Finite Group Databasebeta

← #30650580

Subgroups — #30650580

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550572C211yesmaximal, characteristic, complemented
2#22550230C211yesmaximal, characteristic, complemented
3#22550209C211yesmaximal, characteristic, complemented
4#22550197C211yesmaximal, characteristic, complemented
5#22550176C211yesmaximal, characteristic, complemented
6#22550110C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550088C211yesmaximal, characteristic, complemented
9#22550021C211yesmaximal, characteristic, complemented
10#22550010C211yesmaximal, characteristic, complemented
11C4×#14523233C211yesmaximal, characteristic, complemented
12D4×C127⋊C8C211yesmaximal, characteristic, complemented
13#22549872C211yesmaximal, characteristic, complemented
14C2×#16861130C211yesmaximal, characteristic, complemented
15#546×C127C211yesmaximal, characteristic, complemented
16C2×#14523235C2211yescharacteristic, complemented
17C2×#14523233C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C22×C127⋊C8C2211yescharacteristic
20C22×C127⋊C8C2211yescharacteristic, complemented
21#16861204—21no—
22#16861201C2211yescharacteristic, complemented
23#16861191—21no—
24#16861183C2211yescharacteristic
25#16861183C2211yescharacteristic, complemented
26#16861181—21no—
27C4×D508C2211yescharacteristic, complemented
28(C22⋊D127)⋊C4—21no—
29#16861168C2211yescharacteristic, complemented
30C2×C4⋊Dic127C2211yescharacteristic
31#16861142C2211yescharacteristic, complemented
32#16861142C2211yescharacteristic, 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
47C4×C127⋊C8C2211yescharacteristic, 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—
59C2×D508C2311yescharacteristic
60C2×C22⋊D127—21no—
61C2×C22⋊D127—21no—
62C22×Dic127—21no—
63C2×C4×D127—21no—
64#14523235—41no—
65#14523233—22no—
66#14523233—41no—
67C2×C127⋊C8—41no—
68C2×C127⋊C8—41no—
69C2×C127⋊C8C2311yescharacteristic
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C2×C127⋊C8C2311yescharacteristic
73C4⋊Dic127—21no—
74C4⋊Dic127C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79C22⋊Dic127—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83C4×Dic127—21no—
84C24×C127—21no—
85#41×C127—22no—
86#41×C127—41no—
87C2×Q8×C127C2311yescharacteristic
88C2×D4×C127—22no—
89C2×D4×C127—21no—
90C2×D4×C127—21no—
91C2×D4×C127—21no—
92C2×D4×C127—41no—
93C2×D4×C127D411yescharacteristic
94C2×D4×C127C2311yescharacteristic
95C2×D4×C127C2311yescharacteristic
96C22×C4×C127—21no—
97C22×C4×C127—21no—
98C22×C4×C127C2311yescharacteristic
99C22×C4×C127C2311yescharacteristic
100C4⋊C4×C127—21no—
101C4⋊C4×C127D411yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C22⋊C4×C127—21no—
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C22⋊C4×C127—21no—
108C22⋊C4×C127D411yescharacteristic
109C22⋊C4×C127D411yescharacteristic
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112D508—21no—
113D508—21no—
114C22⋊D127—42no—
115C22⋊D127—81no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—81no—
120C4×D127—81no—
121C127⋊C8—21no—
122C127⋊C8—21no—
123C127⋊C8—21no—
124C127⋊C8—21no—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127—42no—
128C23×C127—42no—
129C23×C127—81no—
130C23×C127C2×D411yescharacteristic
131C23×C127C2×D411yescharacteristic
132Q8×C127—21no—
133Q8×C127—21no—
134D4×C127—22no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—42no—
139D4×C127—42no—
140D4×C127—81no—
141D4×C127#4112yes—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—42no—
149C2×C4×C127—81no—
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2×D411yescharacteristic
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2411yescharacteristic
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—22no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—84no—
167C22×C127#13512yes—
168C22×C127C22×D411yescharacteristic
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—41no—
172C4×C127—21no—
173C4×C127—21no—
174C4×C127—21no—
175C4×C127—21no—
176C4×C127—41no—
177C4×C127—41no—
178C4×C127C2×#4111yescharacteristic
179C4×C127H(5,𝔽2)11yescharacteristic
180Dic127—41no—
181Dic127—41no—
182Dic127—41no—
183C2×C127—22no—
184C2×C127—41no—
185C2×C127—42no—
186C2×C127—81no—
187C2×C127#54611yescharacteristic
188C2×C127C2×#13511yescharacteristic
189C2×C127#57611yescharacteristic
190D127—81no—
191#3297—1271nomaximal, Sylow
192C127#329711yescharacteristic, complemented, Sylow
193#553—1271no—
194#546D12711yescharacteristic, complemented
195#488—1271no—
196#484—1271no—
197#482—1271no—
198#478—1271no—
199#466—1271no—
200#465—1271no—
201#463—1271no—
202#449—1271no—
203#447—1271no—
204C4×D8—1271no—
205C8×D4—1271no—
206#416—1271no—
207C2×#102—1271no—
208C2×#41C2×D12711yescharacteristic
209C22×D4C2×D12711yescharacteristic
210C2×SD16—1271no—
211C2×D8—1271no—
212C2×D8—1271no—
213C22×C8—1271no—
214C22×C8—1271no—
215#126—2541no—
216#125—1271no—
217#124C2×D12711yescharacteristic
218#123—21no—
219#123—2541no—
220C22⋊Q8C2×D12711yescharacteristic
221#121—21no—
222#121—1271no—
223#121—1271no—
224#121—2541no—
225#121C2×D12711yescharacteristic
226#120—21no—
227C4×D4—1271no—
228C4×D4—2541no—
229C4×D4C2×D12711yescharacteristic
230C4×D4C2×D12711yescharacteristic
231#117—1271no—
232C2×C4⋊C4—1271no—
233C2×C22⋊C4—21no—
234#107—1271no—
235#107—1271no—
236#106—1271no—
237C4⋊C8—1271no—
238#103—1271no—
239#103—1271no—
240#103—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248C22⋊C8—1271no—
249C22⋊C8—1271no—
250C4×C8—1271no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D12711yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—1271no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4C22⋊D12711yescharacteristic
264C2×D4C22×D12711yescharacteristic
265C2×D4C22×D12711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—2541no—
269C22×C4—2541no—
270C22×C4C22×D12711yescharacteristic
271C22×C4C22×D12711yescharacteristic
272SD16—5081no—
273D8—5081no—
274D8—2542no—
275C2×C8—1271no—
276C2×C8—1271no—
277C2×C8—1271no—
278C2×C8—1271no—
279C2×C8—5081no—
280C2×C8—5081no—
281C4⋊C4—21no—
282C4⋊C4—1271no—
283C4⋊C4—1271no—
284C4⋊C4—1271no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
288C4⋊C4C22⋊D12711yescharacteristic
289C4⋊C4C22×D12711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C22⋊C4C22⋊D12711yescharacteristic
300C22⋊C4C22⋊D12711yescharacteristic
301C42—2541no—
302C42C22×D12711yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—2541no—
309C23C2×C22⋊D12711yescharacteristic
310C23C2×C22⋊D12711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—10161no—
321D4—5082no—
322D4—2541no—
323D4—2541no—
324D4#1452324712yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—10161no—
334C2×C4—10161no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4C2×C22⋊D12711yescharacteristic
339C2×C4C2×C22⋊D12711yescharacteristic
340C2×C4C2×C22⋊D12711yescharacteristic
341C2×C4C2×C22⋊D12711yescharacteristic
342C2×C4C23×D12711yescharacteristic
343C8—2541no—
344C8—2541no—
345C8—2541no—
346C8—2541no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—5081no—
360C22—5081no—
361C22—5081no—
362C22#1686122912yes—
363C22C22×C22⋊D12711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—5081no—
371C4—5081no—
372C4—5081no—
373C4C2×#1452324711yescharacteristic
374C4#1686125411yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—10161no—
380C2#2255052011yescharacteristic
381C2C2×#1686122911yescharacteristic
382C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#546×C127C2
socleC22×C127C22×D4
hypercenter#546D127
hypocenterC127#3297
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. #22550572C2semidirect
2. #22550230C2semidirect
3. #22550209C2semidirect
4. #22550197C2semidirect
5. #22550176C2semidirect
6. #22550110C2semidirect
7. #22550107C2semidirect
8. #22550088C2semidirect
9. #22550021C2semidirect
10. #22550010C2semidirect
11. C4×#14523233C2semidirect
12. D4×C127⋊C8C2semidirect
13. #22549872C2semidirect
14. C2×#16861130C2semidirect
15. #546×C127C2semidirect
16. C2×#14523235C22semidirect
20. C22×C127⋊C8C22semidirect
22. #16861201C22semidirect
25. #16861183C22semidirect
27. C4×D508C22semidirect
29. #16861168C22semidirect
31. #16861142C22semidirect
32. #16861142C22semidirect
33. #16861140C22semidirect
34. C4⋊(C127⋊C8)C22semidirect
35. #16861133C22semidirect
36. #16861133C22semidirect
37. #16861133C22semidirect
44. #16861129C22semidirect
46. C22⋊(C127⋊C8)C22semidirect
47. C4×C127⋊C8C22semidirect
192. C127#3297semidirect
194. #546D127semidirect

Maximal subgroups

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

subgroupindex
#225505722
#225502302
#225502092
#225501972
#225501762
#225501102
#225501072
#225500882
#225500212
#225500102
C4×#145232332
D4×C127⋊C82
#225498722
C2×#168611302
#546×C1272
#3297127