Finite Group Databasebeta

← #30650565

Subgroups — #30650565

Order 16256.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550566C211yesmaximal, characteristic, complemented
2#22550230C211yesmaximal, characteristic, complemented
3#22550204C211yesmaximal, characteristic, complemented
4#22550197C211yesmaximal, characteristic, complemented
5#22550176C211yesmaximal, characteristic, complemented
6#22550115C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550095C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550027C211yesmaximal, characteristic, complemented
11C4×#14523236C211yesmaximal, characteristic, complemented
12D4×C127⋊C8C211yesmaximal, characteristic, complemented
13#22549872C211yesmaximal, characteristic, complemented
14C2×#16861130C211yesmaximal, characteristic, complemented
15#546×C127C211yesmaximal, characteristic
16C2×#14523238C2211yescharacteristic, complemented
17C2×#14523236C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C22×C127⋊C8C2211yescharacteristic
20C22×C127⋊C8C2211yescharacteristic, complemented
21#16861202—21no—
22#16861201C2211yescharacteristic, complemented
23#16861199—21no—
24#16861190—21no—
25#16861188C2211yescharacteristic
26#16861188C2211yescharacteristic, complemented
27C4×Dic254C2211yescharacteristic, complemented
28#16861168C2211yescharacteristic, complemented
29#16861165—21no—
30C2×C4⋊Dic127C2211yescharacteristic
31#16861142C2211yescharacteristic, complemented
32#16861140C2211yescharacteristic, complemented
33#16861140C2211yescharacteristic, complemented
34C4⋊(C127⋊C8)C2211yescharacteristic, complemented
35#16861133C2211yescharacteristic, complemented
36#16861133C2211yescharacteristic, complemented
37#16861133C2211yescharacteristic, complemented
38#16861132C2211yescharacteristic, complemented
39#16861130C2211yescharacteristic
40#16861130C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861130C2211yescharacteristic
43#16861130C2211yescharacteristic
44#16861130C2211yescharacteristic
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×Dic254C2311yescharacteristic
60C22×Dic127—21no—
61#14523238—41no—
62#14523236—22no—
63#14523236—41no—
64C2×C127⋊C8—41no—
65C2×C127⋊C8—41no—
66C2×C127⋊C8C2311yescharacteristic
67C2×C127⋊C8C2311yescharacteristic
68C2×C127⋊C8C2311yescharacteristic
69C2×C127⋊C8C2311yescharacteristic
70C4⋊Dic127—21no—
71C4⋊Dic127C2311yescharacteristic
72C4⋊Dic127C2311yescharacteristic
73C4⋊Dic127C2311yescharacteristic
74Dic127⋊C4—21no—
75Dic127⋊C4—21no—
76Dic127⋊C4—21no—
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79C22⋊Dic127—21no—
80C22⋊Dic127—21no—
81C22⋊Dic127—21no—
82C4×Dic127—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
111Dic254—21no—
112Dic254—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—42no—
118C127⋊C8—21no—
119C127⋊C8—21no—
120C127⋊C8—21no—
121C127⋊C8—21no—
122C23×C127—21no—
123C23×C127—21no—
124C23×C127—42no—
125C23×C127—42no—
126C23×C127—81no—
127C23×C127C2×D411yescharacteristic
128C23×C127C2×D411yescharacteristic
129Q8×C127—21no—
130Q8×C127—21no—
131D4×C127—22no—
132D4×C127—21no—
133D4×C127—21no—
134D4×C127—21no—
135D4×C127—42no—
136D4×C127—42no—
137D4×C127—81no—
138D4×C127#4112yes—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—42no—
146C2×C4×C127—81no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2411yescharacteristic
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—22no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—84no—
164C22×C127#13512yes—
165C22×C127C22×D411yescharacteristic
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—21no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127C2×#4111yescharacteristic
173C4×C127H(5,𝔽2)11yescharacteristic
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177Dic127—41no—
178C2×C127—22no—
179C2×C127—41no—
180C2×C127—42no—
181C2×C127—81no—
182C2×C127#54611yescharacteristic
183C2×C127C2×#13511yescharacteristic
184C2×C127#57711yescharacteristic
185#3296—1271nomaximal, Sylow
186C127#329611yescharacteristic, complemented, Sylow
187#552—1271no—
188#546D12711yescharacteristic
189#488—1271no—
190#483—1271no—
191#482—1271no—
192#478—1271no—
193#467—1271no—
194#465—1271no—
195#464—1271no—
196#452—1271no—
197#450—1271no—
198C4×SD16—1271no—
199C8×D4—1271no—
200#416—1271no—
201C2×#102—1271no—
202C2×#41C2×D12711yescharacteristic
203C22×D4C2×D12711yescharacteristic
204C2×Q16—1271no—
205C2×SD16—1271no—
206C2×SD16—1271no—
207C22×C8—1271no—
208C22×C8—1271no—
209#126—2541no—
210#125—1271no—
211#125—2541no—
212#124C2×D12711yescharacteristic
213#123—21no—
214#123—2541no—
215C22⋊Q8—1271no—
216C22⋊Q8—1271no—
217C22⋊Q8C2×D12711yescharacteristic
218#121—21no—
219#121C2×D12711yescharacteristic
220#120—21no—
221C4×Q8—1271no—
222C4×D4C2×D12711yescharacteristic
223C4×D4C2×D12711yescharacteristic
224#117—1271no—
225#117—2541no—
226C2×C4⋊C4—1271no—
227C2×C22⋊C4—21no—
228#107—1271no—
229#106—1271no—
230#106—1271no—
231C4⋊C8—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241#102—1271no—
242C22⋊C8—1271no—
243C22⋊C8—1271no—
244C4×C8—1271no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—1271no—
249C2×Q8C22×D12711yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D12711yescharacteristic
256C2×D4C22×D12711yescharacteristic
257C2×D4C22×D12711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2541no—
261C22×C4C22×D12711yescharacteristic
262C22×C4C22×D12711yescharacteristic
263Q16—5081no—
264SD16—5081no—
265SD16—2542no—
266C2×C8—1271no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—5081no—
271C2×C8—5081no—
272C4⋊C4—21no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4C22⋊D12711yescharacteristic
283C4⋊C4C22×D12711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4C22⋊D12711yescharacteristic
293C22⋊C4C22⋊D12711yescharacteristic
294C42—2541no—
295C42—2541no—
296C42C22×D12711yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D12711yescharacteristic
303C23C2×C22⋊D12711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2541no—
307Q8—2541no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1452324712yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—5082no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4C2×C22⋊D12711yescharacteristic
330C2×C4C2×C22⋊D12711yescharacteristic
331C2×C4C2×C22⋊D12711yescharacteristic
332C2×C4C2×C22⋊D12711yescharacteristic
333C2×C4C23×D12711yescharacteristic
334C8—2541no—
335C8—2541no—
336C8—2541no—
337C8—2541no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#1686122912yes—
351C22C22×C22⋊D12711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—5081no—
359C4—5081no—
360C4—5081no—
361C4—5081no—
362C4C2×#1452324711yescharacteristic
363C4#1686125411yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#2255052011yescharacteristic
369C2C2×#1686122911yescharacteristic
370C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#546×C127C2
socleC22×C127C22×D4
hypercenter#546D127
hypocenterC127#3296
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. #22550566C2semidirect
2. #22550230C2semidirect
3. #22550204C2semidirect
4. #22550197C2semidirect
5. #22550176C2semidirect
6. #22550115C2semidirect
7. #22550103C2semidirect
8. #22550095C2semidirect
9. #22550038C2semidirect
10. #22550027C2semidirect
11. C4×#14523236C2semidirect
12. D4×C127⋊C8C2semidirect
13. #22549872C2semidirect
14. C2×#16861130C2semidirect
16. C2×#14523238C22semidirect
20. C22×C127⋊C8C22semidirect
22. #16861201C22semidirect
26. #16861188C22semidirect
27. C4×Dic254C22semidirect
28. #16861168C22semidirect
31. #16861142C22semidirect
32. #16861140C22semidirect
33. #16861140C22semidirect
34. C4⋊(C127⋊C8)C22semidirect
35. #16861133C22semidirect
36. #16861133C22semidirect
37. #16861133C22semidirect
38. #16861132C22semidirect
46. C22⋊(C127⋊C8)C22semidirect
47. C4×C127⋊C8C22semidirect
186. C127#3296semidirect

Maximal subgroups

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

subgroupindex
#225505662
#225502302
#225502042
#225501972
#225501762
#225501152
#225501032
#225500952
#225500382
#225500272
C4×#145232362
D4×C127⋊C82
#225498722
C2×#168611302
#546×C1272
#3296127