Finite Group Databasebeta

← #30648914

Subgroups — #30648914

Order 16256.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550497C211yesmaximal, characteristic, complemented
2#22550217C211yesmaximal, characteristic, complemented
3#22550200C211yesmaximal, characteristic, complemented
4#22550166C211yesmaximal, characteristic, complemented
5#22550156C211yesmaximal, characteristic, complemented
6#22550091C211yesmaximal, characteristic, complemented
7#22550091C211yesmaximal, characteristic, complemented
8#22550027C211yesmaximal, characteristic, complemented
9#22550010C211yesmaximal, characteristic
10#22549986C211yesmaximal, characteristic, complemented
11#22549976C211yesmaximal, characteristic, complemented
12C4×#14523236C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549937C211yesmaximal, characteristic, complemented
15#518×C127C211yesmaximal, characteristic, complemented
16C2×#14523236C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20#16861209C2211yescharacteristic, complemented
21#16861204—21no—
22#16861197C2211yescharacteristic, complemented
23#16861195—21no—
24#16861191—21no—
25#16861188C2211yescharacteristic
26#16861187—21no—
27#16861183C2211yescharacteristic
28C4×Dic254C2211yescharacteristic, complemented
29C4×D508C2211yescharacteristic, complemented
30#16861142C2211yescharacteristic, complemented
31#16861142C2211yescharacteristic, complemented
32#16861140C2211yescharacteristic, complemented
33#16861140C2211yescharacteristic, complemented
34C4⋊(C127⋊C8)C2211yescharacteristic, complemented
35C4⋊(C127⋊C8)C2211yescharacteristic, complemented
36#16861132C2211yescharacteristic, complemented
37#16861132C2211yescharacteristic, complemented
38#16861130C2211yescharacteristic
39#16861130C2211yescharacteristic
40#16861130C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861129C2211yescharacteristic, complemented
43#16861129C2211yescharacteristic, complemented
44C22⋊(C127⋊C8)C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46#16861120C2211yescharacteristic, complemented
47C4×C127⋊C8C2211yescharacteristic, complemented
48C22×D4×C127C2211yescharacteristic
49C4×D4×C127—21no—
50C4×D4×C127—21no—
51C4×D4×C127C2211yescharacteristic
52C4×D4×C127C2211yescharacteristic
53C4×D4×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55#117×C127C2211yescharacteristic
56#117×C127C2211yescharacteristic
57C2×C22⋊C4×C127—21no—
58C2×C22⋊C4×C127—21no—
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C4×D127—21no—
64#14523236—22no—
65#14523236—22no—
66#14523233—22no—
67#14523233—22no—
68C2×C127⋊C8C2311yescharacteristic
69C2×C127⋊C8C2311yescharacteristic
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C4⋊Dic127—21no—
73C4⋊Dic127C2311yescharacteristic
74C4⋊Dic127C2311yescharacteristic
75Dic127⋊C4—21no—
76Dic127⋊C4—21no—
77C22⋊Dic127—21no—
78(C2×D127)⋊C4—21no—
79(C2×D127)⋊C4—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82C4×Dic127—21no—
83C24×C127—21no—
84C2×D4×C127—22no—
85C2×D4×C127—22no—
86C2×D4×C127—21no—
87C2×D4×C127C2311yescharacteristic
88C2×D4×C127C2311yescharacteristic
89C22×C4×C127—21no—
90C22×C4×C127—21no—
91C22×C4×C127—21no—
92C22×C4×C127—21no—
93C22×C4×C127C2311yescharacteristic
94C4⋊C4×C127D411yescharacteristic
95C4⋊C4×C127D411yescharacteristic
96C4⋊C4×C127C2311yescharacteristic
97C4⋊C4×C127C2311yescharacteristic
98C22⋊C4×C127—21no—
99C22⋊C4×C127—21no—
100C22⋊C4×C127—21no—
101C22⋊C4×C127—21no—
102C22⋊C4×C127—21no—
103C22⋊C4×C127D411yescharacteristic
104C22⋊C4×C127D411yescharacteristic
105C42×C127—21no—
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109Dic254—21no—
110Dic254—21no—
111Dic254—81no—
112D508—21no—
113D508—21no—
114C22⋊D127—81no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C4×D127—81no—
119C127⋊C8—21no—
120C127⋊C8—21no—
121C127⋊C8—21no—
122C127⋊C8—21no—
123C23×C127—24no—
124C23×C127—21no—
125C23×C127—21no—
126C23×C127—81no—
127C23×C127—81no—
128C23×C127C2×D411yescharacteristic
129D4×C127—24no—
130D4×C127—21no—
131D4×C127—21no—
132D4×C127#4112yes—
133D4×C127#4112yes—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—81no—
139C2×C4×C127—81no—
140C2×C4×C127—81no—
141C2×C4×C127—81no—
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C22×C127—21no—
149C22×C127—22no—
150C22×C127—22no—
151C22×C127—21no—
152C22×C127—21no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—84no—
158C22×C127C22×D411yescharacteristic
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C4×C127—21no—
163C4×C127—21no—
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127C2×#4111yescharacteristic
170C4×C127C2×#4111yescharacteristic
171Dic127—41no—
172Dic127—41no—
173Dic127—41no—
174C2×C127—41no—
175C2×C127—42no—
176C2×C127—42no—
177C2×C127#52511yescharacteristic
178C2×C127#57611yescharacteristic
179C2×C127#57711yescharacteristic
180D127—81no—
181#3118—1271nomaximal, Sylow
182C127#311811yescharacteristic, complemented, Sylow
183#542—1271no—
184#518D12711yescharacteristic, complemented
185#485—1271no—
186#482—1271no—
187#476—1271no—
188#474—1271no—
189#463—1271no—
190#463—1271no—
191#450—1271no—
192#447—1271no—
193#442—1271no—
194#440—1271no—
195C4×SD16—1271no—
196C4×D8—1271no—
197#433—1271no—
198C22×D4C2×D12711yescharacteristic
199C2×SD16—1271no—
200C2×SD16—1271no—
201C2×D8—1271no—
202C2×D8—1271no—
203C4⋊Q8—1271no—
204#126—2541no—
205#124—1271no—
206#124—2541no—
207#123—2541no—
208C22⋊Q8—1271no—
209C22⋊Q8—2541no—
210#121—1271no—
211C4×Q8—1271no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1271no—
215C4×D4C2×D12711yescharacteristic
216C4×D4C2×D12711yescharacteristic
217C4×D4C2×D12711yescharacteristic
218C4×D4C2×D12711yescharacteristic
219#117C2×D12711yescharacteristic
220#117C2×D12711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—1271no—
224#107—1271no—
225#106—1271no—
226#106—1271no—
227C4⋊C8—1271no—
228C4⋊C8—1271no—
229#103—1271no—
230#103—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237C22⋊C8—1271no—
238C22⋊C8—1271no—
239#97—1271no—
240C4×C8—1271no—
241C24—21no—
242C2×Q8—1271no—
243C2×Q8—2541no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—1271no—
248C2×D4—2541no—
249C2×D4C22×D12711yescharacteristic
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—2541no—
256C22×C4C22×D12711yescharacteristic
257SD16—2542no—
258SD16—2542no—
259D8—2542no—
260D8—2542no—
261C2×C8—1271no—
262C2×C8—1271no—
263C2×C8—1271no—
264C2×C8—1271no—
265C4⋊C4—1271no—
266C4⋊C4—1271no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4C22⋊D12711yescharacteristic
271C4⋊C4C22⋊D12711yescharacteristic
272C4⋊C4C22×D12711yescharacteristic
273C4⋊C4C22×D12711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4C22⋊D12711yescharacteristic
285C22⋊C4C22⋊D12711yescharacteristic
286C42—21no—
287C42—2541no—
288C42C22×D12711yescharacteristic
289C42C22×D12711yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—2541no—
296C23C2×C22⋊D12711yescharacteristic
297Q8—10161no—
298Q8—2541no—
299Q8—2541no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—10161no—
304D4—2541no—
305D4—2541no—
306D4#1452324812yes—
307D4#1452324812yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—10161no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4C2×C22⋊D12711yescharacteristic
321C2×C4C2×C22⋊D12711yescharacteristic
322C2×C4C2×C22⋊D12711yescharacteristic
323C2×C4C2×C22⋊D12711yescharacteristic
324C2×C4C2×C22⋊D12711yescharacteristic
325C2×C4C23×D12711yescharacteristic
326C8—2541no—
327C8—2541no—
328C8—2541no—
329C8—2541no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—5081no—
341C22—5081no—
342C22—5081no—
343C22C22×C22⋊D12711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—5081no—
352C4—5081no—
353C4—5081no—
354C4C2×#1452324811yescharacteristic
355C4C2×#1452324811yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—10161no—
360C2#2255038711yescharacteristic
361C2#2255067911yescharacteristic
362C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#518×C127C2
socleC22×C127C22×D4
prime radical#22550010C2
hypercenter#518D127
hypocenterC127#3118
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #22550497C2semidirect
2. #22550217C2semidirect
3. #22550200C2semidirect
4. #22550166C2semidirect
5. #22550156C2semidirect
6. #22550091C2semidirect
7. #22550091C2semidirect
8. #22550027C2semidirect
10. #22549986C2semidirect
11. #22549976C2semidirect
12. C4×#14523236C2semidirect
13. C4×#14523233C2semidirect
14. #22549937C2semidirect
15. #518×C127C2semidirect
20. #16861209C22semidirect
22. #16861197C22semidirect
28. C4×Dic254C22semidirect
29. C4×D508C22semidirect
30. #16861142C22semidirect
31. #16861142C22semidirect
32. #16861140C22semidirect
33. #16861140C22semidirect
34. C4⋊(C127⋊C8)C22semidirect
35. C4⋊(C127⋊C8)C22semidirect
36. #16861132C22semidirect
37. #16861132C22semidirect
42. #16861129C22semidirect
43. #16861129C22semidirect
46. #16861120C22semidirect
47. C4×C127⋊C8C22semidirect
182. C127#3118semidirect
184. #518D127semidirect

Maximal subgroups

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

subgroupindex
#225504972
#225502172
#225502002
#225501662
#225501562
#225500912
#225500272
#225500102
#225499862
#225499762
C4×#145232362
C4×#145232332
#225499372
#518×C1272
#3118127