Finite Group Databasebeta

← #30650581

Subgroups — #30650581

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#22550570C211yesmaximal, characteristic, complemented
2#22550514C211yesmaximal, characteristic, complemented
3#22550232C211yesmaximal, characteristic, complemented
4#22550210C211yesmaximal, characteristic, complemented
5#22550200C211yesmaximal, characteristic, complemented
6#22550174C211yesmaximal, characteristic, complemented
7#22550112C211yesmaximal, characteristic, complemented
8#22550108C211yesmaximal, characteristic, complemented
9#22550094C211yesmaximal, characteristic, complemented
10#22550021C211yesmaximal, characteristic, complemented
11#22550011C211yesmaximal, characteristic, complemented
12#22549956C211yesmaximal, characteristic, complemented
13C8×C22⋊D127C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#416×C127C211yesmaximal, characteristic, complemented
16C2×D4×D127C2211yescharacteristic
17C2×#14523235C2211yescharacteristic, complemented
18C2×D1016C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861203—21no—
22#16861200C2211yescharacteristic, complemented
23#16861198C2211yescharacteristic
24#16861192—21no—
25#16861191—21no—
26#16861185C2211yescharacteristic
27#16861184C2211yescharacteristic
28#16861183C2211yescharacteristic
29#16861183C2211yescharacteristic, complemented
30#16861182—21no—
31#16861181—21no—
32#16861177—21no—
33C4×D508—21no—
34D4×Dic127C2211yescharacteristic
35#16861171C2211yescharacteristic, complemented
36C4×C22⋊D127C2211yescharacteristic
37C4⋊C4×D127C2211yescharacteristic
38C2×(C2×D127)⋊C4—21no—
39#16861143C2211yescharacteristic, complemented
40#16861142C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861133C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic, complemented
46#16861130C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic, complemented
52C2×#41×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic, complemented
54#117×C127C2211yescharacteristic, complemented
55#103×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C23×D127—21no—
60C2×D508—21no—
61C2×D508C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127D411yescharacteristic
67D4×D127—22no—
68D4×D127—41no—
69C22×Dic127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127C2311yescharacteristic
73#14523235—41no—
74D1016—41no—
75#14523233—22no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C8×D127—41no—
79C4⋊Dic127—21no—
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4D411yescharacteristic
86C22⋊Dic127—21no—
87C22⋊Dic127D411yescharacteristic
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—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127C2311yescharacteristic
97#41×C127—22no—
98#41×C127—41no—
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127—21no—
101C2×D4×C127C2311yescharacteristic
102C22×C4×C127—21no—
103C22×C4×C127C2311yescharacteristic
104C2×C8×C127—41no—
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C42×C127—21no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116C22×D127C2×D411yescharacteristic
117D508—21no—
118D508—21no—
119D508—21no—
120D508—21no—
121D508—42no—
122C22⋊D127—22no—
123C22⋊D127—42no—
124C22⋊D127—81no—
125C22⋊D127—81no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—81no—
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C4×D127—21no—
133C4×D127—21no—
134C4×D127—81no—
135C4×D127—81no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138C23×C127—21no—
139C23×C127C2×D411yescharacteristic
140Q8×C127—21no—
141Q8×C127—21no—
142D4×C127—21no—
143D4×C127—42no—
144D4×C127#4112yes—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—42no—
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2411yescharacteristic
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—21no—
156C22×C127—22no—
157C22×C127—21no—
158C22×C127—21no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127C22×D411yescharacteristic
162C2×D127—21no—
163C2×D127—21no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—84no—
171C2×D127#13512yes—
172C4×C127—21no—
173C4×C127—41no—
174C4×C127—41no—
175C4×C127—41no—
176C4×C127C2×#4111yescharacteristic
177C4×C127H(5,𝔽2)11yescharacteristic
178Dic127—21no—
179Dic127—21no—
180Dic127—21no—
181Dic127—41no—
182Dic127—41no—
183C2×C127—42no—
184C2×C127—41no—
185C2×C127#54611yescharacteristic
186C2×C127C2×#13511yescharacteristic
187C2×C127#57611yescharacteristic
188D127—22no—
189D127—81no—
190D127—81no—
191#3297—1271nomaximal, Sylow
192C127#329711yescharacteristic, complemented, Sylow
193#553—1271no—
194#546—1271no—
195#488—1271no—
196#484—1271no—
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#416D12711yescharacteristic, complemented
207C2×#102—1271no—
208C2×#41C2×D12711yescharacteristic
209C22×D4—1271no—
210C2×SD16—1271no—
211C2×D8—1271no—
212C2×D8—1271no—
213C22×C8—1271no—
214C22×C8C2×D12711yescharacteristic, complemented
215#126—2541no—
216#125—1271no—
217#124—1271no—
218#123—2541no—
219#123—2541no—
220C22⋊Q8—1271no—
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#117C2×D12711yescharacteristic, complemented
232C2×C4⋊C4—1271no—
233C2×C22⋊C4—2541no—
234#107—1271no—
235#107—1271no—
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#102C2×D12711yescharacteristic
247#102C2×D12711yescharacteristic
248C22⋊C8—1271no—
249C22⋊C8—1271no—
250C4×C8—1271no—
251C24—2541no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D12711yescharacteristic
255C2×D4—21no—
256C2×D4—1271no—
257C2×D4—1271no—
258C2×D4—1271no—
259C2×D4—5081no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4—2542no—
265C2×D4C22×D12711yescharacteristic
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—41no—
276C2×C8—1271no—
277C2×C8—1271no—
278C2×C8—5081no—
279C2×C8C22×D12711yescharacteristic
280C2×C8C22×D12711yescharacteristic
281C4⋊C4—1271no—
282C4⋊C4—1271no—
283C4⋊C4—1271no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
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—21no—
302C42—1271no—
303C23—21no—
304C23—1271no—
305C23—10161no—
306C23—5082no—
307C23—5082no—
308C23—2541no—
309C23—2541no—
310C23D4×D12711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—42no—
315D4—10161no—
316D4—10161no—
317D4—5082no—
318D4—5082no—
319D4—2542no—
320D4—2541no—
321D4—2541no—
322D4—2541no—
323D4—2541no—
324D4#1452324812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—1271no—
332C2×C4—1271no—
333C2×C4—1271no—
334C2×C4—10161no—
335C2×C4—10161no—
336C2×C4—10161no—
337C2×C4—2541no—
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—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—1272no—
354C22—10164no—
355C22—5081no—
356C22—5081no—
357C22—5081no—
358C22—5081no—
359C22—5081no—
360C22—5081no—
361C22—2541no—
362C22—2541no—
363C22C2×D4×D12711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—5081no—
369C4—5081no—
370C4—2541no—
371C4—2541no—
372C4—2541no—
373C4C2×#1452324811yescharacteristic
374C4#1686125411yescharacteristic
375C2—41no—
376C2—42no—
377C2—10161no—
378C2—10161no—
379C2—2542no—
380C2#2255051611yescharacteristic
381C2#135×D12711yescharacteristic
382C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
hypercenter#416D127
hypocenterC127#3297
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #22550570C2semidirect
2. #22550514C2semidirect
3. #22550232C2semidirect
4. #22550210C2semidirect
5. #22550200C2semidirect
6. #22550174C2semidirect
7. #22550112C2semidirect
8. #22550108C2semidirect
9. #22550094C2semidirect
10. #22550021C2semidirect
11. #22550011C2semidirect
12. #22549956C2semidirect
13. C8×C22⋊D127C2semidirect
14. #102×D127C2semidirect
15. #416×C127C2semidirect
17. C2×#14523235C22semidirect
22. #16861200C22semidirect
29. #16861183C22semidirect
35. #16861171C22semidirect
39. #16861143C22semidirect
40. #16861142C22semidirect
41. #16861140C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861133C22semidirect
45. #16861131C22semidirect
49. C22⋊(C127⋊C8)C22semidirect
51. C8×Dic127C22semidirect
53. C22×C8×C127C22semidirect
54. #117×C127C22semidirect
55. #103×C127C22semidirect
56. #103×C127C22semidirect
192. C127#3297semidirect
206. #416D127semidirect
214. C22×C8C2×D127semidirect
231. #117C2×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
#225505702
#225505142
#225502322
#225502102
#225502002
#225501742
#225501122
#225501082
#225500942
#225500212
#225500112
#225499562
C8×C22⋊D1272
#102×D1272
#416×C1272
#3297127