Finite Group Databasebeta

← #30650231

Subgroups — #30650231

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×#16861233C211yesmaximal, characteristic, complemented
2C2×SD16×D127C211yesmaximal, characteristic, complemented
3#22550534C211yesmaximal, characteristic, complemented
4D4×D508C211yesmaximal, characteristic, complemented
5#22550266C211yesmaximal, characteristic, complemented
6#22550125C211yesmaximal, characteristic, complemented
7#22550111C211yesmaximal, characteristic, complemented
8#22550077C211yesmaximal, characteristic, complemented
9#22550025C211yesmaximal, characteristic, complemented
10#22550022C211yesmaximal, characteristic, complemented
11#22550016C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549976C211yesmaximal, characteristic, complemented
14#22549946C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic, complemented
17C2×Q8×D127C2211yescharacteristic, complemented
18C22×D508—21no—
19C2×D4×D127C2211yescharacteristic
20C2×D4×D127C2211yescharacteristic
21#16861233—24no—
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic, complemented
24C2×#14523236C2211yescharacteristic
25C2×#14523235C2211yescharacteristic, complemented
26C2×#14523235C2211yescharacteristic, complemented
27SD16×D127—24no—
28C2×#14523233C2211yescharacteristic, complemented
29C2×#14523233C2211yescharacteristic, complemented
30C2×#14523231C2211yescharacteristic, complemented
31C2×C8×D127C2211yescharacteristic, complemented
32#16861206C2211yescharacteristic
33#16861195—21no—
34#16861186—21no—
35#16861185C2211yescharacteristic, complemented
36#16861184C2211yescharacteristic
37#16861184C2211yescharacteristic
38#16861183—21no—
39#16861182—21no—
40#16861180C2211yescharacteristic, complemented
41#16861179—21no—
42#16861179—21no—
43C4×Dic254C2211yescharacteristic, complemented
44C4×D508C2211yescharacteristic
45(C22⋊D127)⋊C4—21no—
46#16861142C2211yescharacteristic, complemented
47#16861132C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic, complemented
50#16861130C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861120C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic
55C2×D8×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63#14523247—22no—
64#14523247—41no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67Q8×D127—22no—
68Q8×D127—41no—
69C2×D508—21no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508—21no—
73C2×D508—44no—
74C2×D508D411yescharacteristic
75C2×D508C2311yescharacteristic
76C2×D508C2311yescharacteristic
77C2×C22⋊D127—21no—
78C2×C22⋊D127—21no—
79C2×C22⋊D127—21no—
80C2×C22⋊D127—21no—
81D4×D127—22no—
82D4×D127—22no—
83D4×D127—41no—
84D4×D127—41no—
85C22×Dic127—21no—
86C2×C4×D127—21no—
87C2×C4×D127C2311yescharacteristic
88C2×C4×D127C2311yescharacteristic
89#14523237—22no—
90#14523237—22no—
91#14523236D414yescomplemented
92#14523235—22no—
93#14523235—41no—
94#14523233—22no—
95#14523233—41no—
96#14523231—22no—
97C2×C127⋊C8C2311yescharacteristic
98C2×C127⋊C8C2311yescharacteristic
99C8×D127—22no—
100C4⋊Dic127D411yescharacteristic
101C4⋊Dic127C2311yescharacteristic
102Dic127⋊C4—21no—
103Dic127⋊C4—21no—
104C22⋊Dic127—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4D411yescharacteristic
109(C2×D127)⋊C4D411yescharacteristic
110C4×Dic127—21no—
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115SD16×C127—22no—
116D8×C127—22no—
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C4⋊C4×C127C2311yescharacteristic
120C22⋊C4×C127—21no—
121C22⋊C4×C127—21no—
122C22⋊C4×C127—21no—
123C42×C127C2311yescharacteristic
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
133C22×D127C2×D411yescharacteristic
134Dic254—21no—
135Dic254—42no—
136Dic254C2×D412yescomplemented
137D508—22no—
138D508—21no—
139D508—21no—
140D508—21no—
141D508—21no—
142D508—42no—
143D508—42no—
144D508—42no—
145D508—81no—
146C22⋊D127—42no—
147C22⋊D127—42no—
148C22⋊D127—42no—
149C22⋊D127—42no—
150C2×Dic127—21no—
151C2×Dic127—21no—
152C2×Dic127—42no—
153C2×Dic127C2×D411yescharacteristic
154C2×Dic127C2×D411yescharacteristic
155C4×D127—21no—
156C4×D127—21no—
157C4×D127—21no—
158C4×D127—21no—
159C4×D127—42no—
160C127⋊C8—21no—
161C127⋊C8C2×D412yescomplemented
162C23×C127—21no—
163C23×C127—21no—
164Q8×C127—21no—
165Q8×C127—21no—
166D4×C127—21no—
167D4×C127—21no—
168D4×C127—21no—
169D4×C127C2×D412yes—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—42no—
173C2×C4×C127C2×D411yescharacteristic
174C2×C4×C127C2×D411yescharacteristic
175C2×C4×C127C2411yescharacteristic
176C8×C127—21no—
177C8×C127—21no—
178C22×C127—22no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127C22×D411yescharacteristic
185C2×D127—21no—
186C2×D127—21no—
187C2×D127—21no—
188C2×D127—21no—
189C2×D127—21no—
190C2×D127—41no—
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—21no—
201C4×C127—41no—
202C4×C127—41no—
203C4×C127C22×D411yescharacteristic
204C4×C127H(5,𝔽2)11yescharacteristic
205Dic127—21no—
206Dic127—21no—
207Dic127—22no—
208Dic127—41no—
209C2×C127—42no—
210C2×C127—81no—
211C2×C127D4211yescharacteristic
212C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C127#57711yescharacteristic
214D127—22no—
215D127—41no—
216D127—81no—
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#463D12711yescharacteristic, complemented
228#460—1271no—
229#450—1271no—
230#449—1271no—
231#448—1271no—
232#447—1271no—
233#440—1271no—
234#435—1271no—
235C2×#41—1271no—
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
248C2×D8—1271no—
249C2×D8—1271no—
250C2×D8C2×D12711yescharacteristic
251C2×M16—1271no—
252C22×C8—1271no—
253#127—1271no—
254#124—2541no—
255#124C2×D12711yescharacteristic, complemented
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—2541no—
268C4×D4C2×D12711yescharacteristic
269#107—1271no—
270C4⋊C8C2×D12711yescharacteristic, complemented
271#103—1271no—
272#103C2×D12711yescharacteristic, complemented
273#102—1271no—
274#102—1271no—
275#102—1271no—
276#102C2×D12711yescharacteristic, complemented
277C22⋊C8—1271no—
278C22⋊C8—1271no—
279#97—1271no—
280C24—2541no—
281C24—2541no—
282#41—5081no—
283#41—2542no—
284C2×Q8—1271no—
285C2×Q8—5081no—
286C2×Q8—2541no—
287C2×Q8—2542no—
288C2×Q8C22×D12711yescharacteristic
289C2×D4—1271no—
290C2×D4—1271no—
291C2×D4—1271no—
292C2×D4—5084no—
293C2×D4—5081no—
294C2×D4—5081no—
295C2×D4—2541no—
296C2×D4—2541no—
297C2×D4—2541no—
298C2×D4—2541no—
299C2×D4—2541no—
300C2×D4—2541no—
301C2×D4—2542no—
302C2×D4—2542no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4C22×D12711yescharacteristic
306C2×D4C22×D12711yescharacteristic
307C22×C4—21no—
308C22×C4—1271no—
309C22×C4—1271no—
310C22×C4—2541no—
311C22×C4—2541no—
312SD16—22no—
313SD16—1274no—
314SD16—5081no—
315SD16—2542no—
316SD16—2542no—
317SD16—2542no—
318D8—22no—
319D8—5081no—
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—21no—
334C22⋊C4—21no—
335C22⋊C4—1271no—
336C22⋊C4—1271no—
337C22⋊C4—2541no—
338C22⋊C4—2541no—
339C22⋊C4—2541no—
340C22⋊C4—2541no—
341C42—2541no—
342C42C22×D12711yescharacteristic
343C23—21no—
344C23—21no—
345C23—1271no—
346C23—1271no—
347C23—10161no—
348C23—10161no—
349C23—5082no—
350C23—5082no—
351C23—5082no—
352C23—5082no—
353C23—2541no—
354C23—2541no—
355Q8—21no—
356Q8—21no—
357Q8—1272no—
358Q8—5082no—
359Q8—2541no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—10161no—
364D4—5082no—
365D4—5082no—
366D4—5082no—
367D4—5082no—
368D4—5082no—
369D4—5082no—
370D4—5082no—
371D4—2542no—
372D4—2541no—
373D4—2541no—
374D4—2541no—
375D4—2541no—
376D4D4×D12712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—1271no—
381C2×C4—1271no—
382C2×C4—5082no—
383C2×C4—5082no—
384C2×C4—2541no—
385C2×C4—2541no—
386C2×C4—2541no—
387C2×C4—2541no—
388C2×C4—2541no—
389C2×C4—2541no—
390C2×C4D4×D12711yescharacteristic
391C2×C4D4×D12711yescharacteristic
392C2×C4C23×D12711yescharacteristic
393C8—21no—
394C8—21no—
395C8—1272no—
396C8—2541no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—1272no—
404C22—10164no—
405C22—10164no—
406C22—5081no—
407C22—5081no—
408C22—5081no—
409C22—5081no—
410C22—5081no—
411C22—5081no—
412C22—2541no—
413C22—2541no—
414C22—2541no—
415C22—2541no—
416C22—2541no—
417C22C2×D4×D12711yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—5081no—
423C4—2541no—
424C4—2541no—
425C4—2542no—
426C4C2×D4×D12711yescharacteristic
427C4#1686125511yescharacteristic
428C2—42no—
429C2—81no—
430C2—10161no—
431C2—10161no—
432C2—5081no—
433C2—2542no—
434C2#2255051011yescharacteristic
435C2AGL(1,ℤ/8)×D12711yescharacteristic
436C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
hypercenter#463D127
hypocenterC127#3265
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. C2×SD16×D127C2semidirect
3. #22550534C2semidirect
4. D4×D508C2semidirect
5. #22550266C2semidirect
6. #22550125C2semidirect
7. #22550111C2semidirect
8. #22550077C2semidirect
9. #22550025C2semidirect
10. #22550022C2semidirect
11. #22550016C2semidirect
12. #22550008C2semidirect
13. #22549976C2semidirect
14. #22549946C2semidirect
15. #463×C127C2semidirect
16. C2×#14523247C22semidirect
17. C2×Q8×D127C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523237C22semidirect
25. C2×#14523235C22semidirect
26. C2×#14523235C22semidirect
28. C2×#14523233C22semidirect
29. C2×#14523233C22semidirect
30. C2×#14523231C22semidirect
31. C2×C8×D127C22semidirect
35. #16861185C22semidirect
40. #16861180C22semidirect
43. C4×Dic254C22semidirect
46. #16861142C22semidirect
47. #16861132C22semidirect
48. #16861131C22semidirect
49. #16861131C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861120C22semidirect
56. #124×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
91. #14523236D4semidirect
136. Dic254C2×D4semidirect
161. C127⋊C8C2×D4semidirect
198. C2×D127AGL(1,ℤ/8)semidirect
219. C127#3265semidirect
227. #463D127semidirect
255. #124C2×D127semidirect
270. C4⋊C8C2×D127semidirect
272. #103C2×D127semidirect
276. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
C2×SD16×D1272
#225505342
D4×D5082
#225502662
#225501252
#225501112
#225500772
#225500252
#225500222
#225500162
#225500082
#225499762
#225499462
#463×C1272
#3265127