Finite Group Databasebeta

← #30650228

Subgroups — #30650228

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×#16861234C211yesmaximal, characteristic, complemented
2C2×SD16×D127C211yesmaximal, characteristic, complemented
3#22550533C211yesmaximal, characteristic, complemented
4#22550510C211yesmaximal, characteristic, complemented
5#22550264C211yesmaximal, characteristic, complemented
6#22550120C211yesmaximal, characteristic, complemented
7#22550111C211yesmaximal, characteristic, complemented
8#22550090C211yesmaximal, characteristic, complemented
9#22550074C211yesmaximal, characteristic, complemented
10#22550029C211yesmaximal, characteristic, complemented
11#22550016C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549979C211yesmaximal, characteristic, complemented
14#22549950C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×Q8×D127C2211yescharacteristic, complemented
17C22×D508C2211yescharacteristic
18C2×D4×D127—21no—
19C2×D4×D127C2211yescharacteristic
20#16861234—24no—
21C2×#14523237C2211yescharacteristic
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523236C2211yescharacteristic, complemented
24C2×#14523235C2211yescharacteristic
25C2×#14523235C2211yescharacteristic, complemented
26SD16×D127—24no—
27C2×#14523233C2211yescharacteristic
28C2×#14523233C2211yescharacteristic, complemented
29C2×#14523232C2211yescharacteristic, complemented
30C2×C8×D127C2211yescharacteristic, complemented
31#16861205C2211yescharacteristic
32#16861196—21no—
33#16861196C2211yescharacteristic, complemented
34#16861188—21no—
35#16861188C2211yescharacteristic, complemented
36#16861184C2211yescharacteristic
37#16861182—21no—
38#16861180—21no—
39#16861180C2211yescharacteristic, complemented
40#16861179—21no—
41#16861177—21no—
42Q8×Dic127C2211yescharacteristic, complemented
43#16861171C2211yescharacteristic
44C4×C22⋊D127—21no—
45C4×C22⋊D127C2211yescharacteristic
46#16861142C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861132C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861119C2211yescharacteristic, complemented
54C2×#41×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic, complemented
56C2×D8×C127C2211yescharacteristic, complemented
57#121×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic, complemented
60C22⋊C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63C2×Dic254—21no—
64C2×Dic254C2311yescharacteristic
65Q8×D127—22no—
66Q8×D127—41no—
67C2×D508—22no—
68C2×D508—21no—
69C2×D508—21no—
70C2×D508—41no—
71C2×D508C2311yescharacteristic
72C2×D508C2311yescharacteristic
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77C2×C22⋊D127D411yescharacteristic
78D4×D127—22no—
79D4×D127—41no—
80D4×D127—44no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127C2311yescharacteristic
84#14523237—22no—
85#14523237—41no—
86#14523236—22no—
87#14523235—22no—
88#14523235D414yescomplemented
89#14523233—22no—
90#14523233—22no—
91#14523232—22no—
92C2×C127⋊C8C2311yescharacteristic
93C2×C127⋊C8C2311yescharacteristic
94C8×D127—22no—
95C4⋊Dic127—21no—
96C4⋊Dic127C2311yescharacteristic
97Dic127⋊C4—21no—
98Dic127⋊C4D411yescharacteristic
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127D411yescharacteristic
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4D411yescharacteristic
107C4×Dic127—21no—
108C4×Dic127C2311yescharacteristic
109#41×C127—22no—
110#41×C127—41no—
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127—21no—
114C2×D4×C127C2311yescharacteristic
115C2×D4×C127C2311yescharacteristic
116C22×C4×C127—21no—
117C22×C4×C127C2311yescharacteristic
118SD16×C127—22no—
119D8×C127—41no—
120C2×C8×C127C2311yescharacteristic
121C2×C8×C127C2311yescharacteristic
122C4⋊C4×C127C2311yescharacteristic
123C22⋊C4×C127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—81no—
131C22×D127—81no—
132C22×D127C2×D411yescharacteristic
133Dic254—21no—
134Dic254—21no—
135Dic254—42no—
136D508—21no—
137D508—21no—
138D508—21no—
139D508—42no—
140D508—42no—
141D508C2×D412yes—
142C22⋊D127—22no—
143C22⋊D127—42no—
144C22⋊D127—42no—
145C22⋊D127—42no—
146C22⋊D127—81no—
147C2×Dic127—21no—
148C2×Dic127—21no—
149C2×Dic127C2×D411yescharacteristic
150C2×Dic127C2×D411yescharacteristic
151C2×Dic127C2×D411yescharacteristic
152C4×D127—21no—
153C4×D127—21no—
154C4×D127—42no—
155C4×D127—42no—
156C127⋊C8—21no—
157C127⋊C8C2×D412yescomplemented
158C23×C127—21no—
159C23×C127—21no—
160C23×C127C2×D411yescharacteristic
161Q8×C127—21no—
162Q8×C127C2×D412yescomplemented
163D4×C127—21no—
164D4×C127—21no—
165D4×C127—21no—
166D4×C127—21no—
167D4×C127—42no—
168D4×C127—42no—
169C2×C4×C127—21no—
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×C127—21no—
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—21no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127C22×D411yescharacteristic
188C2×D127—21no—
189C2×D127—21no—
190C2×D127—22no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—41no—
195C2×D127—41no—
196C2×D127—84no—
197C2×D127—84no—
198C2×D127AGL(1,ℤ/8)12yescomplemented
199C4×C127—21no—
200C4×C127—22no—
201C4×C127—41no—
202C4×C127C22×D411yescharacteristic
203C4×C127H(5,𝔽2)11yescharacteristic
204Dic127—21no—
205Dic127—21no—
206Dic127—21no—
207Dic127—41no—
208Dic127—41no—
209C2×C127—41no—
210C2×C127—81no—
211C2×C127—81no—
212C2×C127D4211yescharacteristic
213C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C127#57711yescharacteristic
215D127—22no—
216D127—42no—
217D127—81no—
218#3265—1271nomaximal, Sylow
219C127#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1271no—
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#449D12711yescharacteristic, complemented
231#448—1271no—
232#447—1271no—
233#440—1271no—
234#435—1271no—
235C2×#41C2×D12711yescharacteristic, complemented
236C22×Q8—1271no—
237C22×D4—1271no—
238C22×D4—1271no—
239C22×D4—2541no—
240AGL(1,ℤ/8)—2544no—
241C2×SD16—1271no—
242C2×SD16—1271no—
243C2×SD16—1271no—
244C2×SD16—1271no—
245C2×SD16—1271no—
246C2×SD16—2544no—
247C2×SD16C2×D12711yescharacteristic, complemented
248C2×D8—1271no—
249C2×D8—1271no—
250C2×D8C2×D12711yescharacteristic, complemented
251C2×M16—1271no—
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—2541no—
261#121—2541no—
262#121C2×D12711yescharacteristic
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#103C2×D12711yescharacteristic, complemented
273#102—1271no—
274#102—1271no—
275#102—1271no—
276#102C2×D12711yescharacteristic, complemented
277C22⋊C8—1271no—
278C22⋊C8C2×D12711yescharacteristic, complemented
279#97—1271no—
280C24—2541no—
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—21no—
290C2×D4—21no—
291C2×D4—1271no—
292C2×D4—1271no—
293C2×D4—1271no—
294C2×D4—5081no—
295C2×D4—5084no—
296C2×D4—5081no—
297C2×D4—2542no—
298C2×D4—2541no—
299C2×D4—2541no—
300C2×D4—2541no—
301C2×D4—2541no—
302C2×D4—2542no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4C22×D12711yescharacteristic
306C2×D4C22×D12711yescharacteristic
307C22×C4—21no—
308C22×C4—1271no—
309C22×C4—2541no—
310C22×C4—2541no—
311C22×C4C22×D12711yescharacteristic
312SD16—22no—
313SD16—1274no—
314SD16—5081no—
315SD16—2542no—
316SD16—2542no—
317SD16—2542no—
318D8—41no—
319D8—2542no—
320D8—2542no—
321M16—2542no—
322C2×C8—1271no—
323C2×C8—1271no—
324C2×C8—2542no—
325C2×C8C22×D12711yescharacteristic
326C2×C8C22×D12711yescharacteristic
327C4⋊C4—1271no—
328C4⋊C4—1271no—
329C4⋊C4—2541no—
330C4⋊C4—2541no—
331C4⋊C4C22×D12711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—1271no—
334C22⋊C4—1271no—
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—1271no—
346C23—10161no—
347C23—10161no—
348C23—5082no—
349C23—5082no—
350C23—5082no—
351C23—5082no—
352C23—2541no—
353C23—2541no—
354C23D4×D12711yescharacteristic
355Q8—21no—
356Q8—5082no—
357Q8—2541no—
358Q8—2541no—
359Q8D4×D12712yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—1272no—
367D4—10161no—
368D4—5082no—
369D4—5082no—
370D4—5082no—
371D4—5082no—
372D4—5082no—
373D4—2542no—
374D4—2541no—
375D4—2541no—
376D4—2541no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—1271no—
383C2×C4—1271no—
384C2×C4—1271no—
385C2×C4—5082no—
386C2×C4—5082no—
387C2×C4—2541no—
388C2×C4—2541no—
389C2×C4—2541no—
390C2×C4—2541no—
391C2×C4D4×D12711yescharacteristic
392C2×C4C23×D12711yescharacteristic
393C8—21no—
394C8—21no—
395C8—1272no—
396C8—2541no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—1272no—
407C22—10164no—
408C22—10164no—
409C22—5081no—
410C22—5081no—
411C22—5081no—
412C22—5081no—
413C22—5081no—
414C22—2542no—
415C22—2541no—
416C22—2541no—
417C22C2×D4×D12711yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—5081no—
422C4—5081no—
423C4—2541no—
424C4—2541no—
425C4—2541no—
426C4C2×D4×D12711yescharacteristic
427C4#1686125411yescharacteristic
428C2—41no—
429C2—81no—
430C2—81no—
431C2—10161no—
432C2—5082no—
433C2—2542no—
434C2#2255050811yescharacteristic
435C2AGL(1,ℤ/8)×D12711yescharacteristic
436C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3265
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×#16861234C2semidirect
2. C2×SD16×D127C2semidirect
3. #22550533C2semidirect
4. #22550510C2semidirect
5. #22550264C2semidirect
6. #22550120C2semidirect
7. #22550111C2semidirect
8. #22550090C2semidirect
9. #22550074C2semidirect
10. #22550029C2semidirect
11. #22550016C2semidirect
12. #22550008C2semidirect
13. #22549979C2semidirect
14. #22549950C2semidirect
15. #449×C127C2semidirect
16. C2×Q8×D127C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523236C22semidirect
25. C2×#14523235C22semidirect
28. C2×#14523233C22semidirect
29. C2×#14523232C22semidirect
30. C2×C8×D127C22semidirect
33. #16861196C22semidirect
35. #16861188C22semidirect
39. #16861180C22semidirect
42. Q8×Dic127C22semidirect
46. #16861142C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861132C22semidirect
50. #16861131C22semidirect
51. #16861130C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861119C22semidirect
54. C2×#41×C127C22semidirect
55. C2×SD16×C127C22semidirect
56. C2×D8×C127C22semidirect
58. #103×C127C22semidirect
59. #102×C127C22semidirect
60. C22⋊C8×C127C22semidirect
88. #14523235D4semidirect
157. C127⋊C8C2×D4semidirect
162. Q8×C127C2×D4semidirect
198. C2×D127AGL(1,ℤ/8)semidirect
219. C127#3265semidirect
230. #449D127semidirect
235. C2×#41C2×D127semidirect
247. C2×SD16C2×D127semidirect
250. C2×D8C2×D127semidirect
272. #103C2×D127semidirect
276. #102C2×D127semidirect
278. C22⋊C8C2×D127semidirect
359. Q8D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×SD16×D1272
#225505332
#225505102
#225502642
#225501202
#225501112
#225500902
#225500742
#225500292
#225500162
#225500082
#225499792
#225499502
#449×C1272
#3265127