Finite Group Databasebeta

← #30650691

Subgroups — #30650691

Order 16256.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550530C211yesmaximal, characteristic, complemented
2#22550235C211yesmaximal, characteristic, complemented
3#22550204C211yesmaximal, characteristic, complemented
4#22550192C211yesmaximal, characteristic, complemented
5#22550166C211yesmaximal, characteristic, complemented
6#22550127C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550088C211yesmaximal, characteristic, complemented
9#22550021C211yesmaximal, characteristic, complemented
10#22550010C211yesmaximal, characteristic, complemented
11#22549986C211yesmaximal, characteristic, complemented
12#22549948C211yesmaximal, characteristic, complemented
13#22549877C211yesmaximal, characteristic, complemented
14C2×#16861130C211yesmaximal, characteristic, complemented
15#546×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×#14523235C2211yescharacteristic, complemented
18C2×#14523233C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523232C2211yescharacteristic, complemented
21C22×C127⋊C8C2211yescharacteristic
22#16861209C2211yescharacteristic, complemented
23#16861193—21no—
24#16861193—21no—
25#16861189—21no—
26#16861183C2211yescharacteristic
27#16861183C2211yescharacteristic, complemented
28C4×D508C2211yescharacteristic, complemented
29C2×C4⋊Dic127C2211yescharacteristic
30C2×C4⋊Dic127C2211yescharacteristic, complemented
31#16861140C2211yescharacteristic, complemented
32#16861140C2211yescharacteristic, complemented
33#16861140C2211yescharacteristic, complemented
34C4⋊(C127⋊C8)C2211yescharacteristic, complemented
35#16861133C2211yescharacteristic, complemented
36#16861133C2211yescharacteristic, complemented
37#16861133C2211yescharacteristic, complemented
38#16861130C2211yescharacteristic
39#16861130C2211yescharacteristic
40#16861130C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861130C2211yescharacteristic
43#16861130C2211yescharacteristic
44#16861129C2211yescharacteristic, complemented
45C22⋊(C127⋊C8)C2211yescharacteristic
46C22⋊(C127⋊C8)C2211yescharacteristic, complemented
47#16861120C2211yescharacteristic, complemented
48C2×#41×C127C2211yescharacteristic
49C22×D4×C127C2211yescharacteristic
50#124×C127C2211yescharacteristic
51#123×C127—21no—
52C22⋊Q8×C127C2211yescharacteristic
53#121×C127—21no—
54#121×C127C2211yescharacteristic
55#120×C127—21no—
56C4×D4×C127C2211yescharacteristic
57C4×D4×C127C2211yescharacteristic
58C2×C22⋊C4×C127—21no—
59#14523247—44no—
60C2×Dic254—21no—
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C22×Dic127—21no—
65C22×Dic127—21no—
66C2×C4×D127—21no—
67#14523235—41no—
68#14523233—22no—
69#14523233—41no—
70#14523232—41no—
71C2×C127⋊C8—41no—
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C4⋊Dic127—21no—
77C4⋊Dic127—21no—
78C4⋊Dic127—21no—
79C4⋊Dic127C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82(C2×D127)⋊C4—21no—
83(C2×D127)⋊C4—21no—
84(C2×D127)⋊C4—21no—
85C24×C127—21no—
86#41×C127—22no—
87#41×C127—41no—
88C2×Q8×C127C2311yescharacteristic
89C2×D4×C127—22no—
90C2×D4×C127—21no—
91C2×D4×C127—21no—
92C2×D4×C127—21no—
93C2×D4×C127—41no—
94C2×D4×C127D411yescharacteristic
95C2×D4×C127C2311yescharacteristic
96C2×D4×C127C2311yescharacteristic
97C22×C4×C127—21no—
98C22×C4×C127—21no—
99C22×C4×C127C2311yescharacteristic
100C22×C4×C127C2311yescharacteristic
101C4⋊C4×C127—21no—
102C4⋊C4×C127D411yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C22⋊C4×C127—21no—
108C22⋊C4×C127—21no—
109C22⋊C4×C127D411yescharacteristic
110C22⋊C4×C127D411yescharacteristic
111C42×C127C2311yescharacteristic
112C22×D127—21no—
113Dic254—42no—
114D508—21no—
115D508—21no—
116C22⋊D127—42no—
117C22⋊D127—42no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—42no—
122C2×Dic127—42no—
123C4×D127—42no—
124C127⋊C8—21no—
125C127⋊C8—21no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C23×C127—21no—
129C23×C127—21no—
130C23×C127—42no—
131C23×C127—42no—
132C23×C127—81no—
133C23×C127C2×D411yescharacteristic
134C23×C127C2×D411yescharacteristic
135Q8×C127—21no—
136Q8×C127—21no—
137D4×C127—22no—
138D4×C127—21no—
139D4×C127—21no—
140D4×C127—21no—
141D4×C127—42no—
142D4×C127—42no—
143D4×C127—81no—
144D4×C127#4112yescomplemented
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—42no—
152C2×C4×C127—81no—
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2×D411yescharacteristic
156C2×C4×C127C2×D411yescharacteristic
157C2×C4×C127C2411yescharacteristic
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—22no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—84no—
170C22×C127AGL(1,ℤ/8)12yes—
171C22×C127C22×D411yescharacteristic
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—41no—
175C4×C127—21no—
176C4×C127—21no—
177C4×C127—21no—
178C4×C127—21no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127C2×#4111yescharacteristic
182C4×C127H(5,𝔽2)11yescharacteristic
183Dic127—41no—
184Dic127—41no—
185Dic127—41no—
186C2×C127—22no—
187C2×C127—41no—
188C2×C127—42no—
189C2×C127—81no—
190C2×C127#54611yescharacteristic
191C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C127#57711yescharacteristic
193D127—81no—
194#3306—1271nomaximal, Sylow
195C127#330611yescharacteristic, complemented, Sylow
196#547—1271no—
197#546D12711yescharacteristic, complemented
198#489—1271no—
199#483—1271no—
200#481—1271no—
201#476—1271no—
202#469—1271no—
203#465—1271no—
204#463—1271no—
205#449—1271no—
206#447—1271no—
207#442—1271no—
208#435—1271no—
209#417—1271no—
210C2×#102—1271no—
211C2×#41—2541no—
212C2×#41C2×D12711yescharacteristic
213C22×D4C2×D12711yescharacteristic
214C2×SD16—1271no—
215C2×D8—1271no—
216C2×D8—1271no—
217C2×M16—1271no—
218C22×C8—1271no—
219C4⋊Q8—1271no—
220#124C2×D12711yescharacteristic
221#123—21no—
222#123—2541no—
223#123—2541no—
224C22⋊Q8—2541no—
225C22⋊Q8C2×D12711yescharacteristic
226#121—21no—
227#121—1271no—
228#121—1271no—
229#121C2×D12711yescharacteristic
230#120—21no—
231C4×D4—1271no—
232C4×D4C2×D12711yescharacteristic
233C4×D4C2×D12711yescharacteristic
234C2×C4⋊C4—1271no—
235C2×C4⋊C4—1271no—
236C2×C22⋊C4—21no—
237#106—1271no—
238#106—1271no—
239#106—1271no—
240C4⋊C8—1271no—
241#103—1271no—
242#103—1271no—
243#103—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102—1271no—
251C22⋊C8—1271no—
252C22⋊C8—1271no—
253#97—1271no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—5084no—
258C2×Q8—2541no—
259C2×Q8C22×D12711yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—1271no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4C22⋊D12711yescharacteristic
269C2×D4C22×D12711yescharacteristic
270C2×D4C22×D12711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—2541no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4C22×D12711yescharacteristic
277C22×C4C22×D12711yescharacteristic
278SD16—5081no—
279D8—5081no—
280D8—2542no—
281M16—5081no—
282C2×C8—1271no—
283C2×C8—1271no—
284C2×C8—1271no—
285C2×C8—1271no—
286C2×C8—5081no—
287C4⋊C4—21no—
288C4⋊C4—1271no—
289C4⋊C4—1271no—
290C4⋊C4—1271no—
291C4⋊C4—2541no—
292C4⋊C4—2541no—
293C4⋊C4—2541no—
294C4⋊C4C22⋊D12711yescharacteristic
295C4⋊C4C22×D12711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4—2541no—
304C22⋊C4C22⋊D12711yescharacteristic
305C22⋊C4C22⋊D12711yescharacteristic
306C42C22×D12711yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—2541no—
313C23C2×C22⋊D12711yescharacteristic
314C23C2×C22⋊D12711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—5082no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—5082no—
326D4—5082no—
327D4—2541no—
328D4—2541no—
329D4#1452324712yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—5082no—
339C2×C4—5082no—
340C2×C4—5082no—
341C2×C4—2541no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4C2×C22⋊D12711yescharacteristic
345C2×C4C2×C22⋊D12711yescharacteristic
346C2×C4C2×C22⋊D12711yescharacteristic
347C2×C4C2×C22⋊D12711yescharacteristic
348C2×C4C23×D12711yescharacteristic
349C8—2541no—
350C8—2541no—
351C8—2541no—
352C8—2541no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—5081no—
366C22—5081no—
367C22—5081no—
368C22#1686123412yes—
369C22C22×C22⋊D12711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—5081no—
377C4—5081no—
378C4—5081no—
379C4C2×#1452324711yescharacteristic
380C4#1686125411yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—10161no—
386C2#2255052011yescharacteristic
387C2C2×#1686123411yescharacteristic
388C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#546×C127C2
socleC22×C127C22×D4
hypercenter#546D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #22550530C2semidirect
2. #22550235C2semidirect
3. #22550204C2semidirect
4. #22550192C2semidirect
5. #22550166C2semidirect
6. #22550127C2semidirect
7. #22550107C2semidirect
8. #22550088C2semidirect
9. #22550021C2semidirect
10. #22550010C2semidirect
11. #22549986C2semidirect
12. #22549948C2semidirect
13. #22549877C2semidirect
14. C2×#16861130C2semidirect
15. #546×C127C2semidirect
17. C2×#14523235C22semidirect
20. C2×#14523232C22semidirect
22. #16861209C22semidirect
27. #16861183C22semidirect
28. C4×D508C22semidirect
30. C2×C4⋊Dic127C22semidirect
31. #16861140C22semidirect
32. #16861140C22semidirect
33. #16861140C22semidirect
34. C4⋊(C127⋊C8)C22semidirect
35. #16861133C22semidirect
36. #16861133C22semidirect
37. #16861133C22semidirect
44. #16861129C22semidirect
46. C22⋊(C127⋊C8)C22semidirect
47. #16861120C22semidirect
144. D4×C127#41semidirect
195. C127#3306semidirect
197. #546D127semidirect
329. D4#14523247semidirect

Maximal subgroups

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

subgroupindex
#225505302
#225502352
#225502042
#225501922
#225501662
#225501272
#225501072
#225500882
#225500212
#225500102
#225499862
#225499482
#225498772
C2×#168611302
#546×C1272
#3306127