Finite Group Databasebeta

← #30650219

Subgroups — #30650219

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×SD16×D127C211yesmaximal, characteristic, complemented
2#22550533C211yesmaximal, characteristic, complemented
3D4×C22⋊D127C211yesmaximal, characteristic, complemented
4#22550264C211yesmaximal, characteristic, complemented
5#22550122C211yesmaximal, characteristic, complemented
6#22550109C211yesmaximal, characteristic, complemented
7#22550089C211yesmaximal, characteristic, complemented
8#22550074C211yesmaximal, characteristic, complemented
9#22550027C211yesmaximal, characteristic, complemented
10#22550021C211yesmaximal, characteristic, complemented
11#22550013C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549977C211yesmaximal, characteristic, complemented
14#22549945C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
16C2×Q8×D127C2211yescharacteristic, complemented
17C22×C22⋊D127—21no—
18C2×D4×D127C2211yescharacteristic
19C2×#14523237C2211yescharacteristic, complemented
20C2×#14523236C2211yescharacteristic
21C2×#14523236C2211yescharacteristic, complemented
22C2×#14523235C2211yescharacteristic, complemented
23SD16×D127—24no—
24C2×#14523233C2211yescharacteristic, complemented
25C2×C8×D127C2211yescharacteristic, complemented
26#16861205C2211yescharacteristic
27#16861196—21no—
28#16861196C2211yescharacteristic, complemented
29#16861188—21no—
30#16861188C2211yescharacteristic, complemented
31#16861183C2211yescharacteristic
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#16861143C2211yescharacteristic, complemented
43Dic127⋊C8C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861133C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50C22⋊(C127⋊C8)C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52#16861119C2211yescharacteristic, 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—
62C2×Dic254—21no—
63C2×Dic254C2311yescharacteristic
64Q8×D127—22no—
65Q8×D127—41no—
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—44no—
74C2×C22⋊D127D411yescharacteristic
75D4×D127—22no—
76D4×D127—41no—
77C22×Dic127—21no—
78C2×C4×D127—21no—
79C2×C4×D127C2311yescharacteristic
80#14523237—22no—
81#14523236—22no—
82#14523236—41no—
83#14523235—22no—
84#14523233—41no—
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C8×D127—22no—
88C4⋊Dic127—21no—
89C4⋊Dic127C2311yescharacteristic
90C4⋊Dic127C2311yescharacteristic
91Dic127⋊C4—21no—
92Dic127⋊C4D411yescharacteristic
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95C22⋊Dic127—21no—
96C22⋊Dic127—21no—
97C22⋊Dic127—21no—
98C22⋊Dic127—21no—
99C22⋊Dic127D411yescharacteristic
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—42no—
126C22×D127—42no—
127C22×D127—81no—
128C22×D127C2×D411yescharacteristic
129Dic254—21no—
130Dic254—21no—
131Dic254—42no—
132D508—21no—
133D508—21no—
134C22⋊D127—22no—
135C22⋊D127—42no—
136C22⋊D127—42no—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—81no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—42no—
145C2×Dic127C2×D411yescharacteristic
146C2×Dic127C2×D411yescharacteristic
147C2×Dic127C2×D411yescharacteristic
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—42no—
151C127⋊C8—21no—
152C127⋊C8—21no—
153C23×C127—21no—
154C23×C127—21no—
155C23×C127—21no—
156C23×C127—42no—
157C23×C127—42no—
158C23×C127—81no—
159C23×C127C2×D411yescharacteristic
160Q8×C127—21no—
161Q8×C127C2×D412yescomplemented
162D4×C127—21no—
163D4×C127—21no—
164D4×C127—21no—
165D4×C127—21no—
166D4×C127—21no—
167D4×C127—42no—
168D4×C127—42no—
169D4×C127C2×D412yes—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—42no—
174C2×C4×C127C2×D411yescharacteristic
175C2×C4×C127C2411yescharacteristic
176C8×C127—21no—
177C8×C127C2×D412yescomplemented
178C22×C127—21no—
179C22×C127—22no—
180C22×C127—21no—
181C22×C127—21no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127—84no—
191C22×C127C22×D411yescharacteristic
192C2×D127—21no—
193C2×D127—21no—
194C2×D127—41no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—84no—
198C2×D127AGL(1,ℤ/8)12yescomplemented
199C4×C127—21no—
200C4×C127—22no—
201C4×C127C22×D411yescharacteristic
202C4×C127H(5,𝔽2)11yescharacteristic
203Dic127—21no—
204Dic127—21no—
205Dic127—21no—
206Dic127—41no—
207Dic127—41no—
208Dic127—41no—
209C2×C127—42no—
210C2×C127—41no—
211C2×C127—81no—
212C2×C127—81no—
213C2×C127D4211yescharacteristic
214C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C127#57711yescharacteristic
216D127—22no—
217D127—81no—
218#3265—1271nomaximal, Sylow
219C127#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
221C22×SD16—1271no—
222#548—1271no—
223D42—1271no—
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×#41C2×D12711yescharacteristic, complemented
236C22×Q8—1271no—
237C22×D4—1271no—
238C22×D4—2541no—
239C22×D4C2×D12711yescharacteristic
240AGL(1,ℤ/8)—24no—
241C2×SD16—1271no—
242C2×SD16—1271no—
243C2×SD16—1271no—
244C2×SD16—1271no—
245C2×SD16—2544no—
246C2×SD16C2×D12711yescharacteristic
247C2×SD16C2×D12711yescharacteristic, complemented
248C2×D8—1271no—
249C2×D8C2×D12711yescharacteristic
250C2×D8C2×D12711yescharacteristic, complemented
251C2×M16C2×D12711yescharacteristic, complemented
252C22×C8—1271no—
253#127—1271no—
254#124—1271no—
255#124—2541no—
256C22⋊Q8—1271no—
257C22⋊Q8—2541no—
258#121—1271no—
259#121—1271no—
260#121—1271no—
261#121—2541no—
262#121—2541no—
263#120—2541no—
264#120—2541no—
265C4×Q8—1271no—
266C4×D4—1271no—
267C4×D4—1271no—
268C4×D4—2541no—
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—2541no—
282#41—22no—
283#41—41no—
284C2×Q8—1271no—
285C2×Q8—5081no—
286C2×Q8—2541no—
287C2×Q8—2542no—
288C2×Q8C22×D12711yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—1271no—
294C2×D4—1271no—
295C2×D4—5084no—
296C2×D4—5081no—
297C2×D4—2541no—
298C2×D4—2541no—
299C2×D4—2541no—
300C2×D4—2541no—
301C2×D4—2541no—
302C2×D4—2541no—
303C2×D4—2542no—
304C2×D4C22×D12711yescharacteristic
305C2×D4C22×D12711yescharacteristic
306C2×D4C22×D12711yescharacteristic
307C22×C4—21no—
308C22×C4—1271no—
309C22×C4—2541no—
310C22×C4—2541no—
311C22×C4C22×D12711yescharacteristic
312SD16—22no—
313SD16—5081no—
314SD16—2542no—
315SD16—2542no—
316SD16—2542no—
317SD16C22⋊D12714yescomplemented
318D8—22no—
319D8—22no—
320D8—5081no—
321M16—22no—
322C2×C8—1271no—
323C2×C8—1271no—
324C2×C8—2542no—
325C2×C8C22×D12711yescharacteristic
326C2×C8C22×D12711yescharacteristic
327C4⋊C4—1271no—
328C4⋊C4—1271no—
329C4⋊C4—1271no—
330C4⋊C4—2541no—
331C4⋊C4—2541no—
332C22⋊C4—1271no—
333C22⋊C4—1271no—
334C22⋊C4—2541no—
335C22⋊C4—2541no—
336C22⋊C4—2541no—
337C22⋊C4—2541no—
338C22⋊C4—2541no—
339C22⋊C4—2541no—
340C22⋊C4—2541no—
341C42—1271no—
342C42—2541no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—1271no—
350C23—10161no—
351C23—5082no—
352C23—5082no—
353C23—2541no—
354C23D4×D12711yescharacteristic
355Q8—21no—
356Q8—5082no—
357Q8—2541no—
358Q8—2541no—
359Q8C2×C22⋊D12712yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—10161no—
368D4—5082no—
369D4—5082no—
370D4—5082no—
371D4—5082no—
372D4—5082no—
373D4—2542no—
374D4—2541no—
375D4—2541no—
376D4C2×C22⋊D12712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—1271no—
382C2×C4—1271no—
383C2×C4—1271no—
384C2×C4—5082no—
385C2×C4—5082no—
386C2×C4—2541no—
387C2×C4—2541no—
388C2×C4—2541no—
389C2×C4—2541no—
390C2×C4—2541no—
391C2×C4D4×D12711yescharacteristic
392C2×C4C23×D12711yescharacteristic
393C8—21no—
394C8—2541no—
395C8—2541no—
396C8C2×C22⋊D12712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—1272no—
411C22—10164no—
412C22—5081no—
413C22—5081no—
414C22—5081no—
415C22—2541no—
416C22—2541no—
417C22C2×D4×D12711yescharacteristic
418C4—21no—
419C4—22no—
420C4—5081no—
421C4—5081no—
422C4—5081no—
423C4—2541no—
424C4—2541no—
425C4—2541no—
426C4C22×C22⋊D12711yescharacteristic
427C4#1686125411yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—10161no—
433C2—2542no—
434C2D4×C22⋊D12711yescharacteristic
435C2AGL(1,ℤ/8)×D12711yescharacteristic
436C2#2255068611yescharacteristic
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#3265
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×SD16×D127C2semidirect
2. #22550533C2semidirect
3. D4×C22⋊D127C2semidirect
4. #22550264C2semidirect
5. #22550122C2semidirect
6. #22550109C2semidirect
7. #22550089C2semidirect
8. #22550074C2semidirect
9. #22550027C2semidirect
10. #22550021C2semidirect
11. #22550013C2semidirect
12. #22550008C2semidirect
13. #22549977C2semidirect
14. #22549945C2semidirect
15. C2×AGL(1,ℤ/8)×C127C2semidirect
16. C2×Q8×D127C22semidirect
19. C2×#14523237C22semidirect
21. C2×#14523236C22semidirect
22. C2×#14523235C22semidirect
24. C2×#14523233C22semidirect
25. C2×C8×D127C22semidirect
28. #16861196C22semidirect
30. #16861188C22semidirect
35. #16861180C22semidirect
38. Q8×Dic127C22semidirect
42. #16861143C22semidirect
43. Dic127⋊C8C22semidirect
44. #16861134C22semidirect
45. #16861133C22semidirect
46. #16861131C22semidirect
48. #16861130C22semidirect
49. #16861130C22semidirect
50. C22⋊(C127⋊C8)C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. #16861119C22semidirect
53. C2×#41×C127C22semidirect
57. C2×SD16×C127C22semidirect
59. C2×D8×C127C22semidirect
60. C2×M16×C127C22semidirect
118. SD16×C127D4semidirect
161. Q8×C127C2×D4semidirect
177. C8×C127C2×D4semidirect
198. C2×D127AGL(1,ℤ/8)semidirect
219. C127#3265semidirect
220. C2×AGL(1,ℤ/8)D127semidirect
235. C2×#41C2×D127semidirect
247. C2×SD16C2×D127semidirect
250. C2×D8C2×D127semidirect
251. C2×M16C2×D127semidirect
317. SD16C22⋊D127semidirect
359. Q8C2×C22⋊D127semidirect
396. C8C2×C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1272
#225505332
D4×C22⋊D1272
#225502642
#225501222
#225501092
#225500892
#225500742
#225500272
#225500212
#225500132
#225500082
#225499772
#225499452
C2×AGL(1,ℤ/8)×C1272
#3265127