Finite Group Databasebeta

← #30648827

Subgroups — #30648827

Order 16256.

444 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550430C211yesmaximal, characteristic, complemented
2C4×D4×D127C211yesmaximal, characteristic
3#22550213C211yesmaximal, characteristic, complemented
4#22550210C211yesmaximal, characteristic, complemented
5#22550155C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550080C211yesmaximal, characteristic, complemented
8#22550079C211yesmaximal, characteristic, complemented
9#22550028C211yesmaximal, characteristic
10#22550025C211yesmaximal, characteristic
11#22549978C211yesmaximal, characteristic, complemented
12#22549976C211yesmaximal, characteristic, complemented
13#22549974C211yesmaximal, characteristic, complemented
14#22549931C211yesmaximal, characteristic, complemented
15#440×C127C211yesmaximal, characteristic, complemented
16C2×D4×D127C2211yescharacteristic
17C22×C4×D127—21no—
18C2×#14523237C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21#16861208C2211yescharacteristic, complemented
22#16861207C2211yescharacteristic, complemented
23#16861199—21no—
24#16861189C2211yescharacteristic
25#16861187—21no—
26#16861185C2211yescharacteristic
27C4×Dic254—21no—
28C4×Dic254C2211yescharacteristic, complemented
29C4×D508C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31#16861171C2211yescharacteristic
32C4×C22⋊D127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861166—21no—
35C4⋊C4×D127C2211yescharacteristic
36C22⋊C4×D127—21no—
37C42×D127C2211yescharacteristic
38#16861143C2211yescharacteristic, complemented
39#16861142C2211yescharacteristic, complemented
40#16861142C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861132C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861120C2211yescharacteristic, complemented
52C2×SD16×C127C2211yescharacteristic
53C4×Q8×C127C2211yescharacteristic, complemented
54C4×D4×C127C2211yescharacteristic
55#107×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C2×Dic254—21no—
61C2×Dic254C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64D4×D127—22no—
65D4×D127—22no—
66C22×Dic127—21no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—44no—
71C2×C4×D127D411yescharacteristic
72C2×C4×D127D411yescharacteristic
73C2×C4×D127C2311yescharacteristic
74#14523237—22no—
75#14523236—22no—
76#14523235—22no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C4⋊Dic127—21no—
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88C22⋊Dic127—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×Dic127D411yescharacteristic
95C4×Dic127D411yescharacteristic
96C4×Dic127C2311yescharacteristic
97C2×Q8×C127C2311yescharacteristic
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127—21no—
100SD16×C127—22no—
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C4⋊C4×C127—21no—
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127—21no—
108C42×C127C2311yescharacteristic
109C22×D127—24no—
110C22×D127—21no—
111C22×D127—42no—
112C22×D127—42no—
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254—21no—
116Dic254—42no—
117D508—21no—
118D508#4112yes—
119C22⋊D127—24no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—42no—
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C4×D127—22no—
128C4×D127—21no—
129C4×D127—21no—
130C4×D127—22no—
131C4×D127—42no—
132C4×D127—42no—
133C4×D127—42no—
134C127⋊C8—21no—
135C127⋊C8—21no—
136C23×C127—21no—
137Q8×C127—21no—
138Q8×C127—21no—
139D4×C127—21no—
140D4×C127#4112yes—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—42no—
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C8×C127—21no—
149C8×C127—21no—
150C22×C127—22no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127C22×D411yescharacteristic
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—22no—
157C2×D127—48no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127#13712yes—
161C4×C127—21no—
162C4×C127—21no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127C2×#4111yescharacteristic
167C4×C127C2×#4111yescharacteristic
168Dic127—21no—
169Dic127—21no—
170Dic127—41no—
171Dic127—41no—
172Dic127—41no—
173Dic127AGL(1,ℤ/8)12yes—
174C2×C127—42no—
175C2×C127#52511yescharacteristic
176C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C127C2×#13711yescharacteristic
178D127—22no—
179D127—42no—
180#3110—1271nomaximal, Sylow
181C127#311011yescharacteristic, complemented, Sylow
182#533—1271no—
183C2×C4×D4—1271no—
184#484—1271no—
185#484—1271no—
186#474—1271no—
187#474—1271no—
188#461—1271no—
189#461—1271no—
190#450—1271no—
191#450—1271no—
192#440—1271no—
193#440—1271no—
194#440—1271no—
195#440D12711yescharacteristic, complemented
196#432—1271no—
197C22×D4—1271no—
198C23×C4—2541no—
199C2×SD16—1271no—
200C2×SD16—1271no—
201C2×SD16—1271no—
202C2×SD16C2×D12711yescharacteristic
203C4⋊Q8—1271no—
204C4⋊Q8—1271no—
205#125—2541no—
206C22⋊Q8—1271no—
207C22⋊Q8—1271no—
208C22⋊Q8—2541no—
209C4×Q8—1271no—
210C4×Q8—2541no—
211C4×Q8C2×D12711yescharacteristic, complemented
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—1271no—
215C4×D4—2541no—
216C4×D4—2541no—
217C4×D4C2×D12711yescharacteristic
218#117—2541no—
219C2×C4⋊C4—1271no—
220C2×C22⋊C4—2541no—
221C2×C42—1271no—
222#107—1271no—
223#107—1271no—
224#107—1271no—
225#107C2×D12711yescharacteristic, complemented
226C4⋊C8—1271no—
227C4⋊C8—1271no—
228#103—1271no—
229#103—1271no—
230#103—1271no—
231#103C2×D12711yescharacteristic, complemented
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102C2×D12711yescharacteristic
236C22⋊C8—1271no—
237C22⋊C8—1271no—
238#97—1271no—
239#97C2×D12711yescharacteristic, complemented
240C24—2541no—
241C2×Q8—1271no—
242C2×Q8—2541no—
243C2×Q8C22×D12711yescharacteristic
244C2×D4—1271no—
245C2×D4—2541no—
246C2×D4—2542no—
247C2×D4—2542no—
248C2×D4C22×D12711yescharacteristic
249C22×C4—21no—
250C22×C4—1271no—
251C22×C4—1271no—
252C22×C4—1271no—
253C22×C4—5084no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4—2541no—
258SD16—22no—
259SD16—2542no—
260SD16—2542no—
261SD16—2542no—
262C2×C8—1271no—
263C2×C8—1271no—
264C2×C8C22×D12711yescharacteristic
265C2×C8C22×D12711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4C22×D12711yescharacteristic
277C4⋊C4C22×D12711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C42—21no—
285C42—1271no—
286C42—1271no—
287C42—1271no—
288C42—2541no—
289C42C22×D12711yescharacteristic
290C23—21no—
291C23—1271no—
292C23—5082no—
293C23—5082no—
294C23—2544no—
295C23—2541no—
296Q8—21no—
297Q8—21no—
298Q8—5082no—
299Q8—2541no—
300Q8—2541no—
301D4—21no—
302D4—1272no—
303D4—2544no—
304D4—2541no—
305D4#1452324812yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—1271no—
311C2×C4—1271no—
312C2×C4—1271no—
313C2×C4—5082no—
314C2×C4—5082no—
315C2×C4—5082no—
316C2×C4—5082no—
317C2×C4—2542no—
318C2×C4—2542no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4D4×D12711yescharacteristic
325C2×C4D4×D12711yescharacteristic
326C2×C4C23×D12711yescharacteristic
327C8—21no—
328C8—21no—
329C8—2541no—
330C8—2541no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—1272no—
335C22—5088no—
336C22—5081no—
337C22—5081no—
338C22—2542no—
339C22—2541no—
340C22—2541no—
341C22C2×D4×D12711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—1272no—
348C4—5081no—
349C4—5081no—
350C4—5081no—
351C4—2541no—
352C4—2541no—
353C4#41×D12711yescharacteristic
354C4C2×#1452324811yescharacteristic
355C2—42no—
356C2—5082no—
357C2—2542no—
358C2#2255038511yescharacteristic
359C2AGL(1,ℤ/8)×D12711yescharacteristic
360C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#440×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×D127C22
hypercenter#440D127
hypocenterC127#3110
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #22550430C2semidirect
3. #22550213C2semidirect
4. #22550210C2semidirect
5. #22550155C2semidirect
6. #22550154C2semidirect
7. #22550080C2semidirect
8. #22550079C2semidirect
11. #22549978C2semidirect
12. #22549976C2semidirect
13. #22549974C2semidirect
14. #22549931C2semidirect
15. #440×C127C2semidirect
21. #16861208C22semidirect
22. #16861207C22semidirect
28. C4×Dic254C22semidirect
38. #16861143C22semidirect
39. #16861142C22semidirect
40. #16861142C22semidirect
41. Dic127⋊C8C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861134C22semidirect
44. #16861133C22semidirect
45. #16861132C22semidirect
51. #16861120C22semidirect
53. C4×Q8×C127C22semidirect
55. #107×C127C22semidirect
56. #103×C127C22semidirect
58. #97×C127C22semidirect
181. C127#3110semidirect
195. #440D127semidirect
211. C4×Q8C2×D127semidirect
225. #107C2×D127semidirect
231. #103C2×D127semidirect
239. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504302
C4×D4×D1272
#225502132
#225502102
#225501552
#225501542
#225500802
#225500792
#225500282
#225500252
#225499782
#225499762
#225499742
#225499312
#440×C1272
#3110127