Finite Group Databasebeta

← #30647586

Subgroups — #30647586

Order 16256.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D127C211yesmaximal, characteristic
2#22550352C211yesmaximal, characteristic, complemented
3#22550348C211yesmaximal, characteristic
4#22549978C211yesmaximal, characteristic, complemented
5#22549977C211yesmaximal, characteristic, complemented
6#22549975C211yesmaximal, characteristic, complemented
7#22549967C211yesmaximal, characteristic, complemented
8C4×#14523235C211yesmaximal, characteristic, complemented
9C4×#14523236C211yesmaximal, characteristic, complemented
10#22549961C211yesmaximal, characteristic, complemented
11#22549913C211yesmaximal, characteristic, complemented
12#22549883C211yesmaximal, characteristic, complemented
13#22549879C211yesmaximal, characteristic
14#22549833C211yesmaximal, characteristic, complemented
15#440×C127C211yesmaximal, characteristic, complemented
16C2×Q8×D127C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21SD16×D127C418yescomplemented
22C2×#14523231C2211yescharacteristic
23C2×C8×D127C2211yescharacteristic
24Q8×Dic127C2211yescharacteristic, complemented
25C4×Dic254—21no—
26C4×Dic254C2211yescharacteristic, complemented
27#16861174—21no—
28#16861174C2211yescharacteristic, complemented
29C4×D508C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31#16861171C2211yescharacteristic
32C4×C22⋊D127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167—21no—
35#16861167C2211yescharacteristic
36#16861167C2211yescharacteristic
37#16861166—21no—
38#16861166C2211yescharacteristic
39C2×(C2×D127)⋊C4—21no—
40C22⋊C4×D127—21no—
41#16861143C2211yescharacteristic, complemented
42#16861140C2211yescharacteristic, complemented
43#16861140C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861133C2211yescharacteristic, complemented
46#16861132C2211yescharacteristic, complemented
47#16861131C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861129C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51C4×C127⋊C8C2211yescharacteristic, complemented
52C8×Dic127C2211yescharacteristic, complemented
53C2×SD16×C127C2211yescharacteristic
54C4×Q8×C127C2211yescharacteristic, complemented
55C4×D4×C127C2211yescharacteristic
56#107×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic
59#97×C127C2211yescharacteristic, complemented
60C23×D127—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63Q8×D127—22no—
64Q8×D127C2×C414yescomplemented
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67D4×D127—22no—
68D4×D127C2×C414yes—
69C22×Dic127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74#14523237C2×C414yescomplemented
75#14523236C2×C414yescomplemented
76#14523235C2×C414yescomplemented
77#14523231—22no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127C2×C414yescomplemented
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4D411yescharacteristic
87Dic127⋊C4D411yescharacteristic
88C22⋊Dic127—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4D411yescharacteristic
94(C2×D127)⋊C4D411yescharacteristic
95C4×Dic127—21no—
96C4×Dic127—21no—
97C4×Dic127—21no—
98C4×Dic127—21no—
99C4×Dic127—21no—
100C4×Dic127C2311yescharacteristic
101C2×Q8×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104SD16×C127C2×C414yescomplemented
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127—21no—
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C42×C127—21no—
112C42×C127C2311yescharacteristic
113C22×D127—21no—
114C22×D127—24no—
115C22×D127—81no—
116C22×D127—81no—
117C22×D127C2×D411yescharacteristic
118Dic254—21no—
119Dic254—24no—
120Dic254C22×C412yes—
121D508—21no—
122D508C22×C412yes—
123C22⋊D127—24no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—81no—
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C4×D127—24no—
132C4×D127—81no—
133C4×D127—81no—
134C4×D127C22×C412yes—
135C4×D127#4112yes—
136C127⋊C8—21no—
137C127⋊C8C22×C412yes—
138C23×C127—21no—
139Q8×C127—21no—
140Q8×C127C22×C412yes—
141D4×C127—21no—
142D4×C127C22×C412yes—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—81no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127C22×C412yes—
152C22×C127—22no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—22no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—84no—
162C2×D127C4×D412yes—
163C4×C127—22no—
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127C23×C411yescharacteristic
169C4×C127C2×#4111yescharacteristic
170Dic127—22no—
171Dic127—21no—
172Dic127—21no—
173Dic127—41no—
174Dic127—41no—
175Dic127C4×D412yes—
176C2×C127—42no—
177C2×C127C2×C4×D411yescharacteristic
178C2×C127#57711yescharacteristic
179C2×C127#57711yescharacteristic
180D127—22no—
181D127—42no—
182#2949—1271nomaximal, Sylow
183C127#294911yescharacteristic, complemented, Sylow
184C22×SD16—1271no—
185#519—1271no—
186#518—1271no—
187#440—1271no—
188#440—1271no—
189#440—1271no—
190#440D12711yescharacteristic, complemented
191C4×SD16—1271no—
192C4×SD16—1271no—
193C4×SD16—1271no—
194C4×SD16—1271no—
195#427—1271no—
196#419—1271no—
197#418—1271no—
198#405—1271no—
199C22×Q8—1271no—
200C22×D4—1271no—
201C2×SD16—1271no—
202C2×SD16—1271no—
203C2×SD16—1278no—
204C2×SD16—1271no—
205C2×SD16C2×D12711yescharacteristic
206C2×M16—1271no—
207C22×C8—1271no—
208C4×Q8—1271no—
209C4×Q8—1271no—
210C4×Q8—1271no—
211C4×Q8—2541no—
212C4×Q8—2541no—
213C4×Q8C2×D12711yescharacteristic, complemented
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4—1271no—
217C4×D4—2541no—
218C4×D4—2541no—
219C4×D4C2×D12711yescharacteristic
220#117—1271no—
221#117—1271no—
222#117—1271no—
223#117—2541no—
224#117—2541no—
225C2×C22⋊C4—2541no—
226C2×C22⋊C4—2541no—
227#107—1271no—
228#107C2×D12711yescharacteristic, complemented
229#106—1271no—
230#106—1271no—
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103C2×D12711yescharacteristic, complemented
235#102—1271no—
236#102—1271no—
237#102—1271no—
238#102C2×D12711yescharacteristic
239#97—1271no—
240#97C2×D12711yescharacteristic, complemented
241C4×C8—1271no—
242C4×C8—1271no—
243C24—2541no—
244C2×Q8—1274no—
245C2×Q8—1271no—
246C2×Q8—2541no—
247C2×Q8—2542no—
248C2×Q8C22×D12711yescharacteristic
249C2×D4—1274no—
250C2×D4—1271no—
251C2×D4—2542no—
252C2×D4—2541no—
253C2×D4C22×D12711yescharacteristic
254C22×C4—21no—
255C22×C4—1271no—
256C22×C4—2541no—
257C22×C4—2541no—
258C22×C4—2541no—
259C22×C4—2541no—
260SD16—1274no—
261SD16—1274no—
262SD16—1274no—
263SD16C4×D12714yescomplemented
264M16—2542no—
265C2×C8—1271no—
266C2×C8—1274no—
267C2×C8—1271no—
268C2×C8C22×D12711yescharacteristic
269C2×C8C22×D12711yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—1271no—
272C4⋊C4—1271no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4C22×D12711yescharacteristic
279C4⋊C4C22×D12711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—1271no—
282C22⋊C4—1271no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C42—21no—
289C42—1271no—
290C42—2541no—
291C42—2541no—
292C42—2541no—
293C42—2541no—
294C42—2541no—
295C42C22×D12711yescharacteristic
296C23—21no—
297C23—1271no—
298C23—10161no—
299C23—10161no—
300C23—2544no—
301C23—2541no—
302Q8—21no—
303Q8—1272no—
304Q8—2544no—
305Q8—2541no—
306Q8C2×C4×D12712yes—
307D4—21no—
308D4—1272no—
309D4—2544no—
310D4—2541no—
311D4C2×C4×D12712yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—1271no—
317C2×C4—1272no—
318C2×C4—1272no—
319C2×C4—1271no—
320C2×C4—1271no—
321C2×C4—10161no—
322C2×C4—10161no—
323C2×C4—10161no—
324C2×C4—2544no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4D4×D12711yescharacteristic
329C2×C4D4×D12711yescharacteristic
330C2×C4C23×D12711yescharacteristic
331C8—21no—
332C8—1272no—
333C8—2541no—
334C8C2×C4×D12712yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—1272no—
339C22—10164no—
340C22—5081no—
341C22—5081no—
342C22—2542no—
343C22—2541no—
344C22—2541no—
345C22C2×D4×D12711yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—1272no—
352C4—5081no—
353C4—5081no—
354C4—2542no—
355C4—2541no—
356C4—2541no—
357C4C22×C4×D12711yescharacteristic
358C4#41×D12711yescharacteristic
359C2—42no—
360C2—5082no—
361C2—2542no—
362C2C4×D4×D12711yescharacteristic
363C2#2255068511yescharacteristic
364C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
Fitting#440×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×D127C22
cosocleC2×C4×C127C24
hypercenter#440D127
hypocenterC127#2949
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #22550352C2semidirect
4. #22549978C2semidirect
5. #22549977C2semidirect
6. #22549975C2semidirect
7. #22549967C2semidirect
8. C4×#14523235C2semidirect
9. C4×#14523236C2semidirect
10. #22549961C2semidirect
11. #22549913C2semidirect
12. #22549883C2semidirect
14. #22549833C2semidirect
15. #440×C127C2semidirect
21. SD16×D127C4semidirect
24. Q8×Dic127C22semidirect
26. C4×Dic254C22semidirect
28. #16861174C22semidirect
41. #16861143C22semidirect
42. #16861140C22semidirect
43. #16861140C22semidirect
44. #16861134C22semidirect
45. #16861133C22semidirect
46. #16861132C22semidirect
50. #16861119C22semidirect
51. C4×C127⋊C8C22semidirect
52. C8×Dic127C22semidirect
54. C4×Q8×C127C22semidirect
56. #107×C127C22semidirect
57. #103×C127C22semidirect
59. #97×C127C22semidirect
64. Q8×D127C2×C4semidirect
74. #14523237C2×C4semidirect
75. #14523236C2×C4semidirect
76. #14523235C2×C4semidirect
80. C8×D127C2×C4semidirect
104. SD16×C127C2×C4semidirect
183. C127#2949semidirect
190. #440D127semidirect
213. C4×Q8C2×D127semidirect
228. #107C2×D127semidirect
234. #103C2×D127semidirect
240. #97C2×D127semidirect
263. SD16C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1272
#225503522
#225503482
#225499782
#225499772
#225499752
#225499672
C4×#145232352
C4×#145232362
#225499612
#225499132
#225498832
#225498792
#225498332
#440×C1272
#2949127