Finite Group Databasebeta

← #30650578

Subgroups — #30650578

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550574C211yesmaximal, characteristic, complemented
2#22550517C211yesmaximal, characteristic, complemented
3#22550232C211yesmaximal, characteristic, complemented
4#22550203C211yesmaximal, characteristic, complemented
5#22550179C211yesmaximal, characteristic, complemented
6#22550113C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550094C211yesmaximal, characteristic, complemented
9#22550024C211yesmaximal, characteristic, complemented
10#22550012C211yesmaximal, characteristic, complemented
11#22549958C211yesmaximal, characteristic, complemented
12(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14C2×#16861129C211yesmaximal, characteristic, complemented
15#484×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C2×#14523235C2211yescharacteristic, complemented
19C2×D1016C2211yescharacteristic
20C2×D1016C2211yescharacteristic
21C22×C127⋊C8C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic, complemented
23#16861204—21no—
24#16861200C2211yescharacteristic, complemented
25#16861198C2211yescharacteristic
26#16861194—21no—
27#16861194—21no—
28#16861184—21no—
29#16861184C2211yescharacteristic
30#16861184C2211yescharacteristic, complemented
31#16861183—21no—
32#16861183C2211yescharacteristic
33#16861179—21no—
34#16861171C2211yescharacteristic
35#16861171C2211yescharacteristic, complemented
36C4×C22⋊D127—21no—
37C4×C22⋊D127C2211yescharacteristic
38#16861167C2211yescharacteristic, complemented
39C22⋊C4×D127—21no—
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861129C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22⋊Q8×C127C2211yescharacteristic
53C2×C4⋊C4×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic, complemented
55#107×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic
59C23×D127—21no—
60#14523249—22no—
61#14523249—41no—
62C2×D508—22no—
63C2×D508—21no—
64C2×D508—21no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508—41no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127D411yescharacteristic
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127C2311yescharacteristic
77#14523235—41no—
78D1016—22no—
79D1016—41no—
80C2×C127⋊C8—41no—
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C8×D127—41no—
84C4⋊Dic127—21no—
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87Dic127⋊C4D411yescharacteristic
88C22⋊Dic127D411yescharacteristic
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
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×C127—21no—
102C22×C4×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127—21no—
106C4⋊C4×C127—21no—
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—42no—
115C22×D127—42no—
116C22×D127—81no—
117C22×D127C2×D411yescharacteristic
118D508—21no—
119D508—21no—
120D508—21no—
121D508—21no—
122D508—21no—
123D508—42no—
124D508—42no—
125D508—81no—
126D508—81no—
127D508#4112yes—
128C22⋊D127—22no—
129C22⋊D127—42no—
130C2×Dic127—21no—
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C2×Dic127C2×D411yescharacteristic
134C4×D127—21no—
135C4×D127—21no—
136C4×D127—42no—
137C4×D127—81no—
138C4×D127—81no—
139C127⋊C8—21no—
140C127⋊C8—21no—
141C23×C127C2×D411yescharacteristic
142Q8×C127—21no—
143Q8×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—81no—
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2411yescharacteristic
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127#13512yes—
158C22×C127C22×D411yescharacteristic
159C2×D127—21no—
160C2×D127—22no—
161C2×D127—21no—
162C2×D127—21no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—84no—
172C4×C127—21no—
173C4×C127—41no—
174C4×C127—41no—
175C4×C127—41no—
176C4×C127—41no—
177C4×C127C2×#4111yescharacteristic
178C4×C127H(5,𝔽2)11yescharacteristic
179Dic127—21no—
180Dic127—21no—
181Dic127—21no—
182Dic127—41no—
183C2×C127—22no—
184C2×C127#54611yescharacteristic
185C2×C127C2×#13511yescharacteristic
186C2×C127#57611yescharacteristic
187D127—41no—
188D127—42no—
189D127—81no—
190D127—81no—
191#3297—1271nomaximal, Sylow
192C127#329711yescharacteristic, complemented, Sylow
193#553—1271no—
194#546—1271no—
195#488—1271no—
196#484D12711yescharacteristic, complemented
197#482—1271no—
198#478—1271no—
199#466—1271no—
200#465—1271no—
201#463—1271no—
202#449—1271no—
203#447—1271no—
204C4×D8—1271no—
205C8×D4—1271no—
206#416—1271no—
207C2×#102—1271no—
208C2×#41—1271no—
209C22×D4—1271no—
210C2×SD16—1271no—
211C2×D8—1271no—
212C2×D8—1271no—
213C22×C8—1271no—
214C22×C8—1271no—
215#126—2541no—
216#125—1271no—
217#124—1271no—
218#123—2541no—
219#123—2541no—
220C22⋊Q8C2×D12711yescharacteristic
221#121—1271no—
222#121—1271no—
223#121—1271no—
224#121—2541no—
225#121—2541no—
226#120—2541no—
227C4×D4—1271no—
228C4×D4—1271no—
229C4×D4—1271no—
230C4×D4—2541no—
231#117—1271no—
232C2×C4⋊C4C2×D12711yescharacteristic
233C2×C22⋊C4—2541no—
234#107C2×D12711yescharacteristic, complemented
235#107C2×D12711yescharacteristic, complemented
236#106—1271no—
237C4⋊C8—1271no—
238#103—1271no—
239#103C2×D12711yescharacteristic, complemented
240#103C2×D12711yescharacteristic, complemented
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248C22⋊C8—1271no—
249C22⋊C8C2×D12711yescharacteristic
250C4×C8—1271no—
251C24—2541no—
252#41—5081no—
253#41—2542no—
254C2×Q8C22×D12711yescharacteristic
255C2×D4—1271no—
256C2×D4—1271no—
257C2×D4—1271no—
258C2×D4—1271no—
259C2×D4—5081no—
260C2×D4—2542no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4—2541no—
265C2×D4—2541no—
266C22×C4—21no—
267C22×C4—1271no—
268C22×C4—2541no—
269C22×C4—2541no—
270C22×C4—2541no—
271C22×C4C22×D12711yescharacteristic
272SD16—5081no—
273D8—5081no—
274D8—2542no—
275C2×C8—1271no—
276C2×C8—1271no—
277C2×C8—5081no—
278C2×C8—5081no—
279C2×C8C22×D12711yescharacteristic
280C2×C8C22×D12711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—1271no—
284C4⋊C4—1271no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4C22×D12711yescharacteristic
288C4⋊C4C22×D12711yescharacteristic
289C4⋊C4C22×D12711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—1271no—
292C22⋊C4—1271no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C22⋊C4—2541no—
300C22⋊C4—2541no—
301C42—1271no—
302C42—2541no—
303C23—1271no—
304C23—10161no—
305C23—5082no—
306C23—5082no—
307C23—2541no—
308C23—2541no—
309C23—2541no—
310C23D4×D12711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—1272no—
314D4—10161no—
315D4—10161no—
316D4—5082no—
317D4—5082no—
318D4—5082no—
319D4—2542no—
320D4—2541no—
321D4—2541no—
322D4—2541no—
323D4—2541no—
324D4—2541no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—1271no—
333C2×C4—1271no—
334C2×C4—1271no—
335C2×C4—10161no—
336C2×C4—10161no—
337C2×C4—5082no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4D4×D12711yescharacteristic
342C2×C4C23×D12711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2541no—
346C8—2541no—
347C22—21no—
348C22—21no—
349C22—10164no—
350C22—5081no—
351C22—5081no—
352C22—5081no—
353C22—5081no—
354C22—5081no—
355C22—5081no—
356C22—5081no—
357C22—5081no—
358C22—2541no—
359C22—2541no—
360C22—2542no—
361C22—2541no—
362C22#1686123212yes—
363C22C2×D4×D12711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—5081no—
370C4—2541no—
371C4—2541no—
372C4—2541no—
373C4#41×D12711yescharacteristic
374C4#1686125511yescharacteristic
375C2—22no—
376C2—10161no—
377C2—10161no—
378C2—5082no—
379C2—5081no—
380C2#2255051811yescharacteristic
381C2C2×#1686123211yescharacteristic
382C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#484×C127C2
socleC22×C127C22×D4
hypercenter#484D127
hypocenterC127#3297
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #22550574C2semidirect
2. #22550517C2semidirect
3. #22550232C2semidirect
4. #22550203C2semidirect
5. #22550179C2semidirect
6. #22550113C2semidirect
7. #22550107C2semidirect
8. #22550094C2semidirect
9. #22550024C2semidirect
10. #22550012C2semidirect
11. #22549958C2semidirect
12. (C22⋊D127)⋊C8C2semidirect
13. #22549871C2semidirect
14. C2×#16861129C2semidirect
15. #484×C127C2semidirect
18. C2×#14523235C22semidirect
22. C2×C8×D127C22semidirect
24. #16861200C22semidirect
30. #16861184C22semidirect
35. #16861171C22semidirect
38. #16861167C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. #16861133C22semidirect
49. #16861129C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. C8×Dic127C22semidirect
54. #107×C127C22semidirect
55. #107×C127C22semidirect
56. #103×C127C22semidirect
57. #103×C127C22semidirect
192. C127#3297semidirect
196. #484D127semidirect
234. #107C2×D127semidirect
235. #107C2×D127semidirect
239. #103C2×D127semidirect
240. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505742
#225505172
#225502322
#225502032
#225501792
#225501132
#225501072
#225500942
#225500242
#225500122
#225499582
(C22⋊D127)⋊C82
#225498712
C2×#168611292
#484×C1272
#3297127