Finite Group Databasebeta

← #30650224

Subgroups — #30650224

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
4#22550511C211yesmaximal, characteristic, complemented
5#22550119C211yesmaximal, characteristic, complemented
6#22550109C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550073C211yesmaximal, characteristic, complemented
9#22550025C211yesmaximal, characteristic, complemented
10#22550018C211yesmaximal, characteristic, complemented
11#22550013C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549978C211yesmaximal, characteristic, complemented
14#22549949C211yesmaximal, characteristic, complemented
15#496×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic, complemented
17C2×Q8×D127C2211yescharacteristic, complemented
18C2×D4×D127—21no—
19C2×D4×D127C2211yescharacteristic
20C2×D4×D127C2211yescharacteristic
21#16861233—24no—
22C2×#14523237C2211yescharacteristic
23C2×#14523236C2211yescharacteristic, complemented
24C2×#14523236C2211yescharacteristic, complemented
25C2×#14523235C2211yescharacteristic
26SD16×D127—24no—
27C2×#14523233C2211yescharacteristic
28C2×#14523231C2211yescharacteristic, complemented
29C2×C8×D127C2211yescharacteristic, complemented
30#16861195—21no—
31#16861186—21no—
32#16861185C2211yescharacteristic, complemented
33#16861182—21no—
34#16861180—21no—
35#16861180C2211yescharacteristic
36#16861180C2211yescharacteristic
37#16861180C2211yescharacteristic, complemented
38#16861177—21no—
39#16861177—21no—
40C4×Dic254C2211yescharacteristic, complemented
41C4×D508C2211yescharacteristic
42C4×D508C2211yescharacteristic
43(C22⋊D127)⋊C4—21no—
44#16861143C2211yescharacteristic, complemented
45C4⋊(C127⋊C8)C2211yescharacteristic, complemented
46#16861134C2211yescharacteristic, complemented
47#16861133C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic
49#16861130C2211yescharacteristic, complemented
50#16861130C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53(C2×D127)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic, complemented
56C2×D8×C127C2211yescharacteristic, complemented
57C2×D8×C127C2211yescharacteristic, complemented
58#127×C127C2211yescharacteristic
59#124×C127C2211yescharacteristic, complemented
60#97×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×D508D411yescharacteristic
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77D4×D127—22no—
78D4×D127—22no—
79D4×D127—41no—
80D4×D127—41no—
81D4×D127—44no—
82C22×Dic127—21no—
83C2×C4×D127—21no—
84C2×C4×D127—21no—
85C2×C4×D127C2311yescharacteristic
86C2×C4×D127C2311yescharacteristic
87#14523237D414yescomplemented
88#14523236—22no—
89#14523236—22no—
90#14523235—22no—
91#14523233—22no—
92#14523231—22no—
93C2×C127⋊C8C2311yescharacteristic
94C2×C127⋊C8C2311yescharacteristic
95C8×D127—22no—
96C4⋊Dic127D411yescharacteristic
97C4⋊Dic127C2311yescharacteristic
98C4⋊Dic127C2311yescharacteristic
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4D411yescharacteristic
107(C2×D127)⋊C4D411yescharacteristic
108C4×Dic127—21no—
109C2×Q8×C127C2311yescharacteristic
110C2×D4×C127—21no—
111C2×D4×C127—21no—
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115SD16×C127—22no—
116SD16×C127—41no—
117D8×C127—22no—
118D8×C127—41no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C22⋊C4×C127—21no—
123C42×C127C2311yescharacteristic
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—81no—
130C22×D127—81no—
131C22×D127C2×D411yescharacteristic
132C22×D127C2×D411yescharacteristic
133Dic254—21no—
134Dic254—42no—
135Dic254C2×D412yescomplemented
136D508—22no—
137D508—21no—
138D508C2×D412yes—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C22⋊D127—42no—
142C22⋊D127—42no—
143C22⋊D127—42no—
144C22⋊D127—42no—
145C2×Dic127—21no—
146C2×Dic127—21no—
147C2×Dic127—21no—
148C2×Dic127—42no—
149C2×Dic127C2×D411yescharacteristic
150C2×Dic127C2×D411yescharacteristic
151C4×D127—21no—
152C4×D127—21no—
153C4×D127—21no—
154C4×D127—21no—
155C4×D127—42no—
156C4×D127—42no—
157C127⋊C8—21no—
158C127⋊C8—21no—
159C23×C127—21no—
160C23×C127—21no—
161C23×C127—21no—
162Q8×C127—21no—
163Q8×C127—21no—
164D4×C127—21no—
165D4×C127—21no—
166D4×C127—21no—
167D4×C127—21no—
168D4×C127—21no—
169D4×C127—21no—
170D4×C127—42no—
171D4×C127—81no—
172C2×C4×C127—21no—
173C2×C4×C127C2×D411yescharacteristic
174C2×C4×C127C2×D411yescharacteristic
175C2×C4×C127C2411yescharacteristic
176C8×C127—21no—
177C8×C127C2×D412yescomplemented
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
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—21no—
191C2×D127—22no—
192C2×D127—21no—
193C2×D127—21no—
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×C127C22×D411yescharacteristic
203C4×C127H(5,𝔽2)11yescharacteristic
204Dic127—21no—
205Dic127—21no—
206Dic127—22no—
207Dic127—41no—
208Dic127—41no—
209C2×C127—81no—
210C2×C127—81no—
211C2×C127—81no—
212C2×C127D4211yescharacteristic
213C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C127#57711yescharacteristic
215D127—22no—
216D127—41no—
217D127—42no—
218#3265—1271nomaximal, Sylow
219C127#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1271no—
221C22×SD16—1271no—
222#548—1271no—
223D42—1271no—
224#496D12711yescharacteristic, complemented
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×#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—2544no—
246C2×SD16C2×D12711yescharacteristic, complemented
247C2×SD16C2×D12711yescharacteristic, complemented
248C2×D8—1271no—
249C2×D8C2×D12711yescharacteristic, complemented
250C2×D8C2×D12711yescharacteristic, complemented
251C2×M16—1271no—
252C22×C8—1271no—
253#127C2×D12711yescharacteristic
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—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#97C2×D12711yescharacteristic, complemented
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—21no—
290C2×D4—21no—
291C2×D4—1271no—
292C2×D4—1271no—
293C2×D4—5084no—
294C2×D4—5081no—
295C2×D4—5081no—
296C2×D4—2541no—
297C2×D4—2541no—
298C2×D4—2542no—
299C2×D4—2542no—
300C2×D4—2541no—
301C2×D4—2541no—
302C2×D4—2541no—
303C2×D4—2541no—
304C2×D4C22×D12711yescharacteristic
305C2×D4C22×D12711yescharacteristic
306C2×D4C22×D12711yescharacteristic
307C22×C4—1271no—
308C22×C4—1271no—
309C22×C4—2541no—
310C22×C4—2541no—
311C22×C4—2541no—
312SD16—22no—
313SD16—41no—
314SD16—1274no—
315SD16—2542no—
316SD16—2542no—
317SD16—2542no—
318D8—22no—
319D8—41no—
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—1271no—
330C4⋊C4—2541no—
331C4⋊C4—2541no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—1271no—
335C22⋊C4—1271no—
336C22⋊C4—2541no—
337C22⋊C4—2541no—
338C22⋊C4—2541no—
339C22⋊C4—2541no—
340C22⋊C4—2541no—
341C42—2541no—
342C42C22×D12711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—1271no—
347C23—1271no—
348C23—10161no—
349C23—10161no—
350C23—5082no—
351C23—5082no—
352C23—5082no—
353C23—5082no—
354C23—2541no—
355Q8—21no—
356Q8—21no—
357Q8—1272no—
358Q8—5082no—
359Q8—2541no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—1272no—
369D4—5082no—
370D4—5082no—
371D4—5082no—
372D4—5082no—
373D4—5082no—
374D4—5082no—
375D4—2542no—
376D4—2541no—
377C2×C4—21no—
378C2×C4—1271no—
379C2×C4—1271no—
380C2×C4—5082no—
381C2×C4—5082no—
382C2×C4—5082no—
383C2×C4—2541no—
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—2541no—
395C8—2541no—
396C8D4×D12712yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—1272no—
407C22—10164no—
408C22—10164no—
409C22—5081no—
410C22—5081no—
411C22—2541no—
412C22—2541no—
413C22—2541no—
414C22—2542no—
415C22—2541no—
416C22—2541no—
417C22C2×D4×D12711yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—5081no—
422C4—5081no—
423C4—2541no—
424C4—2541no—
425C4—2542no—
426C4C2×D4×D12711yescharacteristic
427C4#1686125411yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—5082no—
432C2—5081no—
433C2—2542no—
434C2#2255051111yescharacteristic
435C2AGL(1,ℤ/8)×D12711yescharacteristic
436C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#496×C127C2
socleC22×C127C22×D4
hypercenter#496D127
hypocenterC127#3265
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. C2×SD16×D127C2semidirect
3. #22550534C2semidirect
4. #22550511C2semidirect
5. #22550119C2semidirect
6. #22550109C2semidirect
7. #22550088C2semidirect
8. #22550073C2semidirect
9. #22550025C2semidirect
10. #22550018C2semidirect
11. #22550013C2semidirect
12. #22550008C2semidirect
13. #22549978C2semidirect
14. #22549949C2semidirect
15. #496×C127C2semidirect
16. C2×#14523247C22semidirect
17. C2×Q8×D127C22semidirect
23. C2×#14523236C22semidirect
24. C2×#14523236C22semidirect
28. C2×#14523231C22semidirect
29. C2×C8×D127C22semidirect
32. #16861185C22semidirect
37. #16861180C22semidirect
40. C4×Dic254C22semidirect
44. #16861143C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. #16861134C22semidirect
47. #16861133C22semidirect
49. #16861130C22semidirect
50. #16861130C22semidirect
51. #16861130C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. (C2×D127)⋊C8C22semidirect
54. C2×SD16×C127C22semidirect
55. C2×SD16×C127C22semidirect
56. C2×D8×C127C22semidirect
57. C2×D8×C127C22semidirect
59. #124×C127C22semidirect
60. #97×C127C22semidirect
87. #14523237D4semidirect
135. Dic254C2×D4semidirect
177. C8×C127C2×D4semidirect
198. C2×D127AGL(1,ℤ/8)semidirect
219. C127#3265semidirect
224. #496D127semidirect
246. C2×SD16C2×D127semidirect
247. C2×SD16C2×D127semidirect
249. C2×D8C2×D127semidirect
250. C2×D8C2×D127semidirect
255. #124C2×D127semidirect
279. #97C2×D127semidirect
396. C8D4×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
#225505112
#225501192
#225501092
#225500882
#225500732
#225500252
#225500182
#225500132
#225500082
#225499782
#225499492
#496×C1272
#3265127