Finite Group Databasebeta

← #30648342

Subgroups — #30648342

Order 16256.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2C22×#14523231C211yesmaximal, characteristic, complemented
3#22550389C211yesmaximal, characteristic, complemented
4#121×D127C211yesmaximal, characteristic, complemented
5#22550130C211yesmaximal, characteristic, complemented
6#22550128C211yesmaximal, characteristic, complemented
7#22550125C211yesmaximal, characteristic, complemented
8#22550119C211yesmaximal, characteristic, complemented
9#22550112C211yesmaximal, characteristic, complemented
10#22550110C211yesmaximal, characteristic, complemented
11#22550107C211yesmaximal, characteristic, complemented
12#22549915C211yesmaximal, characteristic, complemented
13#22549879C211yesmaximal, characteristic, complemented
14#22549876C211yesmaximal, characteristic, complemented
15#465×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×D4×D127—21no—
18C2×D4×D127C2211yescharacteristic
19C22×C4×D127C2211yescharacteristic
20#16861236—24no—
21C2×#14523235C2211yescharacteristic, complemented
22C2×D1016C2211yescharacteristic, complemented
23C2×#14523233C2211yescharacteristic, complemented
24C2×#14523231—24no—
25C2×#14523231C2211yescharacteristic
26C2×#14523231C2211yescharacteristic
27C2×#14523231C2211yescharacteristic, complemented
28C2×#14523231C2211yescharacteristic, complemented
29C22×C127⋊C8C2211yescharacteristic, complemented
30#16861194—21no—
31#16861185C2211yescharacteristic, complemented
32#16861184C2211yescharacteristic
33#16861184C2211yescharacteristic, complemented
34#16861183C2211yescharacteristic
35#16861183C2211yescharacteristic, complemented
36#16861182—21no—
37#16861181—21no—
38#16861180C2211yescharacteristic
39#16861179—21no—
40#16861171—21no—
41(C22⋊D127)⋊C4—21no—
42#16861167C2211yescharacteristic
43C4⋊C4×D127C2211yescharacteristic
44C22⋊C4×D127—21no—
45#16861143C2211yescharacteristic, complemented
46#16861142C2211yescharacteristic, complemented
47#16861140C2211yescharacteristic, complemented
48#16861133C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic, complemented
52#16861129C2211yescharacteristic, complemented
53#16861129C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C22×C8×C127C2211yescharacteristic, complemented
56C22⋊Q8×C127C2211yescharacteristic, complemented
57#121×C127C2211yescharacteristic
58#106×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127D411yescharacteristic
63#14523249—22no—
64#14523249—22no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508—21no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74D4×D127—22no—
75D4×D127—22no—
76D4×D127—44no—
77C22×Dic127D411yescharacteristic
78C2×C4×D127—22no—
79C2×C4×D127—22no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127D411yescharacteristic
83C2×C4×D127D411yescharacteristic
84C2×C4×D127C2311yescharacteristic
85C2×C4×D127C2311yescharacteristic
86#14523235—22no—
87D1016—22no—
88#14523233—22no—
89#14523231—22no—
90#14523231—22no—
91#14523231—22no—
92#14523231D414yescomplemented
93C2×C127⋊C8—22no—
94C2×C127⋊C8C2311yescharacteristic
95C2×C127⋊C8C2311yescharacteristic
96C4⋊Dic127C2311yescharacteristic
97C4⋊Dic127C2311yescharacteristic
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100C22⋊Dic127—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107C4×Dic127—21no—
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127—21no—
110C2×D4×C127C2311yescharacteristic
111C22×C4×C127C2311yescharacteristic
112SD16×C127—22no—
113C2×C8×C127—22no—
114C2×C8×C127C2311yescharacteristic
115C2×C8×C127C2311yescharacteristic
116C4⋊C4×C127—21no—
117C4⋊C4×C127C2311yescharacteristic
118C4⋊C4×C127C2311yescharacteristic
119C22⋊C4×C127—21no—
120C22⋊C4×C127—21no—
121C22×D127—22no—
122C22×D127—22no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—41no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127C2×D411yescharacteristic
130C22×D127C2×D411yescharacteristic
131D508—21no—
132D508—21no—
133D508—21no—
134D508—21no—
135D508—42no—
136D508—42no—
137D508—42no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C22⋊D127—42no—
142C2×Dic127—22no—
143C2×Dic127—21no—
144C2×Dic127—21no—
145C2×Dic127C2×D411yescharacteristic
146C2×Dic127C2×D411yescharacteristic
147C4×D127—22no—
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—22no—
151C4×D127—42no—
152C4×D127—42no—
153C4×D127C2×D412yes—
154C4×D127#4112yes—
155C127⋊C8—21no—
156C127⋊C8C2×D412yescomplemented
157C23×C127—21no—
158C23×C127C2×D411yescharacteristic
159Q8×C127—21no—
160Q8×C127—21no—
161D4×C127—21no—
162D4×C127—21no—
163D4×C127—42no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127C2×D411yescharacteristic
170C2×C4×C127C2411yescharacteristic
171C8×C127—21no—
172C8×C127C2×D412yescomplemented
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127AGL(1,ℤ/8)12yescomplemented
179C22×C127C22×D411yescharacteristic
180C2×D127—21no—
181C2×D127—21no—
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C2×D127—44no—
186C2×D127—41no—
187C2×D127—41no—
188C2×D127—41no—
189C2×D127—41no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—84no—
193C2×D127#12112yes—
194C4×C127—21no—
195C4×C127—41no—
196C4×C127—41no—
197C4×C127—41no—
198C4×C127C22×D411yescharacteristic
199C4×C127C2×#4111yescharacteristic
200Dic127—21no—
201Dic127—41no—
202Dic127—41no—
203Dic127#12112yes—
204C2×C127—22no—
205C2×C127—81no—
206C2×C127C2×#12111yescharacteristic
207C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C127#57511yescharacteristic
209D127—22no—
210D127—41no—
211D127—81no—
212D127—81no—
213#3055—1271nomaximal, Sylow
214C127#305511yescharacteristic, complemented, Sylow
215C2×AGL(1,ℤ/8)—1271no—
216C22×M16—1271no—
217#525—1271no—
218C2×#121—1271no—
219#469—1271no—
220#469—1271no—
221#468—1271no—
222#468—1271no—
223#466—1271no—
224#466—1271no—
225#465—1271no—
226#465D12711yescharacteristic, complemented
227#428—1271no—
228#418—1271no—
229#417—1271no—
230C2×#41—1271no—
231C22×D4—1271no—
232C22×D4—2541no—
233C23×C4—1271no—
234AGL(1,ℤ/8)—2544no—
235C2×SD16—1271no—
236C2×SD16C2×D12711yescharacteristic, complemented
237C2×D8—1271no—
238C2×D8—1271no—
239C2×M16—1271no—
240C2×M16—1271no—
241C2×M16—1271no—
242C2×M16—1271no—
243C2×M16—2544no—
244C22×C8—1271no—
245C22×C8C2×D12711yescharacteristic, complemented
246#123—2541no—
247C22⋊Q8—1271no—
248C22⋊Q8C2×D12711yescharacteristic, complemented
249#121—1271no—
250#121—1271no—
251#121—1271no—
252#121—1271no—
253#121—1271no—
254#121—2541no—
255#121—2541no—
256#121C2×D12711yescharacteristic
257#120—2541no—
258C4×D4—2541no—
259C4×D4—2541no—
260#117—1271no—
261C2×C4⋊C4—1271no—
262C2×C22⋊C4—2541no—
263#107—1271no—
264#107—1271no—
265#106—1271no—
266#106C2×D12711yescharacteristic, complemented
267#103—1271no—
268#103C2×D12711yescharacteristic, complemented
269#102—1271no—
270#102—1271no—
271#102—1271no—
272#102—1271no—
273#102—1271no—
274#102C2×D12711yescharacteristic, complemented
275C24—1271no—
276C24—2541no—
277#41—2542no—
278#41—2542no—
279C2×Q8C22×D12711yescharacteristic
280C2×D4—21no—
281C2×D4—1271no—
282C2×D4—1271no—
283C2×D4—5084no—
284C2×D4—2541no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2542no—
288C2×D4—2542no—
289C2×D4—2541no—
290C2×D4—2541no—
291C2×D4—2541no—
292C2×D4—2541no—
293C2×D4C22×D12711yescharacteristic
294C22×C4—1271no—
295C22×C4—1271no—
296C22×C4—1271no—
297C22×C4—1271no—
298C22×C4—1271no—
299C22×C4—2541no—
300C22×C4—2542no—
301C22×C4—2542no—
302C22×C4—2541no—
303C22×C4C22×D12711yescharacteristic
304SD16—22no—
305SD16—2542no—
306D8—2542no—
307D8—2542no—
308M16—1274no—
309M16—2542no—
310M16—2542no—
311M16—2542no—
312C2×C8—22no—
313C2×C8—1271no—
314C2×C8—1271no—
315C2×C8—2542no—
316C2×C8C22×D12711yescharacteristic
317C2×C8C22×D12711yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—1271no—
320C4⋊C4—1271no—
321C4⋊C4—2541no—
322C4⋊C4—2541no—
323C4⋊C4C22×D12711yescharacteristic
324C4⋊C4C22×D12711yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C22⋊C4—2541no—
331C22⋊C4—2541no—
332C22⋊C4—2541no—
333C22⋊C4—2541no—
334C42—2541no—
335C23—21no—
336C23—1271no—
337C23—1271no—
338C23—5081no—
339C23—5082no—
340C23—5082no—
341C23—5082no—
342C23—2542no—
343C23—2541no—
344C23—2541no—
345C23—2542no—
346C23D4×D12711yescharacteristic
347Q8—21no—
348Q8—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—5082no—
353D4—5082no—
354D4—5082no—
355D4—5082no—
356D4—5082no—
357D4—5082no—
358D4—5082no—
359D4—2541no—
360D4—2541no—
361D4—2541no—
362D4—2541no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—1271no—
369C2×C4—1272no—
370C2×C4—1272no—
371C2×C4—1271no—
372C2×C4—5082no—
373C2×C4—5082no—
374C2×C4—2542no—
375C2×C4—2542no—
376C2×C4—2542no—
377C2×C4—2541no—
378C2×C4—2541no—
379C2×C4—2541no—
380C2×C4—2541no—
381C2×C4D4×D12711yescharacteristic
382C2×C4C23×D12711yescharacteristic
383C8—21no—
384C8—1272no—
385C8—2541no—
386C8D4×D12712yescomplemented
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—1272no—
393C22—10164no—
394C22—5084no—
395C22—5081no—
396C22—5081no—
397C22—5081no—
398C22—5081no—
399C22—5081no—
400C22—5081no—
401C22—2541no—
402C22—2541no—
403C22—2541no—
404C22—2541no—
405C22—2541no—
406C22#1686123612yescomplemented
407C22C2×D4×D12711yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—1272no—
413C4—5081no—
414C4—5081no—
415C4—2541no—
416C4C2×D4×D12711yescharacteristic
417C4#41×D12711yescharacteristic
418C2—22no—
419C2—81no—
420C2—10161no—
421C2—10161no—
422C2—5081no—
423C2—2542no—
424C2#121×D12711yescharacteristic
425C2C2×#1686123611yescharacteristic
426C2#2255067311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#465×C127C2
socleC22×C127C22×D4
hypercenter#465D127
hypocenterC127#3055
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C22×#14523231C2semidirect
3. #22550389C2semidirect
4. #121×D127C2semidirect
5. #22550130C2semidirect
6. #22550128C2semidirect
7. #22550125C2semidirect
8. #22550119C2semidirect
9. #22550112C2semidirect
10. #22550110C2semidirect
11. #22550107C2semidirect
12. #22549915C2semidirect
13. #22549879C2semidirect
14. #22549876C2semidirect
15. #465×C127C2semidirect
21. C2×#14523235C22semidirect
22. C2×D1016C22semidirect
23. C2×#14523233C22semidirect
27. C2×#14523231C22semidirect
28. C2×#14523231C22semidirect
29. C22×C127⋊C8C22semidirect
31. #16861185C22semidirect
33. #16861184C22semidirect
35. #16861183C22semidirect
45. #16861143C22semidirect
46. #16861142C22semidirect
47. #16861140C22semidirect
48. #16861133C22semidirect
49. #16861131C22semidirect
50. #16861131C22semidirect
51. #16861130C22semidirect
52. #16861129C22semidirect
53. #16861129C22semidirect
54. C2×SD16×C127C22semidirect
55. C22×C8×C127C22semidirect
56. C22⋊Q8×C127C22semidirect
58. #106×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
92. #14523231D4semidirect
156. C127⋊C8C2×D4semidirect
172. C8×C127C2×D4semidirect
178. C22×C127AGL(1,ℤ/8)semidirect
214. C127#3055semidirect
226. #465D127semidirect
236. C2×SD16C2×D127semidirect
245. C22×C8C2×D127semidirect
248. C22⋊Q8C2×D127semidirect
266. #106C2×D127semidirect
268. #103C2×D127semidirect
274. #102C2×D127semidirect
386. C8D4×D127semidirect
406. C22#16861236semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C22×#145232312
#225503892
#121×D1272
#225501302
#225501282
#225501252
#225501192
#225501122
#225501102
#225501072
#225499152
#225498792
#225498762
#465×C1272
#3055127