Finite Group Databasebeta

← #30650656

Subgroups — #30650656

Order 16256.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550589C211yesmaximal, characteristic, complemented
2#22550528C211yesmaximal, characteristic, complemented
3#22550225C211yesmaximal, characteristic, complemented
4#22550219C211yesmaximal, characteristic, complemented
5#22550165C211yesmaximal, characteristic, complemented
6#22550107C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550080C211yesmaximal, characteristic, complemented
9#22550024C211yesmaximal, characteristic, complemented
10#22550023C211yesmaximal, characteristic, complemented
11#22549958C211yesmaximal, characteristic, complemented
12(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14#22549870C211yesmaximal, characteristic, complemented
15#485×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×D1016C2211yescharacteristic
21C22×C127⋊C8C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861208C2211yescharacteristic
24#16861207C2211yescharacteristic
25#16861197—21no—
26#16861195—21no—
27#16861194—21no—
28#16861191—21no—
29#16861189C2211yescharacteristic
30#16861184C2211yescharacteristic
31#16861183C2211yescharacteristic
32#16861182—21no—
33#16861182—21no—
34#16861171C2211yescharacteristic
35#16861171C2211yescharacteristic
36C4×C22⋊D127C2211yescharacteristic
37#16861167C2211yescharacteristic
38#16861166—21no—
39C4⋊C4×D127—21no—
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861132C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C22⋊Q8×C127C2211yescharacteristic
53#117×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59#14523249—22no—
60#14523249—41no—
61#14523248—22no—
62#14523248—41no—
63C2×Dic254—21no—
64C2×Dic254C2311yescharacteristic
65C2×D508—21no—
66C2×D508—21no—
67C2×D508C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127D411yescharacteristic
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127C2311yescharacteristic
77#14523237—41no—
78#14523235—41no—
79D1016—22no—
80C2×C127⋊C8—41no—
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C8×D127—41no—
84C4⋊Dic127—21no—
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127D411yescharacteristic
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4D411yescharacteristic
98C4×Dic127—21no—
99C4×Dic127C2311yescharacteristic
100C2×Q8×C127C2311yescharacteristic
101C22×C4×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C4⋊C4×C127—21no—
105C4⋊C4×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C22⋊C4×C127—21no—
109C22⋊C4×C127—21no—
110C42×C127—21no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254—21no—
116Dic254—81no—
117D508—21no—
118D508—21no—
119D508—21no—
120D508—42no—
121D508—81no—
122D508#4112yes—
123C22⋊D127—22no—
124C22⋊D127—42no—
125C22⋊D127—81no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C4×D127—21no—
132C4×D127—21no—
133C4×D127—42no—
134C4×D127—42no—
135C4×D127—42no—
136C4×D127—42no—
137C127⋊C8—21no—
138C127⋊C8—21no—
139C23×C127C2×D411yescharacteristic
140Q8×C127—21no—
141Q8×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—22no—
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127C2×#4111yescharacteristic
171C4×C127H(5,𝔽2)11yescharacteristic
172Dic127—21no—
173Dic127—21no—
174Dic127—41no—
175Dic127—41no—
176Dic127#13512yes—
177C2×C127—41no—
178C2×C127#54611yescharacteristic
179C2×C127C2×#13511yescharacteristic
180C2×C127#57711yescharacteristic
181D127—41no—
182D127—42no—
183D127—81no—
184#3303—1271nomaximal, Sylow
185C127#330311yescharacteristic, complemented, Sylow
186#555—1271no—
187#547—1271no—
188#487—1271no—
189#485—1271no—
190#485D12711yescharacteristic, complemented
191#476—1271no—
192#465—1271no—
193#465—1271no—
194#461—1271no—
195#449—1271no—
196#449—1271no—
197C4×D8—1271no—
198C8×D4—1271no—
199#416—1271no—
200#416—1271no—
201C2×#41—1271no—
202C2×#41—1271no—
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C22×C8—1271no—
207C22×C8—1271no—
208C4⋊Q8—1271no—
209C4⋊Q8—1271no—
210#124—2541no—
211#124—2541no—
212#123—2541no—
213#123—2541no—
214C22⋊Q8—1271no—
215C22⋊Q8C2×D12711yescharacteristic
216#121—1271no—
217#121—1271no—
218#121—2541no—
219#121—2541no—
220C4×D4—1271no—
221C4×D4—1271no—
222C4×D4—1271no—
223#117—1271no—
224#117—2541no—
225#117C2×D12711yescharacteristic
226C2×C4⋊C4—2541no—
227#107C2×D12711yescharacteristic
228#106—1271no—
229#106C2×D12711yescharacteristic
230C4⋊C8—1271no—
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103C2×D12711yescharacteristic
236#103C2×D12711yescharacteristic
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241C22⋊C8—1271no—
242C22⋊C8C2×D12711yescharacteristic
243C4×C8—1271no—
244#41—5081no—
245#41—5081no—
246#41—2542no—
247#41—2542no—
248C2×Q8—1271no—
249C2×Q8—2541no—
250C2×Q8C22×D12711yescharacteristic
251C2×D4—1271no—
252C2×D4—1271no—
253C2×D4—1271no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4—2541no—
257C2×D4—2541no—
258C22×C4—1271no—
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4C22×D12711yescharacteristic
264SD16—5081no—
265SD16—5081no—
266D8—2542no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8—5081no—
270C2×C8—5081no—
271C2×C8C22×D12711yescharacteristic
272C2×C8C22×D12711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4C22×D12711yescharacteristic
281C4⋊C4C22×D12711yescharacteristic
282C4⋊C4C22×D12711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—1271no—
286C22⋊C4—1271no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C42—21no—
294C42—1271no—
295C42—2541no—
296C23—1271no—
297C23—2541no—
298C23—2541no—
299C23D4×D12711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—10161no—
303Q8—2541no—
304Q8—2541no—
305D4—1272no—
306D4—10161no—
307D4—10161no—
308D4—5082no—
309D4—5082no—
310D4—2542no—
311D4—2541no—
312D4—2541no—
313D4—2541no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—1271no—
321C2×C4—1271no—
322C2×C4—1271no—
323C2×C4—5082no—
324C2×C4—5082no—
325C2×C4—5082no—
326C2×C4—5082no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4D4×D12711yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—21no—
334C8—21no—
335C8—2541no—
336C8—2541no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—5081no—
341C22—5081no—
342C22—5081no—
343C22—5081no—
344C22—5081no—
345C22—2541no—
346C22—2541no—
347C22—2542no—
348C22—2541no—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—1272no—
356C4—5081no—
357C4—5081no—
358C4—2541no—
359C4—2541no—
360C4#41×D12711yescharacteristic
361C4#1686125511yescharacteristic
362C2—41no—
363C2—10161no—
364C2—5082no—
365C2—5081no—
366C2#2255051811yescharacteristic
367C2#135×D12711yescharacteristic
368C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#485×C127C2
socleC22×C127C22×D4
hypercenter#485D127
hypocenterC127#3303
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
1. #22550589C2semidirect
2. #22550528C2semidirect
3. #22550225C2semidirect
4. #22550219C2semidirect
5. #22550165C2semidirect
6. #22550107C2semidirect
7. #22550106C2semidirect
8. #22550080C2semidirect
9. #22550024C2semidirect
10. #22550023C2semidirect
11. #22549958C2semidirect
12. (C22⋊D127)⋊C8C2semidirect
13. #22549871C2semidirect
14. #22549870C2semidirect
15. #485×C127C2semidirect
185. C127#3303semidirect
190. #485D127semidirect

Maximal subgroups

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

subgroupindex
#225505892
#225505282
#225502252
#225502192
#225501652
#225501072
#225501062
#225500802
#225500242
#225500232
#225499582
(C22⋊D127)⋊C82
#225498712
#225498702
#485×C1272
#3303127