Finite Group Databasebeta

← #30648648

Subgroups — #30648648

Order 16256.

420 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
1#22550389C211yesmaximal, characteristic
2#117×D127C211yesmaximal, characteristic, complemented
3#22550214C211yesmaximal, characteristic, complemented
4#22550210C211yesmaximal, characteristic, complemented
5#22550205C211yesmaximal, characteristic, complemented
6#22550201C211yesmaximal, characteristic, complemented
7#22550200C211yesmaximal, characteristic, complemented
8#22550196C211yesmaximal, characteristic, complemented
9#22550191C211yesmaximal, characteristic, complemented
10#22549913C211yesmaximal, characteristic, complemented
11#22549911C211yesmaximal, characteristic, complemented
12#22549871C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14C22⋊C8×D127C211yesmaximal, characteristic, complemented
15#485×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18C2×C8×D127C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20#16861194—21no—
21#16861185C2211yescharacteristic
22#16861184C2211yescharacteristic
23#16861183C2211yescharacteristic
24#16861179—21no—
25#16861171—21no—
26(C22⋊D127)⋊C4—21no—
27#16861168C2211yescharacteristic
28#16861167C2211yescharacteristic
29#16861167C2211yescharacteristic
30#16861167C2211yescharacteristic
31#16861166—21no—
32#16861165—21no—
33C4⋊C4×D127C2211yescharacteristic
34C4⋊C4×D127C2211yescharacteristic
35C22⋊C4×D127—21no—
36C42×D127—21no—
37#16861143C2211yescharacteristic, complemented
38#16861142C2211yescharacteristic, complemented
39#16861142C2211yescharacteristic, complemented
40#16861141C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42#16861140C2211yescharacteristic, complemented
43#16861133C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic, complemented
47#16861129C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic, complemented
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52C22⋊Q8×C127C2211yescharacteristic
53#117×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic
59C23×D127D411yescharacteristic
60#14523249—22no—
61#14523249—22no—
62C2×D508—21no—
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C22×Dic127D411yescharacteristic
67C2×C4×D127—22no—
68C2×C4×D127—22no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127D411yescharacteristic
73C2×C4×D127D411yescharacteristic
74C2×C4×D127C2311yescharacteristic
75C2×C4×D127C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C8×D127—22no—
79C8×D127—22no—
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
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—
93C4×Dic127—21no—
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
97C2×Q8×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C2×C8×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C4⋊C4×C127—21no—
102C4⋊C4×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C42×C127—21no—
108C22×D127—22no—
109C22×D127—22no—
110C22×D127—21no—
111C22×D127—41no—
112C22×D127C2×D411yescharacteristic
113C22×D127C2×D411yescharacteristic
114D508—21no—
115D508—21no—
116D508—42no—
117D508—42no—
118C22⋊D127—42no—
119C2×Dic127—22no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C4×D127—22no—
126C4×D127—21no—
127C4×D127—21no—
128C4×D127—22no—
129C4×D127—42no—
130C4×D127—42no—
131C4×D127—42no—
132C4×D127#4112yes—
133C4×D127#4112yes—
134C127⋊C8—21no—
135C127⋊C8—21no—
136C23×C127C2×D411yescharacteristic
137Q8×C127—21no—
138Q8×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2411yescharacteristic
147C8×C127—21no—
148C8×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127C22×D411yescharacteristic
153C2×D127—21no—
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—21no—
157C2×D127—44no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127#12312yes—
162C2×D127#13512yes—
163C4×C127—21no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127C2×#4111yescharacteristic
169C4×C127C2×#4111yescharacteristic
170Dic127—21no—
171Dic127—41no—
172Dic127—41no—
173Dic127—41no—
174Dic127#12312yes—
175C2×C127—41no—
176C2×C127C2×#12311yescharacteristic
177C2×C127C2×#13511yescharacteristic
178C2×C127#57511yescharacteristic
179D127—22no—
180D127—22no—
181D127—81no—
182#3091—1271nomaximal, Sylow
183C127#309111yescharacteristic, complemented, Sylow
184#525—1271no—
185C2×#117—1271no—
186#485—1271no—
187#485D12711yescharacteristic, complemented
188#484—1271no—
189#483—1271no—
190#482—1271no—
191#482—1271no—
192#481—1271no—
193#480—1271no—
194#427—1271no—
195#427—1271no—
196#416—1271no—
197#416—1271no—
198C2×C22⋊C8—1271no—
199C2×#41—1271no—
200C23×C4—1271no—
201C22×C8—1271no—
202C22×C8—1271no—
203#123—2541no—
204C22⋊Q8—1271no—
205C22⋊Q8C2×D12711yescharacteristic
206#121—1271no—
207#121—1271no—
208#120—2541no—
209C4×D4—2541no—
210C4×D4—2541no—
211#117—1271no—
212#117—1271no—
213#117—1271no—
214#117—1271no—
215#117—2541no—
216#117—2541no—
217#117C2×D12711yescharacteristic
218C2×C4⋊C4—1271no—
219C2×C4⋊C4—1271no—
220C2×C22⋊C4—2541no—
221C2×C42—2541no—
222#107—1271no—
223#107—1271no—
224#107—1271no—
225#107C2×D12711yescharacteristic, complemented
226#106—1271no—
227#106—1271no—
228#106—1271no—
229#106C2×D12711yescharacteristic, complemented
230#103—1271no—
231#103—1271no—
232#103C2×D12711yescharacteristic, complemented
233#103C2×D12711yescharacteristic, complemented
234#102—1271no—
235#102—1271no—
236#102—1271no—
237#102—1271no—
238C22⋊C8—1271no—
239C22⋊C8—1271no—
240C22⋊C8—1271no—
241C22⋊C8C2×D12711yescharacteristic
242C24—1271no—
243#41—2542no—
244#41—2542no—
245C2×Q8C22×D12711yescharacteristic
246C2×D4—1271no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4—2541no—
250C22×C4—1271no—
251C22×C4—1271no—
252C22×C4—1271no—
253C22×C4—1271no—
254C22×C4—1271no—
255C22×C4—2541no—
256C22×C4—2542no—
257C22×C4—2542no—
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4C22×D12711yescharacteristic
261C2×C8—1271no—
262C2×C8—1271no—
263C2×C8—2542no—
264C2×C8—2542no—
265C2×C8C22×D12711yescharacteristic
266C2×C8C22×D12711yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4—1271no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4C22×D12711yescharacteristic
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C42—21no—
287C42—2541no—
288C42—2541no—
289C42—2541no—
290C42—2541no—
291C23—1271no—
292C23—1271no—
293C23—5081no—
294C23—2542no—
295C23—2542no—
296C23—2541no—
297C23D4×D12711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—5082no—
301D4—5082no—
302D4—5082no—
303D4—2541no—
304D4—2541no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1271no—
312C2×C4—1272no—
313C2×C4—1272no—
314C2×C4—1271no—
315C2×C4—5082no—
316C2×C4—5082no—
317C2×C4—5082no—
318C2×C4—2542no—
319C2×C4—2542no—
320C2×C4—2542no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4D4×D12711yescharacteristic
327C2×C4C23×D12711yescharacteristic
328C8—21no—
329C8—21no—
330C8—2541no—
331C8—2541no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—1272no—
336C22—1272no—
337C22—5084no—
338C22—5081no—
339C22—5081no—
340C22—5081no—
341C22—2541no—
342C22—2541no—
343C22—2541no—
344C22—2541no—
345C22C2×D4×D12711yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—1272no—
352C4—5081no—
353C4—5081no—
354C4—5081no—
355C4—2541no—
356C4#41×D12711yescharacteristic
357C4#41×D12711yescharacteristic
358C2—41no—
359C2—10161no—
360C2—2542no—
361C2—2542no—
362C2#123×D12711yescharacteristic
363C2#135×D12711yescharacteristic
364C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#485×C127C2
socleC22×C127C22×D4
prime radical#22550389C2
hypercenter#485D127
hypocenterC127#3091
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #117×D127C2semidirect
3. #22550214C2semidirect
4. #22550210C2semidirect
5. #22550205C2semidirect
6. #22550201C2semidirect
7. #22550200C2semidirect
8. #22550196C2semidirect
9. #22550191C2semidirect
10. #22549913C2semidirect
11. #22549911C2semidirect
12. #22549871C2semidirect
13. #22549871C2semidirect
14. C22⋊C8×D127C2semidirect
15. #485×C127C2semidirect
37. #16861143C22semidirect
38. #16861142C22semidirect
39. #16861142C22semidirect
40. #16861141C22semidirect
41. #16861140C22semidirect
42. #16861140C22semidirect
43. #16861133C22semidirect
44. #16861133C22semidirect
45. #16861131C22semidirect
46. #16861131C22semidirect
47. #16861129C22semidirect
48. #16861129C22semidirect
54. #107×C127C22semidirect
55. #106×C127C22semidirect
56. #103×C127C22semidirect
57. #103×C127C22semidirect
183. C127#3091semidirect
187. #485D127semidirect
225. #107C2×D127semidirect
229. #106C2×D127semidirect
232. #103C2×D127semidirect
233. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503892
#117×D1272
#225502142
#225502102
#225502052
#225502012
#225502002
#225501962
#225501912
#225499132
#225499112
#225498712
C22⋊C8×D1272
#485×C1272
#3091127