Finite Group Databasebeta

← #30648967

Subgroups — #30648967

Order 16256.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550479C211yesmaximal, characteristic, complemented
2#22550351C211yesmaximal, characteristic, complemented
3#22550201C211yesmaximal, characteristic, complemented
4#22550197C211yesmaximal, characteristic, complemented
5#22550178C211yesmaximal, characteristic, complemented
6#22550174C211yesmaximal, characteristic, complemented
7#22550069C211yesmaximal, characteristic, complemented
8#22550069C211yesmaximal, characteristic, complemented
9#22550010C211yesmaximal, characteristic, complemented
10#22549988C211yesmaximal, characteristic, complemented
11#22549985C211yesmaximal, characteristic, complemented
12D8×Dic127C211yesmaximal, characteristic, complemented
13#22549956C211yesmaximal, characteristic, complemented
14#22549934C211yesmaximal, characteristic, complemented
15#447×C127C211yesmaximal, characteristic, complemented
16C2×#14523233C2211yescharacteristic
17C2×#14523233C2211yescharacteristic
18#16861205C2211yescharacteristic, complemented
19#16861200C2211yescharacteristic, complemented
20#16861193—21no—
21#16861191—21no—
22#16861183C2211yescharacteristic
23#16861181—21no—
24#16861180—21no—
25D4×Dic127—21no—
26D4×Dic127—21no—
27D4×Dic127C2211yescharacteristic
28D4×Dic127C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31D4×Dic127C2211yescharacteristic, complemented
32#16861171C2211yescharacteristic, complemented
33#16861168C2211yescharacteristic
34#16861168C2211yescharacteristic
35C2×C22⋊Dic127—21no—
36C2×C22⋊Dic127—21no—
37#16861143C2211yescharacteristic, complemented
38#16861142C2211yescharacteristic, complemented
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic, complemented
48#16861130C2211yescharacteristic, complemented
49C22⋊(C127⋊C8)C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C2×D8×C127C2211yescharacteristic
54C2×D8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59C2×D508C2311yescharacteristic
60C2×C22⋊D127—21no—
61C2×C22⋊D127—21no—
62C2×C22⋊D127—21no—
63C22×Dic127—21no—
64C22×Dic127—21no—
65C22×Dic127—21no—
66C22×Dic127—21no—
67C22×Dic127—21no—
68C2×C4×D127—21no—
69#14523233—22no—
70#14523233—22no—
71C2×C127⋊C8C2311yescharacteristic
72C2×C127⋊C8C2311yescharacteristic
73C4⋊Dic127—21no—
74C4⋊Dic127D411yescharacteristic
75C4⋊Dic127D411yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79Dic127⋊C4—21no—
80C22⋊Dic127—21no—
81C22⋊Dic127—21no—
82C22⋊Dic127—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127D411yescharacteristic
87C22⋊Dic127D411yescharacteristic
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90C4×Dic127—21no—
91C4×Dic127C2311yescharacteristic
92C4×Dic127C2311yescharacteristic
93C24×C127—21no—
94C2×D4×C127—22no—
95C2×D4×C127—22no—
96C2×D4×C127—21no—
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127C2311yescharacteristic
102D8×C127—22no—
103D8×C127—22no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C22×D127—21no—
109D508—21no—
110D508—21no—
111C22⋊D127—81no—
112C22⋊D127—81no—
113C22⋊D127—81no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—81no—
118C2×Dic127—81no—
119C2×Dic127—81no—
120C2×Dic127—81no—
121C2×Dic127—81no—
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C4×D127—81no—
127C127⋊C8—21no—
128C127⋊C8—21no—
129C23×C127—24no—
130C23×C127—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—81no—
134C23×C127—81no—
135C23×C127C2×D411yescharacteristic
136D4×C127—24no—
137D4×C127—21no—
138D4×C127—21no—
139D4×C127—21no—
140D4×C127—21no—
141D4×C127—81no—
142D4×C127#4112yes—
143D4×C127#4112yes—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2411yescharacteristic
149C8×C127—21no—
150C8×C127—21no—
151C22×C127—21no—
152C22×C127—22no—
153C22×C127—22no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—84no—
164C22×C127C22×D411yescharacteristic
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C4×C127—21no—
169C4×C127—41no—
170C4×C127C2×#4111yescharacteristic
171C4×C127C2×#4111yescharacteristic
172Dic127—21no—
173Dic127—21no—
174Dic127—21no—
175Dic127—21no—
176Dic127—41no—
177Dic127—41no—
178Dic127—41no—
179C2×C127—41no—
180C2×C127—42no—
181C2×C127—42no—
182C2×C127—81no—
183C2×C127#52511yescharacteristic
184C2×C127#57611yescharacteristic
185C2×C127#57611yescharacteristic
186D127—81no—
187#3122—1271nomaximal, Sylow
188C127#312211yescharacteristic, complemented, Sylow
189#540—1271no—
190#518—1271no—
191#482—1271no—
192#482—1271no—
193#478—1271no—
194#478—1271no—
195#459—1271no—
196#459—1271no—
197#447—1271no—
198#447D12711yescharacteristic, complemented
199#442—1271no—
200#442—1271no—
201C4×D8—1271no—
202C4×D8—1271no—
203#433—1271no—
204C22×D4C2×D12711yescharacteristic
205C2×D8—1271no—
206C2×D8—1271no—
207C2×D8C2×D12711yescharacteristic
208C2×D8C2×D12711yescharacteristic
209#127—1271no—
210#125—1271no—
211#123—2541no—
212#123—2541no—
213#121—1271no—
214#121—2541no—
215#121—2541no—
216#121C2×D12711yescharacteristic
217C4×D4—1271no—
218C4×D4—1271no—
219C4×D4—1271no—
220C4×D4—1271no—
221C4×D4—1271no—
222C4×D4—1271no—
223C4×D4—2541no—
224C4×D4—2541no—
225#117—1271no—
226#117—1271no—
227C2×C22⋊C4—2541no—
228C2×C22⋊C4—2541no—
229#107—1271no—
230#107—1271no—
231#106—1271no—
232#106—1271no—
233C4⋊C8—1271no—
234C4⋊C8—1271no—
235#102—1271no—
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241#102C2×D12711yescharacteristic
242#102C2×D12711yescharacteristic
243C22⋊C8—1271no—
244C22⋊C8C2×D12711yescharacteristic
245#97—1271no—
246C4×C8—1271no—
247C24—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—1271no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4C22×D12711yescharacteristic
257C2×D4C22×D12711yescharacteristic
258C2×D4C22×D12711yescharacteristic
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4—2541no—
264C22×C4—2541no—
265C22×C4C22×D12711yescharacteristic
266D8—22no—
267D8—22no—
268D8—2542no—
269D8—2542no—
270C2×C8—1271no—
271C2×C8—1271no—
272C2×C8C22×D12711yescharacteristic
273C2×C8C22×D12711yescharacteristic
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—1271no—
277C4⋊C4—1271no—
278C4⋊C4—1271no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4C22×D12711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—1271no—
284C22⋊C4—1271no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C42—1271no—
294C42—1271no—
295C42—2541no—
296C23—24no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—81no—
301C23—81no—
302C23—2541no—
303C23D4×D12711yescharacteristic
304D4—24no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—10161no—
311D4—10161no—
312D4—10161no—
313D4—2541no—
314D4—2541no—
315D4#1452324712yes—
316D4#1452324712yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—1271no—
321C2×C4—1271no—
322C2×C4—1271no—
323C2×C4—1271no—
324C2×C4—10161no—
325C2×C4—10161no—
326C2×C4—10161no—
327C2×C4—10161no—
328C2×C4—10161no—
329C2×C4—10161no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4—2541no—
333C2×C4D4×D12711yescharacteristic
334C2×C4C23×D12711yescharacteristic
335C8—21no—
336C8—21no—
337C8—2541no—
338C8—2541no—
339C22—21no—
340C22—22no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—84no—
352C22—5081no—
353C22—5081no—
354C22—5081no—
355C22C2×D4×D12711yescharacteristic
356C4—21no—
357C4—41no—
358C4—5081no—
359C4—5081no—
360C4—5081no—
361C4—2541no—
362C4—2541no—
363C4—2541no—
364C4—2541no—
365C4C2×#1452324711yescharacteristic
366C4C2×#1452324711yescharacteristic
367C2—41no—
368C2—42no—
369C2—42no—
370C2—81no—
371C2—10161no—
372C2#2255038211yescharacteristic
373C2#2255067711yescharacteristic
374C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#447×C127C2
socleC22×C127C22×D4
hypercenter#447D127
hypocenterC127#3122
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22550479C2semidirect
2. #22550351C2semidirect
3. #22550201C2semidirect
4. #22550197C2semidirect
5. #22550178C2semidirect
6. #22550174C2semidirect
7. #22550069C2semidirect
8. #22550069C2semidirect
9. #22550010C2semidirect
10. #22549988C2semidirect
11. #22549985C2semidirect
12. D8×Dic127C2semidirect
13. #22549956C2semidirect
14. #22549934C2semidirect
15. #447×C127C2semidirect
18. #16861205C22semidirect
19. #16861200C22semidirect
31. D4×Dic127C22semidirect
32. #16861171C22semidirect
37. #16861143C22semidirect
38. #16861142C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861131C22semidirect
44. #16861131C22semidirect
47. #16861130C22semidirect
48. #16861130C22semidirect
50. #16861119C22semidirect
51. C8×Dic127C22semidirect
188. C127#3122semidirect
198. #447D127semidirect

Maximal subgroups

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

subgroupindex
#225504792
#225503512
#225502012
#225501972
#225501782
#225501742
#225500692
#225500102
#225499882
#225499852
D8×Dic1272
#225499562
#225499342
#447×C1272
#3122127