Finite Group Databasebeta

← #30648681

Subgroups — #30648681

Order 16256.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550442C211yesmaximal, characteristic
2#22550357C211yesmaximal, characteristic, complemented
3#22550203C211yesmaximal, characteristic, complemented
4#22550201C211yesmaximal, characteristic, complemented
5#22550198C211yesmaximal, characteristic, complemented
6#22550197C211yesmaximal, characteristic, complemented
7#22550194C211yesmaximal, characteristic, complemented
8#22550193C211yesmaximal, characteristic, complemented
9#22550191C211yesmaximal, characteristic, complemented
10#22549917C211yesmaximal, characteristic, complemented
11#22549913C211yesmaximal, characteristic, complemented
12#22549879C211yesmaximal, characteristic, complemented
13#102×D127C211yesmaximal, characteristic, complemented
14#22549841C211yesmaximal, characteristic, complemented
15#480×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20#16861205—21no—
21#16861195—21no—
22#16861184C2211yescharacteristic
23#16861183C2211yescharacteristic
24#16861180C2211yescharacteristic
25#16861179—21no—
26#16861177—21no—
27#16861174—21no—
28#16861171—21no—
29C4×C22⋊D127—21no—
30C4×C22⋊D127—21no—
31#16861168C2211yescharacteristic, complemented
32#16861167C2211yescharacteristic
33#16861167C2211yescharacteristic
34#16861167C2211yescharacteristic, complemented
35C4⋊C4×D127C2211yescharacteristic
36C4⋊C4×D127C2211yescharacteristic, complemented
37#16861143C2211yescharacteristic, complemented
38#16861142C2211yescharacteristic, complemented
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42#16861140C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52#121×C127C2211yescharacteristic
53C2×C4⋊C4×C127C2211yescharacteristic, complemented
54#107×C127C2211yescharacteristic, complemented
55#107×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523248—22no—
61#14523248—22no—
62C2×Dic254D411yescharacteristic
63C2×D508—21no—
64C2×D508D411yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127D411yescharacteristic
70C2×C22⋊D127D411yescharacteristic
71D4×D127—22no—
72D4×D127—22no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127C2311yescharacteristic
76C2×C4×D127C2311yescharacteristic
77#14523231—22no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—22no—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—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—
93C4×Dic127—21no—
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127—21no—
100C22×C4×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C4⋊C4×C127—21no—
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C22×D127—21no—
109C22×D127—42no—
110C22×D127—81no—
111C22×D127—81no—
112C22×D127C2×D411yescharacteristic
113C22×D127C2×D411yescharacteristic
114Dic254—22no—
115D508—22no—
116D508—21no—
117D508—21no—
118D508—81no—
119C22⋊D127—22no—
120C22⋊D127—22no—
121C22⋊D127—42no—
122C22⋊D127—81no—
123C22⋊D127—81no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—21no—
130C4×D127—21no—
131C4×D127—81no—
132C4×D127—81no—
133C4×D127#4112yes—
134C4×D127#4112yes—
135C127⋊C8—21no—
136C127⋊C8—21no—
137C23×C127—21no—
138C23×C127C2×D411yescharacteristic
139D4×C127—21no—
140D4×C127—21no—
141D4×C127—81no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—81no—
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127C22×D411yescharacteristic
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—21no—
162C2×D127—21no—
163C2×D127—21no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—84no—
168C2×D127#12312yes—
169C4×C127—21no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127—41no—
173C4×C127C2×#4111yescharacteristic
174C4×C127C2×#4111yescharacteristic
175Dic127—21no—
176Dic127—41no—
177Dic127—41no—
178Dic127—41no—
179Dic127#12312yes—
180C2×C127—41no—
181C2×C127—81no—
182C2×C127C2×#12311yescharacteristic
183C2×C127#57611yescharacteristic
184C2×C127#57711yescharacteristic
185D127—22no—
186D127—41no—
187D127—81no—
188#3096—1271nomaximal, Sylow
189C127#309611yescharacteristic, complemented, Sylow
190#535—1271no—
191#520—1271no—
192#482—1271no—
193#482—1271no—
194#482—1271no—
195#482—1271no—
196#481—1271no—
197#481—1271no—
198#480—1271no—
199#480D12711yescharacteristic, complemented
200#428—1271no—
201#427—1271no—
202#418—1271no—
203C2×#102—1271no—
204#408—1271no—
205C2×#41—1271no—
206C22×D4—1271no—
207C2×M16—1271no—
208C22×C8—1271no—
209#127—2541no—
210#124—2541no—
211#121—1271no—
212#121—1271no—
213#121—1271no—
214#121C2×D12711yescharacteristic
215#120—2541no—
216#120—2541no—
217C4×Q8—2541no—
218C4×D4—2541no—
219C4×D4—2541no—
220C4×D4—2541no—
221#117—1271no—
222#117—1271no—
223#117—1271no—
224#117—1271no—
225C2×C4⋊C4—1271no—
226C2×C4⋊C4—1271no—
227C2×C4⋊C4C2×D12711yescharacteristic, complemented
228#107—1271no—
229#107—1271no—
230#107C2×D12711yescharacteristic, complemented
231#107C2×D12711yescharacteristic, complemented
232#106—1271no—
233#106—1271no—
234#106—1271no—
235#106—1271no—
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241#102—1271no—
242#102C2×D12711yescharacteristic
243#102C2×D12711yescharacteristic
244C22⋊C8—1271no—
245C22⋊C8—1271no—
246C22⋊C8—1271no—
247C22⋊C8C2×D12711yescharacteristic, complemented
248C24—2541no—
249#41—2542no—
250#41—2542no—
251C2×Q8—1271no—
252C2×D4—21no—
253C2×D4—1271no—
254C2×D4—1271no—
255C2×D4—1271no—
256C2×D4—1271no—
257C2×D4—2541no—
258C2×D4—2542no—
259C2×D4—2542no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4C22×D12711yescharacteristic
264C22×C4—21no—
265C22×C4—1271no—
266C22×C4—1271no—
267C22×C4—2541no—
268C22×C4—2541no—
269C22×C4C22×D12711yescharacteristic
270M16—2542no—
271C2×C8—1271no—
272C2×C8—1271no—
273C2×C8—2542no—
274C2×C8C22×D12711yescharacteristic
275C2×C8C22×D12711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—1271no—
278C4⋊C4—1271no—
279C4⋊C4—1271no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4C22×D12711yescharacteristic
283C4⋊C4C22×D12711yescharacteristic
284C4⋊C4C22×D12711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C42—2541no—
294C42—2541no—
295C42—2541no—
296C42—2541no—
297C23—21no—
298C23—1271no—
299C23—1271no—
300C23—10161no—
301C23—10161no—
302C23—5082no—
303C23—2541no—
304C23D4×D12711yescharacteristic
305Q8—2542no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—10161no—
310D4—10161no—
311D4—10161no—
312D4—5082no—
313D4—2542no—
314D4—2542no—
315D4—2542no—
316D4—2541no—
317D4—2541no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—1271no—
325C2×C4—1272no—
326C2×C4—1272no—
327C2×C4—1271no—
328C2×C4—10161no—
329C2×C4—10161no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4—2541no—
333C2×C4—2541no—
334C2×C4—2541no—
335C2×C4D4×D12711yescharacteristic
336C2×C4C23×D12711yescharacteristic
337C8—21no—
338C8—21no—
339C8—2541no—
340C8—2541no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—1272no—
348C22—10164no—
349C22—5081no—
350C22—5081no—
351C22—5081no—
352C22—2541no—
353C22—2541no—
354C22—2541no—
355C22—2541no—
356C22—2541no—
357C22C2×D4×D12711yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—1272no—
363C4—5081no—
364C4—5081no—
365C4—5081no—
366C4—2541no—
367C4#41×D12711yescharacteristic
368C4#41×D12711yescharacteristic
369C2—41no—
370C2—81no—
371C2—10161no—
372C2—5081no—
373C2—2542no—
374C2#123×D12711yescharacteristic
375C2#2255067711yescharacteristic
376C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#480×C127C2
socleC22×C127C22×D4
prime radical#22550442C2
hypercenter#480D127
hypocenterC127#3096
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #22550357C2semidirect
3. #22550203C2semidirect
4. #22550201C2semidirect
5. #22550198C2semidirect
6. #22550197C2semidirect
7. #22550194C2semidirect
8. #22550193C2semidirect
9. #22550191C2semidirect
10. #22549917C2semidirect
11. #22549913C2semidirect
12. #22549879C2semidirect
13. #102×D127C2semidirect
14. #22549841C2semidirect
15. #480×C127C2semidirect
31. #16861168C22semidirect
34. #16861167C22semidirect
36. C4⋊C4×D127C22semidirect
37. #16861143C22semidirect
38. #16861142C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. #16861140C22semidirect
42. #16861140C22semidirect
49. C22⋊(C127⋊C8)C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. (C2×D127)⋊C8C22semidirect
53. C2×C4⋊C4×C127C22semidirect
54. #107×C127C22semidirect
55. #107×C127C22semidirect
58. C22⋊C8×C127C22semidirect
189. C127#3096semidirect
199. #480D127semidirect
227. C2×C4⋊C4C2×D127semidirect
230. #107C2×D127semidirect
231. #107C2×D127semidirect
247. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504422
#225503572
#225502032
#225502012
#225501982
#225501972
#225501942
#225501932
#225501912
#225499172
#225499132
#225498792
#102×D1272
#225498412
#480×C1272
#3096127