Finite Group Databasebeta

← #30648873

Subgroups — #30648873

Order 16256.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550411C211yesmaximal, characteristic, complemented
2C4×D4×D127C211yesmaximal, characteristic, complemented
3#22550208C211yesmaximal, characteristic, complemented
4#22550191C211yesmaximal, characteristic, complemented
5#22550173C211yesmaximal, characteristic, complemented
6#22550092C211yesmaximal, characteristic, complemented
7#22550091C211yesmaximal, characteristic, complemented
8#22550028C211yesmaximal, characteristic, complemented
9#22550011C211yesmaximal, characteristic
10#22549987C211yesmaximal, characteristic, complemented
11#22549985C211yesmaximal, characteristic, complemented
12#22549975C211yesmaximal, characteristic, complemented
13#22549973C211yesmaximal, characteristic, complemented
14#22549932C211yesmaximal, characteristic, complemented
15#478×C127C211yesmaximal, characteristic, complemented
16C2×D4×D127C2211yescharacteristic
17C22×C4×D127—21no—
18C2×#14523237C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×D1016C2211yescharacteristic
21C2×#14523233C2211yescharacteristic
22#16861204—21no—
23#16861197C2211yescharacteristic, complemented
24#16861194—21no—
25#16861189C2211yescharacteristic
26#16861184C2211yescharacteristic
27#16861174C2211yescharacteristic, complemented
28C4×D508C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30#16861171—21no—
31#16861171C2211yescharacteristic
32#16861171C2211yescharacteristic, complemented
33C4×C22⋊D127—21no—
34(C22⋊D127)⋊C4—21no—
35#16861167—21no—
36C4⋊C4×D127C2211yescharacteristic
37C22⋊C4×D127—21no—
38C42×D127C2211yescharacteristic
39#16861142C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41C4⋊(C127⋊C8)C2211yescharacteristic, complemented
42#16861134C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic, complemented
45#16861129C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic, complemented
48(C2×D127)⋊C8C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51#16861119C2211yescharacteristic, complemented
52#125×C127C2211yescharacteristic, complemented
53C4×D4×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56C4⋊C8×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C23×D127—21no—
60C2×Dic254C2311yescharacteristic
61C2×D508—21no—
62C2×D508C2311yescharacteristic
63C2×D508C2311yescharacteristic
64C2×C22⋊D127—21no—
65D4×D127—22no—
66D4×D127—22no—
67C22×Dic127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—44no—
73C2×C4×D127D411yescharacteristic
74C2×C4×D127D411yescharacteristic
75C2×C4×D127C2311yescharacteristic
76#14523237—22no—
77#14523236—22no—
78D1016—22no—
79#14523233—22no—
80C2×C127⋊C8C2311yescharacteristic
81C2×C127⋊C8C2311yescharacteristic
82C4⋊Dic127—21no—
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86C22⋊Dic127—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94C4×Dic127—21no—
95C4×Dic127D411yescharacteristic
96C4×Dic127D411yescharacteristic
97C4×Dic127C2311yescharacteristic
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127—21no—
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127—21no—
103C4⋊C4×C127—21no—
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C42×C127C2311yescharacteristic
109C22×D127—24no—
110C22×D127—21no—
111C22×D127—21no—
112C22×D127—42no—
113C22×D127—42no—
114C22×D127C2×D411yescharacteristic
115Dic254—21no—
116Dic254—21no—
117D508—21no—
118D508—21no—
119D508—21no—
120D508—42no—
121D508#4112yes—
122C22⋊D127—24no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—42no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—22no—
130C4×D127—21no—
131C4×D127—21no—
132C4×D127—41no—
133C4×D127—42no—
134C4×D127—42no—
135C4×D127—42no—
136C4×D127—42no—
137C127⋊C8—21no—
138C127⋊C8—21no—
139C23×C127—21no—
140D4×C127—21no—
141D4×C127#4112yes—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—42no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2411yescharacteristic
149C8×C127—21no—
150C8×C127—21no—
151C22×C127—22no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127C22×D411yescharacteristic
155C2×D127—21no—
156C2×D127—21no—
157C2×D127—22no—
158C2×D127—48no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127AGL(1,ℤ/8)12yes—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127C2×#4111yescharacteristic
171C4×C127C2×#4111yescharacteristic
172Dic127—21no—
173Dic127—21no—
174Dic127—21no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—42no—
178C2×C127#52511yescharacteristic
179C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C127#57511yescharacteristic
181D127—22no—
182D127—42no—
183D127—81no—
184#3114—1271nomaximal, Sylow
185C127#311411yescharacteristic, complemented, Sylow
186#529—1271no—
187C2×C4×D4—1271no—
188#483—1271no—
189#480—1271no—
190#478—1271no—
191#478D12711yescharacteristic, complemented
192#463—1271no—
193#463—1271no—
194#450—1271no—
195#447—1271no—
196#442—1271no—
197#442—1271no—
198#440—1271no—
199#440—1271no—
200#432—1271no—
201C22×D4—1271no—
202C23×C4—2541no—
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C2×D8—1271no—
207#126—2541no—
208#125C2×D12711yescharacteristic, complemented
209#124—1271no—
210#123—2541no—
211C22⋊Q8—1271no—
212#121—1271no—
213C4×Q8—1271no—
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4—1271no—
217C4×D4—1271no—
218C4×D4—2541no—
219C4×D4—2541no—
220C4×D4—2541no—
221C4×D4C2×D12711yescharacteristic
222#117—2541no—
223C2×C4⋊C4—1271no—
224C2×C22⋊C4—2541no—
225C2×C42—1271no—
226#107—1271no—
227#107C2×D12711yescharacteristic, complemented
228#106—1271no—
229#106C2×D12711yescharacteristic, complemented
230C4⋊C8—1271no—
231C4⋊C8C2×D12711yescharacteristic, complemented
232#103—1271no—
233#103—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237#102—1271no—
238#102C2×D12711yescharacteristic
239#102C2×D12711yescharacteristic
240C22⋊C8—1271no—
241C22⋊C8—1271no—
242#97—1271no—
243#97—1271no—
244C24—2541no—
245C2×Q8—1271no—
246C2×D4—1271no—
247C2×D4—1271no—
248C2×D4—2541no—
249C2×D4—2541no—
250C2×D4—2542no—
251C2×D4—2542no—
252C2×D4C22×D12711yescharacteristic
253C22×C4—21no—
254C22×C4—1271no—
255C22×C4—1271no—
256C22×C4—1271no—
257C22×C4—5084no—
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4—2541no—
263SD16—2542no—
264SD16—2542no—
265D8—2542no—
266D8—2542no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8C22×D12711yescharacteristic
270C2×C8C22×D12711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—1271no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4C22×D12711yescharacteristic
278C4⋊C4C22×D12711yescharacteristic
279C4⋊C4C22×D12711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C42—1271no—
290C42—1271no—
291C42—1271no—
292C42—2541no—
293C42C22×D12711yescharacteristic
294C23—21no—
295C23—1271no—
296C23—5082no—
297C23—5082no—
298C23—2544no—
299C23—2541no—
300C23—2541no—
301Q8—2541no—
302Q8—2541no—
303D4—21no—
304D4—1272no—
305D4—5082no—
306D4—2544no—
307D4—2541no—
308D4—2541no—
309D4—2541no—
310D4#1452324912yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—1271no—
316C2×C4—1271no—
317C2×C4—1271no—
318C2×C4—5082no—
319C2×C4—5081no—
320C2×C4—5082no—
321C2×C4—5082no—
322C2×C4—5082no—
323C2×C4—5082no—
324C2×C4—2542no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4D4×D12711yescharacteristic
330C2×C4D4×D12711yescharacteristic
331C2×C4C23×D12711yescharacteristic
332C8—21no—
333C8—21no—
334C8—2541no—
335C8—2541no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—1272no—
340C22—5088no—
341C22—5081no—
342C22—5081no—
343C22—5081no—
344C22—5081no—
345C22—5081no—
346C22—2542no—
347C22—2541no—
348C22—2541no—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—5081no—
356C4—5081no—
357C4—2541no—
358C4—2541no—
359C4—2541no—
360C4#41×D12711yescharacteristic
361C4C2×#1452324911yescharacteristic
362C2—42no—
363C2—10161no—
364C2—5082no—
365C2—2542no—
366C2#2255038911yescharacteristic
367C2AGL(1,ℤ/8)×D12711yescharacteristic
368C2#2255067311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#478×C127C2
socleC22×C127C22×D4
prime radical#22550011C2
hypercenter#478D127
hypocenterC127#3114
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #22550411C2semidirect
2. C4×D4×D127C2semidirect
3. #22550208C2semidirect
4. #22550191C2semidirect
5. #22550173C2semidirect
6. #22550092C2semidirect
7. #22550091C2semidirect
8. #22550028C2semidirect
10. #22549987C2semidirect
11. #22549985C2semidirect
12. #22549975C2semidirect
13. #22549973C2semidirect
14. #22549932C2semidirect
15. #478×C127C2semidirect
23. #16861197C22semidirect
27. #16861174C22semidirect
32. #16861171C22semidirect
39. #16861142C22semidirect
40. #16861140C22semidirect
41. C4⋊(C127⋊C8)C22semidirect
42. #16861134C22semidirect
43. #16861132C22semidirect
44. #16861131C22semidirect
47. #16861129C22semidirect
50. #16861119C22semidirect
51. #16861119C22semidirect
52. #125×C127C22semidirect
54. #107×C127C22semidirect
55. #106×C127C22semidirect
56. C4⋊C8×C127C22semidirect
185. C127#3114semidirect
191. #478D127semidirect
208. #125C2×D127semidirect
227. #107C2×D127semidirect
229. #106C2×D127semidirect
231. C4⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504112
C4×D4×D1272
#225502082
#225501912
#225501732
#225500922
#225500912
#225500282
#225500112
#225499872
#225499852
#225499752
#225499732
#225499322
#478×C1272
#3114127