Finite Group Databasebeta

← #30650676

Subgroups — #30650676

Order 16256.

408 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550589C211yesmaximal, characteristic, complemented
2#22550545C211yesmaximal, characteristic, complemented
3#22550221C211yesmaximal, characteristic, complemented
4#22550200C211yesmaximal, characteristic, complemented
5#22550198C211yesmaximal, characteristic, complemented
6#22550169C211yesmaximal, characteristic, complemented
7#22550112C211yesmaximal, characteristic, complemented
8#22550111C211yesmaximal, characteristic, complemented
9#22550070C211yesmaximal, characteristic, complemented
10#22550024C211yesmaximal, characteristic, complemented
11#22550021C211yesmaximal, characteristic, complemented
12#22549961C211yesmaximal, characteristic, complemented
13C8×C22⋊D127C211yesmaximal, characteristic, complemented
14#22549871C211yesmaximal, characteristic, complemented
15#416×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18C2×D1016C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861208C2211yescharacteristic
22#16861205—21no—
23#16861205C2211yescharacteristic
24#16861197—21no—
25#16861195—21no—
26#16861184C2211yescharacteristic
27#16861183—21no—
28#16861183C2211yescharacteristic
29#16861183C2211yescharacteristic
30#16861182—21no—
31#16861180—21no—
32#16861180C2211yescharacteristic
33Q8×Dic127C2211yescharacteristic
34#16861171C2211yescharacteristic
35C4×C22⋊D127—21no—
36C4×C22⋊D127C2211yescharacteristic
37#16861167C2211yescharacteristic
38#16861166—21no—
39#16861143C2211yescharacteristic
40#16861142C2211yescharacteristic
41#16861140C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×#41×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54#117×C127C2211yescharacteristic
55#103×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523249—22no—
60#14523249—41no—
61C2×Dic254—21no—
62C2×D508—21no—
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127D411yescharacteristic
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2311yescharacteristic
75#14523235—22no—
76D1016—41no—
77#14523233—41no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—41no—
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
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)⋊C4D411yescharacteristic
94C4×Dic127—21no—
95C4×Dic127—21no—
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×D127C2×D411yescharacteristic
114Dic254—81no—
115D508—21no—
116D508—21no—
117D508—21no—
118D508—21no—
119D508—42no—
120D508—81no—
121C22⋊D127—22no—
122C22⋊D127—42no—
123C22⋊D127—81no—
124C22⋊D127—81no—
125C22⋊D127—81no—
126C22⋊D127—81no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C4×D127—21no—
133C4×D127—21no—
134C4×D127—42no—
135C4×D127—42no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138C23×C127—21no—
139C23×C127C2×D411yescharacteristic
140Q8×C127—21no—
141Q8×C127#4112yes—
142D4×C127—21no—
143D4×C127—21no—
144D4×C127—42no—
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—21no—
157C22×C127—21no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127C22×D411yescharacteristic
162C2×D127—21no—
163C2×D127—21no—
164C2×D127—21no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C4×C127—21no—
172C4×C127—22no—
173C4×C127—41no—
174C4×C127—41no—
175C4×C127C2×#4111yescharacteristic
176C4×C127H(5,𝔽2)11yescharacteristic
177Dic127—21no—
178Dic127—21no—
179Dic127—41no—
180Dic127—41no—
181Dic127#13512yes—
182C2×C127—41no—
183C2×C127—81no—
184C2×C127#54611yescharacteristic
185C2×C127C2×#13511yescharacteristic
186C2×C127#57611yescharacteristic
187D127—41no—
188D127—81no—
189D127—81no—
190#3305—1271nomaximal, Sylow
191C127#330511yescharacteristic, complemented, Sylow
192#555—1271no—
193#550—1271no—
194#486—1271no—
195#482—1271no—
196#482—1271no—
197#477—1271no—
198#466—1271no—
199#466—1271no—
200#459—1271no—
201#449—1271no—
202#449—1271no—
203C4×SD16—1271no—
204C8×D4—1271no—
205#416—1271no—
206#416D12711yescharacteristic, complemented
207C2×#41—1271no—
208C2×#41C2×D12711yescharacteristic
209C2×SD16—1271no—
210C2×D8—1271no—
211C2×D8—1271no—
212C22×C8—1271no—
213C22×C8C2×D12711yescharacteristic
214C4⋊Q8—1271no—
215#127—1271no—
216#127—2541no—
217#124—2541no—
218#124—2541no—
219#121—1271no—
220#121—1271no—
221#121—1271no—
222#121—1271no—
223#121—2541no—
224#121—2541no—
225#121—2541no—
226C4×Q8—1271no—
227C4×D4—1271no—
228C4×D4—1271no—
229C4×D4—2541no—
230#117—1271no—
231#117—2541no—
232#117C2×D12711yescharacteristic
233#107—1271no—
234#107—1271no—
235#106—1271no—
236C4⋊C8—1271no—
237#103—1271no—
238#103C2×D12711yescharacteristic
239#103C2×D12711yescharacteristic
240#102—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102—1271no—
245#102C2×D12711yescharacteristic
246#102C2×D12711yescharacteristic
247C22⋊C8—1271no—
248C22⋊C8—1271no—
249C4×C8—1271no—
250#41—22no—
251#41—41no—
252#41—5081no—
253#41—2542no—
254C2×Q8—2541no—
255C2×Q8C22×D12711yescharacteristic
256C2×D4—21no—
257C2×D4—1271no—
258C2×D4—1271no—
259C2×D4—1271no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4—2541no—
265C2×D4—2541no—
266C2×D4—2541no—
267C2×D4C22×D12711yescharacteristic
268C22×C4—21no—
269C22×C4—1271no—
270C22×C4—2541no—
271C22×C4—2541no—
272C22×C4C22×D12711yescharacteristic
273SD16—2542no—
274D8—5081no—
275D8—5081no—
276C2×C8—41no—
277C2×C8—1271no—
278C2×C8—1271no—
279C2×C8—5081no—
280C2×C8C22×D12711yescharacteristic
281C2×C8C22×D12711yescharacteristic
282C4⋊C4—1271no—
283C4⋊C4—1271no—
284C4⋊C4—1271no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4C22×D12711yescharacteristic
288C4⋊C4C22×D12711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—1271no—
291C22⋊C4—1271no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C42—21no—
299C42—1271no—
300C42—2541no—
301C42—2541no—
302C23—21no—
303C23—1271no—
304C23—2541no—
305C23—2541no—
306C23D4×D12711yescharacteristic
307Q8—21no—
308Q8—10161no—
309Q8#1452324812yes—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—10161no—
314D4—10161no—
315D4—10161no—
316D4—10161no—
317D4—10161no—
318D4—5082no—
319D4—5082no—
320D4—2542no—
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—42no—
331C2×C4—1271no—
332C2×C4—1271no—
333C2×C4—1271no—
334C2×C4—5082no—
335C2×C4—5082no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4D4×D12711yescharacteristic
341C2×C4C23×D12711yescharacteristic
342C8—21no—
343C8—21no—
344C8—2541no—
345C8—2541no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—5081no—
353C22—5081no—
354C22—5081no—
355C22—5081no—
356C22—5081no—
357C22—5081no—
358C22—2541no—
359C22—2541no—
360C22—2541no—
361C22C2×D4×D12711yescharacteristic
362C4—21no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—1272no—
367C4—5081no—
368C4—5081no—
369C4—2541no—
370C4—2541no—
371C4C2×#1452324811yescharacteristic
372C4#1686125411yescharacteristic
373C2—41no—
374C2—81no—
375C2—10161no—
376C2—10161no—
377C2—5081no—
378C2#2255051611yescharacteristic
379C2#135×D12711yescharacteristic
380C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
hypercenter#416D127
hypocenterC127#3305
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. #22550589C2semidirect
2. #22550545C2semidirect
3. #22550221C2semidirect
4. #22550200C2semidirect
5. #22550198C2semidirect
6. #22550169C2semidirect
7. #22550112C2semidirect
8. #22550111C2semidirect
9. #22550070C2semidirect
10. #22550024C2semidirect
11. #22550021C2semidirect
12. #22549961C2semidirect
13. C8×C22⋊D127C2semidirect
14. #22549871C2semidirect
15. #416×C127C2semidirect
191. C127#3305semidirect
206. #416D127semidirect

Maximal subgroups

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

subgroupindex
#225505892
#225505452
#225502212
#225502002
#225501982
#225501692
#225501122
#225501112
#225500702
#225500242
#225500212
#225499612
C8×C22⋊D1272
#225498712
#416×C1272
#3305127