Finite Group Databasebeta

← #30650735

Subgroups — #30650735

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#22550579C211yesmaximal, characteristic, complemented
2#22550546C211yesmaximal, characteristic, complemented
3#22550198C211yesmaximal, characteristic, complemented
4#22550188C211yesmaximal, characteristic, complemented
5#22550180C211yesmaximal, characteristic, complemented
6#22550128C211yesmaximal, characteristic, complemented
7#22550111C211yesmaximal, characteristic, complemented
8#22550068C211yesmaximal, characteristic, complemented
9#22550022C211yesmaximal, characteristic, complemented
10#22550022C211yesmaximal, characteristic, complemented
11#22549978C211yesmaximal, characteristic, complemented
12#22549949C211yesmaximal, characteristic, complemented
13#22549874C211yesmaximal, characteristic, complemented
14#22549869C211yesmaximal, characteristic, complemented
15#490×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C2×#14523231C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861205—21no—
24#16861204—21no—
25#16861194—21no—
26#16861184—21no—
27#16861184C2211yescharacteristic
28#16861184C2211yescharacteristic
29#16861181—21no—
30#16861181—21no—
31#16861180C2211yescharacteristic
32#16861180C2211yescharacteristic
33C4×Dic254C2211yescharacteristic
34C4×D508C2211yescharacteristic
35C4×D508C2211yescharacteristic
36D4×Dic127—21no—
37#16861171—21no—
38#16861167C2211yescharacteristic
39C4⋊C4×D127C2211yescharacteristic
40#16861143C2211yescharacteristic
41#16861142C2211yescharacteristic
42#16861140C2211yescharacteristic
43C4⋊(C127⋊C8)C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52#127×C127C2211yescharacteristic
53#125×C127C2211yescharacteristic
54#102×C127C2211yescharacteristic
55#102×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—22no—
61#14523247—41no—
62#14523247—41no—
63C2×Dic254C2311yescharacteristic
64C2×D508—21no—
65C2×D508—21no—
66C2×D508D411yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C22×Dic127—21no—
73C22×Dic127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127C2311yescharacteristic
77C2×C4×D127C2311yescharacteristic
78#14523237—22no—
79#14523233—41no—
80#14523233—41no—
81#14523231—41no—
82C2×C127⋊C8C2311yescharacteristic
83C2×C127⋊C8C2311yescharacteristic
84C8×D127—41no—
85C4⋊Dic127D411yescharacteristic
86C4⋊Dic127C2311yescharacteristic
87C4⋊Dic127C2311yescharacteristic
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4D411yescharacteristic
97(C2×D127)⋊C4D411yescharacteristic
98C4×Dic127—21no—
99C4×Dic127—21no—
100C2×D4×C127—21no—
101C2×D4×C127—21no—
102C2×D4×C127C2311yescharacteristic
103C2×D4×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127—21no—
107C4⋊C4×C127—21no—
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127C2×D411yescharacteristic
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254#4112yes—
116D508—21no—
117D508—21no—
118D508—41no—
119D508—42no—
120D508—81no—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C22⋊D127—81no—
124C22⋊D127—81no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—42no—
129C2×Dic127—42no—
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C4×D127—21no—
133C4×D127—21no—
134C4×D127—21no—
135C4×D127—21no—
136C4×D127—81no—
137C4×D127—81no—
138C127⋊C8—21no—
139C127⋊C8—21no—
140C23×C127—21no—
141C23×C127—21no—
142D4×C127—21no—
143D4×C127—21no—
144D4×C127—21no—
145D4×C127—21no—
146D4×C127—42no—
147D4×C127—81no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2411yescharacteristic
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127C22×D411yescharacteristic
162C2×D127—21no—
163C2×D127—21no—
164C2×D127—21no—
165C2×D127—21no—
166C2×D127—21no—
167C2×D127—21no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C4×C127—21no—
172C4×C127—21no—
173C4×C127—41no—
174C4×C127—41no—
175C4×C127C2×#4111yescharacteristic
176C4×C127H(5,𝔽2)11yescharacteristic
177Dic127—21no—
178Dic127—21no—
179Dic127—22no—
180Dic127—41no—
181Dic127—41no—
182C2×C127—81no—
183C2×C127—81no—
184C2×C127#54611yescharacteristic
185C2×C127#57511yescharacteristic
186C2×C127#57611yescharacteristic
187D127—41no—
188D127—41no—
189D127—81no—
190#3309—1271nomaximal, Sylow
191C127#330911yescharacteristic, complemented, Sylow
192#553—1271no—
193#550—1271no—
194#490D12711yescharacteristic, complemented
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—2541no—
215#127C2×D12711yescharacteristic
216#126—2541no—
217#125C2×D12711yescharacteristic
218#123—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—
230C4×D4—2541no—
231#117—1271no—
232C2×C4⋊C4—1271no—
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#102C2×D12711yescharacteristic
244#102C2×D12711yescharacteristic
245#102C2×D12711yescharacteristic
246#102C2×D12711yescharacteristic
247C22⋊C8—1271no—
248C22⋊C8—1271no—
249#97C2×D12711yescharacteristic
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—1271no—
268C22×C4—1271no—
269C22×C4—2541no—
270C22×C4—2541no—
271C22×C4—2541no—
272C22×C4—2541no—
273SD16—2542no—
274D8—5081no—
275D8—5081no—
276M16—5081no—
277C2×C8—1271no—
278C2×C8—1271no—
279C2×C8—5081no—
280C2×C8C22×D12711yescharacteristic
281C2×C8C22×D12711yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—1271no—
285C4⋊C4—1271no—
286C4⋊C4—1271no—
287C4⋊C4—2541no—
288C4⋊C4—2541no—
289C4⋊C4C22×D12711yescharacteristic
290C4⋊C4C22×D12711yescharacteristic
291C22⋊C4—1271no—
292C22⋊C4—1271no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C42—2541no—
300C42—2541no—
301C42C22×D12711yescharacteristic
302C23—21no—
303C23—21no—
304C23—1271no—
305C23—1271no—
306C23—2541no—
307Q8—1272no—
308Q8—2541no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—81no—
315D4—10161no—
316D4—10161no—
317D4—10161no—
318D4—5081no—
319D4—5082no—
320D4—5082no—
321D4—5082no—
322D4—2541no—
323D4—2541no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—1271no—
327C2×C4—1271no—
328C2×C4—10161no—
329C2×C4—10161no—
330C2×C4—5082no—
331C2×C4—5082no—
332C2×C4—2541no—
333C2×C4—2541no—
334C2×C4—2541no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4D4×D12711yescharacteristic
340C2×C4D4×D12711yescharacteristic
341C2×C4C23×D12711yescharacteristic
342C8—21no—
343C8—21no—
344C8—2541no—
345C8—2541no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—5081no—
353C22—5081no—
354C22—5081no—
355C22—2541no—
356C22—2541no—
357C22—2541no—
358C22—2541no—
359C22—2541no—
360C22—2541no—
361C22C2×D4×D12711yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—5081no—
367C4—5081no—
368C4—2541no—
369C4—2541no—
370C4—2542no—
371C4#41×D12711yescharacteristic
372C4#1686125511yescharacteristic
373C2—81no—
374C2—81no—
375C2—10161no—
376C2—5081no—
377C2—5081no—
378C2#2255051911yescharacteristic
379C2#2255067211yescharacteristic
380C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#490×C127C2
socleC22×C127C22×D4
hypercenter#490D127
hypocenterC127#3309
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #22550579C2semidirect
2. #22550546C2semidirect
3. #22550198C2semidirect
4. #22550188C2semidirect
5. #22550180C2semidirect
6. #22550128C2semidirect
7. #22550111C2semidirect
8. #22550068C2semidirect
9. #22550022C2semidirect
10. #22550022C2semidirect
11. #22549978C2semidirect
12. #22549949C2semidirect
13. #22549874C2semidirect
14. #22549869C2semidirect
15. #490×C127C2semidirect
191. C127#3309semidirect
194. #490D127semidirect

Maximal subgroups

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

subgroupindex
#225505792
#225505462
#225501982
#225501882
#225501802
#225501282
#225501112
#225500682
#225500222
#225499782
#225499492
#225498742
#225498692
#490×C1272
#3309127