×C4×C127C2311yescharacteristic
117SD16×C127—22no—
118SD16×C127D414yescomplemented
119D8×C127—22no—
120D8×C127—22no—
121M16×C127—22no—
122C2×C8×C127C2311yescharacteristic
123C2×C8×C127C2311yescharacteristic
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—81no—
129C22×D127C2×D411yescharacteristic
130D508—21no—
131D508—21no—
132D508—21no—
133D508—21no—
134D508—42no—
135C22⋊D127—22no—
136C22⋊D127—42no—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C22⋊D127—81no—
142C22⋊D127—81no—
143C22⋊D127—81no—
144C22⋊D127—81no—
145C2×Dic127—21no—
146C2×Dic127—21no—
147C2×Dic127—42no—
148C2×Dic127C2×D411yescharacteristic
149C2×Dic127C2×D411yescharacteristic
150C2×Dic127C2×D411yescharacteristic
151C4×D127—21no—
152C4×D127—21no—
153C4×D127—42no—
154C127⋊C8—21no—
155C127⋊C8—21no—
156C23×C127—21no—
157C23×C127—21no—
158C23×C127—21no—
159C23×C127—42no—
160C23×C127—42no—
161C23×C127—81no—
162C23×C127C2×D411yescharacteristic
163Q8×C127—21no—
164Q8×C127C2×D412yescomplemented
165D4×C127—21no—
166D4×C127—21no—
167D4×C127—21no—
168D4×C127—21no—
169D4×C127—21no—
170D4×C127—42no—
171D4×C127—42no—
172D4×C127C2×D412yes—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—42no—
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127C2×D412yescomplemented
181C22×C127—21no—
182C22×C127—22no—
183C22×C127—21no—
184C22×C127—21no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127—41no—
191C22×C127—41no—
192C22×C127—41no—
193C22×C127—84no—
194C22×C127C22×D411yescharacteristic
195C2×D127—21no—
196C2×D127—21no—
197C2×D127—21no—
198C2×D127—41no—
199C2×D127—41no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—41no—
203C2×D127—41no—
204C2×D127—84no—
205C4×C127—21no—
206C4×C127—22no—
207C4×C127C22×D411yescharacteristic
208C4×C127H(5,𝔽2)11yescharacteristic
209Dic127—21no—
210Dic127—21no—
211Dic127—41no—
212Dic127—41no—
213Dic127AGL(1,ℤ/8)12yescomplemented
214C2×C127—42no—
215C2×C127—41no—
216C2×C127—81no—
217C2×C127—81no—
218C2×C127D4211yescharacteristic
219C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C127#57611yescharacteristic
221D127—41no—
222D127—81no—
223D127—81no—
224#3274—1271nomaximal, Sylow
225C127#327411yescharacteristic, complemented, Sylow
226C2×AGL(1,ℤ/8)—1271no—
227C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
228#550—1271no—
229D42—1271no—
230#493—1271no—
231#469—1271no—
232#469—1271no—
233#460—1271no—
234#459—1271no—
235#449—1271no—
236#449—1271no—
237#447—1271no—
238#447—1271no—
239C4×SD16—1271no—
240#436—1271no—
241C2×#41—1271no—
242C2×#41C2×D12711yescharacteristic, complemented
243C22×D4—1271no—
244C22×D4—2541no—
245C22×D4C2×D12711yescharacteristic
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—2544no—
248C2×SD16—1271no—
249C2×SD16C2×D12711yescharacteristic
250C2×SD16C2×D12711yescharacteristic, complemented
251C2×D8—1271no—
252C2×D8—1271no—
253C2×D8—1271no—
254C2×D8—1271no—
255C2×D8C2×D12711yescharacteristic
256C2×D8C2×D12711yescharacteristic, complemented
257C2×M16—1271no—
258C2×M16C2×D12711yescharacteristic, complemented
259#127—1271no—
260#127—1271no—
261#127—2541no—
262#121—1271no—
263#121—1271no—
264#121—1271no—
265#121—1271no—
266#121—2541no—
267#121—2541no—
268#121—2541no—
269#120—2541no—
270#120—2541no—
271C4×Q8—1271no—
272C4×D4—1271no—
273C4×D4—1271no—
274C4×D4—2541no—
275#106—1271no—
276C4⋊C8—1271no—
277#103—1271no—
278#102—1271no—
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102—1271no—
283C22⋊C8—1271no—
284C22⋊C8—1271no—
285C4×C8—1271no—
286C24—21no—
287C24—2541no—
288#41—22no—
289#41—41no—
290#41—5081no—
291#41—2542no—
292C2×Q8C22×D12711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1271no—
298C2×D4—1271no—
299C2×D4—1271no—
300C2×D4—5084no—
301C2×D4—5081no—
302C2×D4—2541no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4—2541no—
306C2×D4—2541no—
307C2×D4—2541no—
308C2×D4—2541no—
309C2×D4—2541no—
310C2×D4—2541no—
311C2×D4—2541no—
312C2×D4—2542no—
313C2×D4C22×D12711yescharacteristic
314C2×D4C22×D12711yescharacteristic
315C2×D4C22×D12711yescharacteristic
316C22×C4—21no—
317C22×C4—1271no—
318C22×C4—2541no—
319C22×C4—2541no—
320C22×C4C22×D12711yescharacteristic
321SD16—22no—
322SD16—2542no—
323SD16C22⋊D12714yescomplemented
324D8—22no—
325D8—22no—
326D8—5081no—
327D8—5081no—
328D8—2542no—
329D8—2542no—
330M16—22no—
331M16—2542no—
332C2×C8—1271no—
333C2×C8—1271no—
334C2×C8C22×D12711yescharacteristic
335C2×C8C22×D12711yescharacteristic
336C4⋊C4—1271no—
337C4⋊C4—1271no—
338C4⋊C4—1271no—
339C4⋊C4—2541no—
340C22⋊C4—1271no—
341C22⋊C4—1271no—
342C22⋊C4—2541no—
343C22⋊C4—2541no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C42—1271no—
348C42—2541no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—1271no—
356C23—10161no—
357C23—5082no—
358C23—5082no—
359C23—2541no—
360C23—2541no—
361C23D4×D12711yescharacteristic
362Q8—21no—
363Q8C2×C22⋊D12712yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—42no—
371D4—10161no—
372D4—10161no—
373D4—10161no—
374D4—10161no—
375D4—5082no—
376D4—5082no—
377D4—5082no—
378D4—5082no—
379D4—5082no—
380D4—5082no—
381D4—2542no—
382D4—2541no—
383D4—2541no—
384D4—2541no—
385D4—2541no—
386D4C2×C22⋊D12712yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—1271no—
392C2×C4—1271no—
393C2×C4—1271no—
394C2×C4—5082no—
395C2×C4—5082no—
396C2×C4—2541no—
397C2×C4—2541no—
398C2×C4—2541no—
399C2×C4—2541no—
400C2×C4D4×D12711yescharacteristic
401C2×C4C23×D12711yescharacteristic
402C8—21no—
403C8—2541no—
404C8—2541no—
405C8C2×C22⋊D12712yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—10164no—
420C22—5081no—
421C22—5081no—
422C22—5081no—
423C22—5081no—
424C22—5081no—
425C22—5081no—
426C22—2541no—
427C22—2541no—
428C22—2541no—
429C22C2×D4×D12711yescharacteristic
430C4—21no—
431C4—22no—
432C4—1272no—
433C4—5081no—
434C4—5081no—
435C4—2541no—
436C4—2541no—
437C4C22×C22⋊D12711yescharacteristic
438C4#1686125411yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—10161no—
444C2—10161no—
445C2—5081no—
446C2D4×C22⋊D12711yescharacteristic
447C2AGL(1,ℤ/8)×D12711yescharacteristic
448C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3274
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. #22550545C2semidirect
3. D4×C22⋊D127C2semidirect
4. #22550251C2semidirect
5. #22550129C2semidirect
6. #22550126C2semidirect
7. #22550074C2semidirect
8. #22550069C2semidirect
9. #22550021C2semidirect
10. #22550020C2semidirect
11. #22550010C2semidirect
12. #22550008C2semidirect
13. SD16×Dic127C2semidirect
14. #22549952C2semidirect
15. C2×AGL(1,ℤ/8)×C127C2semidirect
16. C2×#14523249C22semidirect
20. C2×#14523235C22semidirect
21. C2×D1016C22semidirect
23. C2×#14523233C22semidirect
24. C2×#14523233C22semidirect
25. C2×#14523231C22semidirect
28. #16861205C22semidirect
31. #16861183C22semidirect
35. #16861180C22semidirect
38. Q8×Dic127C22semidirect
42. #16861141C22semidirect
43. Dic127⋊C8C22semidirect
44. #16861133C22semidirect
45. #16861131C22semidirect
46. #16861131C22semidirect
48. #16861130C22semidirect
49. #16861130C22semidirect
50. C22⋊(C127⋊C8)C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. C8×Dic127C22semidirect
53. C2×#41×C127C22semidirect
57. C2×SD16×C127C22semidirect
59. C2×D8×C127C22semidirect
60. C2×M16×C127C22semidirect
118. SD16×C127D4semidirect
164. Q8×C127C2×D4semidirect
180. C8×C127C2×D4semidirect
213. Dic127AGL(1,ℤ/8)semidirect
225. C127#3274semidirect
227. C2×AGL(1,ℤ/8)D127semidirect
242. C2×#41C2×D127semidirect
250. C2×SD16C2×D127semidirect
256. C2×D8C2×D127semidirect
258. C2×M16C2×D127semidirect
323. SD16C22⋊D127semidirect
363. Q8C2×C22⋊D127semidirect
405. C8C2×C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
#225505452
D4×C22⋊D1272
#225502512
#225501292
#225501262
#225500742
#225500692
#225500212
#225500202
#225500102
#225500082
SD16×Dic1272
#225499522
C2×AGL(1,ℤ/8)×C1272
#3274127