Finite Group Databasebeta

← #30650730

Subgroups — #30650730

Order 16256.

404 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#22550547C211yesmaximal, characteristic, complemented
2#22550242C211yesmaximal, characteristic, complemented
3#22550200C211yesmaximal, characteristic, complemented
4#22550189C211yesmaximal, characteristic, complemented
5#22550180C211yesmaximal, characteristic, complemented
6#22550127C211yesmaximal, characteristic, complemented
7#22550110C211yesmaximal, characteristic, complemented
8#22550071C211yesmaximal, characteristic, complemented
9#22550019C211yesmaximal, characteristic, complemented
10#22550019C211yesmaximal, characteristic, complemented
11#22549976C211yesmaximal, characteristic, complemented
12#22549948C211yesmaximal, characteristic, complemented
13#22549875C211yesmaximal, characteristic, complemented
14#22549870C211yesmaximal, characteristic, complemented
15#553×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C2×#14523232C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23#16861206C2211yescharacteristic
24#16861205—21no—
25#16861183C2211yescharacteristic
26#16861183C2211yescharacteristic
27#16861182—21no—
28#16861182—21no—
29C4×Dic254C2211yescharacteristic
30#16861171—21no—
31#16861142C2211yescharacteristic
32#16861142C2211yescharacteristic
33#16861140C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35#16861132C2211yescharacteristic
36#16861132C2211yescharacteristic
37#16861132C2211yescharacteristic
38#16861130C2211yescharacteristic
39#16861129C2211yescharacteristic
40#16861129C2211yescharacteristic
41#16861129C2211yescharacteristic
42#16861129C2211yescharacteristic
43#16861129C2211yescharacteristic
44#16861129C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46C22⋊(C127⋊C8)C2211yescharacteristic
47#16861120C2211yescharacteristic
48#126×C127—21no—
49#125×C127C2211yescharacteristic
50#123×C127—21no—
51#121×C127—21no—
52#121×C127C2211yescharacteristic
53#121×C127C2211yescharacteristic
54C4×D4×C127—21no—
55C4×D4×C127C2211yescharacteristic
56C4×D4×C127C2211yescharacteristic
57#117×C127C2211yescharacteristic
58C2×C4⋊C4×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—22no—
61#14523248—41no—
62#14523248—41no—
63C2×Dic254C2311yescharacteristic
64C2×D508—21no—
65C2×D508—21no—
66C2×D508C2311yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74#14523236—22no—
75#14523233—41no—
76#14523233—41no—
77#14523232—41no—
78C2×C127⋊C8—41no—
79C2×C127⋊C8C2311yescharacteristic
80C2×C127⋊C8C2311yescharacteristic
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87C4×Dic127—21no—
88C2×D4×C127—21no—
89C2×D4×C127—21no—
90C2×D4×C127D411yescharacteristic
91C2×D4×C127C2311yescharacteristic
92C22×C4×C127—21no—
93C22×C4×C127—21no—
94C22×C4×C127C2311yescharacteristic
95C22×C4×C127C2311yescharacteristic
96C4⋊C4×C127—21no—
97C4⋊C4×C127—21no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127D411yescharacteristic
100C4⋊C4×C127C2311yescharacteristic
101C4⋊C4×C127C2311yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C22⋊C4×C127—21no—
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C22⋊C4×C127D411yescharacteristic
108C22⋊C4×C127D411yescharacteristic
109C42×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127—21no—
113Dic254—21no—
114Dic254#4112yes—
115D508—21no—
116D508—21no—
117D508—21no—
118D508—21no—
119D508—42no—
120D508—81no—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C22⋊D127—81no—
124C22⋊D127—81no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C4×D127—42no—
128C4×D127—42no—
129C127⋊C8—21no—
130C127⋊C8—21no—
131C127⋊C8—21no—
132C127⋊C8—21no—
133C23×C127—21no—
134C23×C127C2×D411yescharacteristic
135C23×C127C2×D411yescharacteristic
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—41no—
139D4×C127—42no—
140D4×C127—81no—
141C2×C4×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×C127—81no—
149C2×C4×C127—81no—
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2×D411yescharacteristic
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2411yescharacteristic
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127C22×D411yescharacteristic
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C4×C127—21no—
172C4×C127—21no—
173C4×C127—21no—
174C4×C127—21no—
175C4×C127—41no—
176C4×C127—41no—
177C4×C127—41no—
178C4×C127C2×#4111yescharacteristic
179C4×C127H(5,𝔽2)11yescharacteristic
180Dic127—22no—
181Dic127—41no—
182C2×C127—41no—
183C2×C127—41no—
184C2×C127—81no—
185C2×C127#54611yescharacteristic
186C2×C127#57511yescharacteristic
187C2×C127#57611yescharacteristic
188D127—81no—
189D127—81no—
190#3309—1271nomaximal, Sylow
191C127#330911yescharacteristic, complemented, Sylow
192#553D12711yescharacteristic, complemented
193#550—1271no—
194#490—1271no—
195#482—1271no—
196#480—1271no—
197#479—1271no—
198#469—1271no—
199#466—1271no—
200#459—1271no—
201#449—1271no—
202#449—1271no—
203#440—1271no—
204#435—1271no—
205#417—1271no—
206#416—1271no—
207C2×#41—1271no—
208C2×#41—1271no—
209C2×SD16—1271no—
210C2×D8—1271no—
211C2×D8—1271no—
212C2×M16—1271no—
213C22×C8—1271no—
214#127—1271no—
215#127—2541no—
216#126—21no—
217#125C2×D12711yescharacteristic
218#123—21no—
219#121—21no—
220#121—1271no—
221#121—1271no—
222#121—2541no—
223#121—2541no—
224#121C2×D12711yescharacteristic
225#121C2×D12711yescharacteristic
226C4×Q8—1271no—
227C4×D4—21no—
228C4×D4—2541no—
229C4×D4C2×D12711yescharacteristic
230C4×D4C2×D12711yescharacteristic
231#117C2×D12711yescharacteristic
232C2×C4⋊C4C2×D12711yescharacteristic
233#107—1271no—
234#107—1271no—
235#106—1271no—
236C4⋊C8—1271no—
237#103—1271no—
238#103—1271no—
239#103—1271no—
240#102—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247C22⋊C8—1271no—
248C22⋊C8—1271no—
249#97—1271no—
250#41—5081no—
251#41—5081no—
252#41—2542no—
253#41—2542no—
254C2×Q8—1271no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—1271no—
258C2×D4—1271no—
259C2×D4—2541no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4—2541no—
265C2×D4C22⋊D12711yescharacteristic
266C2×D4C22×D12711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2541no—
270C22×C4—2541no—
271C22×C4C22×D12711yescharacteristic
272C22×C4C22×D12711yescharacteristic
273SD16—2542no—
274D8—5081no—
275D8—5081no—
276M16—5081no—
277C2×C8—1271no—
278C2×C8—1271no—
279C2×C8—1271no—
280C2×C8—1271no—
281C2×C8—5081no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—1271no—
286C4⋊C4—2541no—
287C4⋊C4C22⋊D12711yescharacteristic
288C4⋊C4C22×D12711yescharacteristic
289C4⋊C4C22×D12711yescharacteristic
290C4⋊C4C22×D12711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4C22⋊D12711yescharacteristic
298C22⋊C4C22⋊D12711yescharacteristic
299C42—21no—
300C42—2541no—
301C42C22×D12711yescharacteristic
302C23—21no—
303C23—2541no—
304C23—2541no—
305C23C2×C22⋊D12711yescharacteristic
306C23C2×C22⋊D12711yescharacteristic
307Q8—1272no—
308Q8—2541no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—42no—
313D4—81no—
314D4—10161no—
315D4—10161no—
316D4—10161no—
317D4—5082no—
318D4—5082no—
319D4—5082no—
320D4—2541no—
321D4—2541no—
322D4—2541no—
323D4—2541no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—5082no—
334C2×C4—5082no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4C2×C22⋊D12711yescharacteristic
338C2×C4C2×C22⋊D12711yescharacteristic
339C2×C4C2×C22⋊D12711yescharacteristic
340C2×C4C2×C22⋊D12711yescharacteristic
341C2×C4C23×D12711yescharacteristic
342C8—2541no—
343C8—2541no—
344C8—2541no—
345C8—2541no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—5081no—
356C22—5081no—
357C22—5081no—
358C22—5081no—
359C22—5081no—
360C22—5081no—
361C22C22×C22⋊D12711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—5081no—
370C4—2542no—
371C4#41×D12711yescharacteristic
372C4#1686125511yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—10161no—
377C2—10161no—
378C2#2255051411yescharacteristic
379C2#2255067411yescharacteristic
380C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#553×C127C2
socleC22×C127C22×D4
hypercenter#553D127
hypocenterC127#3309
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #22550547C2semidirect
2. #22550242C2semidirect
3. #22550200C2semidirect
4. #22550189C2semidirect
5. #22550180C2semidirect
6. #22550127C2semidirect
7. #22550110C2semidirect
8. #22550071C2semidirect
9. #22550019C2semidirect
10. #22550019C2semidirect
11. #22549976C2semidirect
12. #22549948C2semidirect
13. #22549875C2semidirect
14. #22549870C2semidirect
15. #553×C127C2semidirect
191. C127#3309semidirect
192. #553D127semidirect

Maximal subgroups

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

subgroupindex
#225505472
#225502422
#225502002
#225501892
#225501802
#225501272
#225501102
#225500712
#225500192
#225499762
#225499482
#225498752
#225498702
#553×C1272
#3309127