Finite Group Databasebeta

← #30650439

Subgroups — #30650439

Order 16256.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861230C211yesmaximal, characteristic, complemented
2C2×SD16×D127C211yesmaximal, characteristic, complemented
3#22550538C211yesmaximal, characteristic, complemented
4#22550521C211yesmaximal, characteristic, complemented
5#22550260C211yesmaximal, characteristic, complemented
6#22550106C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550098C211yesmaximal, characteristic, complemented
9#22550091C211yesmaximal, characteristic, complemented
10#22550039C211yesmaximal, characteristic, complemented
11#22550033C211yesmaximal, characteristic, complemented
12#22550024C211yesmaximal, characteristic, complemented
13#22550011C211yesmaximal, characteristic, complemented
14C8×D508C211yesmaximal, characteristic, complemented
15C4×SD16×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic, complemented
17C2×#14523247C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19C2×D4×D127C2211yescharacteristic
20#16861230—24no—
21C2×Dic508C2211yescharacteristic, complemented
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic
24C2×#14523237C2211yescharacteristic, complemented
25C2×#14523236C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27SD16×D127—24no—
28C2×D1016C2211yescharacteristic
29C2×#14523233C2211yescharacteristic
30C2×C8×D127C2211yescharacteristic
31C2×C8×D127C2211yescharacteristic, complemented
32#16861197—21no—
33#16861197C2211yescharacteristic
34#16861197C2211yescharacteristic, complemented
35#16861193—21no—
36#16861189—21no—
37#16861189C2211yescharacteristic
38#16861189C2211yescharacteristic, complemented
39#16861184—21no—
40#16861184C2211yescharacteristic
41#16861184C2211yescharacteristic
42#16861183—21no—
43#16861179—21no—
44C4×D508—21no—
45C4×D508C2211yescharacteristic
46C2×(C2×D127)⋊C4—21no—
47C4⋊(C127⋊C8)C2211yescharacteristic, complemented
48#16861132C2211yescharacteristic, complemented
49#16861132C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic
51#16861129C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic
53(C2×D127)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic
55C4×Q8×C127C2211yescharacteristic, complemented
56C4×D4×C127C2211yescharacteristic
57#106×C127C2211yescharacteristic, complemented
58#103×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic
60C4×C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—41no—
64#14523247—22no—
65#14523247—41no—
66C2×Dic254—21no—
67C2×Dic254C2311yescharacteristic
68C2×Dic254C2311yescharacteristic
69Q8×D127—22no—
70Q8×D127—41no—
71C2×D508—21no—
72C2×D508—21no—
73C2×D508—21no—
74C2×D508D411yescharacteristic
75C2×D508C2311yescharacteristic
76C2×D508C2311yescharacteristic
77C2×C22⋊D127—21no—
78C2×C22⋊D127—21no—
79D4×D127—22no—
80D4×D127—41no—
81C22×Dic127—21no—
82C2×C4×D127—21no—
83C2×C4×D127—21no—
84C2×C4×D127C2311yescharacteristic
85C2×C4×D127C2311yescharacteristic
86Dic508—41no—
87#14523238—22no—
88#14523237—22no—
89#14523237—22no—
90#14523236—22no—
91#14523235—22no—
92D1016—41no—
93#14523233—22no—
94C2×C127⋊C8C2311yescharacteristic
95C2×C127⋊C8C2311yescharacteristic
96C8×D127—22no—
97C8×D127—22no—
98C4⋊Dic127—21no—
99C4⋊Dic127—21no—
100C4⋊Dic127D411yescharacteristic
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4—21no—
109(C2×D127)⋊C4D411yescharacteristic
110(C2×D127)⋊C4D411yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127—21no—
114SD16×C127D414yescomplemented
115C2×C8×C127C2311yescharacteristic
116C2×C8×C127C2311yescharacteristic
117C4⋊C4×C127—21no—
118C4⋊C4×C127C2311yescharacteristic
119C4⋊C4×C127C2311yescharacteristic
120C22⋊C4×C127—21no—
121C42×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—81no—
128C22×D127C2×D411yescharacteristic
129C22×D127C2×D411yescharacteristic
130Dic254—21no—
131Dic254—21no—
132Dic254—21no—
133Dic254—21no—
134Dic254—42no—
135D508—22no—
136D508—21no—
137D508—21no—
138D508—21no—
139D508—21no—
140D508—42no—
141D508—42no—
142D508—81no—
143C22⋊D127—42no—
144C22⋊D127—42no—
145C2×Dic127—21no—
146C2×Dic127—21no—
147C2×Dic127—42no—
148C2×Dic127C2×D411yescharacteristic
149C2×Dic127C2×D411yescharacteristic
150C4×D127—21no—
151C4×D127—21no—
152C4×D127—21no—
153C4×D127—21no—
154C4×D127—42no—
155C4×D127—42no—
156C127⋊C8—21no—
157C127⋊C8—21no—
158C23×C127—21no—
159Q8×C127—21no—
160Q8×C127C2×D412yes—
161D4×C127—21no—
162D4×C127C2×D412yes—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—81no—
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127C2×D412yes—
172C22×C127—22no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127C22×D411yescharacteristic
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—21no—
179C2×D127—21no—
180C2×D127—21no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—84no—
188C2×D127#13512yescomplemented
189C4×C127—22no—
190C4×C127—21no—
191C4×C127—21no—
192C4×C127—41no—
193C4×C127—41no—
194C4×C127C22×D411yescharacteristic
195C4×C127H(5,𝔽2)11yescharacteristic
196Dic127—21no—
197Dic127—21no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—42no—
201C2×C127D4211yescharacteristic
202C2×C127C2×#13511yescharacteristic
203C2×C127#57711yescharacteristic
204D127—22no—
205D127—41no—
206D127—81no—
207D127—81no—
208#3282—1271nomaximal, Sylow
209C127#328211yescharacteristic, complemented, Sylow
210C2×#135—1271no—
211C22×SD16—1271no—
212#548—1271no—
213#546—1271no—
214#495—1271no—
215#465—1271no—
216#465—1271no—
217#464—1271no—
218#463—1271no—
219#452—1271no—
220#451—1271no—
221#449—1271no—
222#447—1271no—
223C4×SD16D12711yescharacteristic, complemented
224C8×D4—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C22×Q8—1271no—
228C22×D4—1271no—
229#135—2544no—
230C2×Q16—1271no—
231C2×Q16—1271no—
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16—2544no—
237C2×SD16C2×D12711yescharacteristic
238C2×D8—1271no—
239C2×D8—1271no—
240C22×C8—1271no—
241C22×C8—1271no—
242#124—1271no—
243#124—1271no—
244#124—2541no—
245#123—2541no—
246C22⋊Q8—1271no—
247C22⋊Q8—1271no—
248C22⋊Q8—2541no—
249#121—1271no—
250#121—1271no—
251#121—2541no—
252#121—2541no—
253#120—2541no—
254C4×Q8C2×D12711yescharacteristic, complemented
255C4×D4—1271no—
256C4×D4—2541no—
257C4×D4C2×D12711yescharacteristic
258C2×C22⋊C4—2541no—
259#106C2×D12711yescharacteristic, complemented
260C4⋊C8—1271no—
261#103—1271no—
262#103—1271no—
263#103C2×D12711yescharacteristic, complemented
264#102—1271no—
265#102—1271no—
266#102C2×D12711yescharacteristic
267C22⋊C8—1271no—
268C22⋊C8—1271no—
269C4×C8C2×D12711yescharacteristic, complemented
270C24—2541no—
271#41—5081no—
272#41—5081no—
273#41—2542no—
274#41—2542no—
275C2×Q8—1271no—
276C2×Q8—1271no—
277C2×Q8—5081no—
278C2×Q8—2541no—
279C2×Q8—2542no—
280C2×Q8C22×D12711yescharacteristic
281C2×D4—1271no—
282C2×D4—1271no—
283C2×D4—1271no—
284C2×D4—5081no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4—2542no—
291C2×D4C22×D12711yescharacteristic
292C22×C4—21no—
293C22×C4—1271no—
294C22×C4—1271no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4—2541no—
298Q16—5081no—
299Q16—2542no—
300SD16—2542no—
301SD16—2542no—
302SD16—2542no—
303SD16—2542no—
304SD16D50814yescomplemented
305D8—5081no—
306D8—2542no—
307C2×C8—1271no—
308C2×C8—1271no—
309C2×C8—2542no—
310C2×C8—2542no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—1271no—
315C4⋊C4—2541no—
316C4⋊C4—2541no—
317C4⋊C4C22×D12711yescharacteristic
318C4⋊C4C22×D12711yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—1271no—
321C22⋊C4—1271no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C42—21no—
331C42C22×D12711yescharacteristic
332C23—21no—
333C23—1271no—
334C23—1271no—
335C23—10161no—
336C23—5082no—
337C23—5082no—
338C23—2541no—
339C23—2541no—
340Q8—21no—
341Q8—5082no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345Q8—2541no—
346Q8C2×D50812yes—
347D4—21no—
348D4—10161no—
349D4—5082no—
350D4—5082no—
351D4—5082no—
352D4—5082no—
353D4—2542no—
354D4—2541no—
355D4—2541no—
356D4—2541no—
357D4—2541no—
358D4C2×D50812yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—81no—
363C2×C4—1271no—
364C2×C4—1271no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—2541no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4D4×D12711yescharacteristic
375C2×C4D4×D12711yescharacteristic
376C2×C4C23×D12711yescharacteristic
377C8—21no—
378C8—2541no—
379C8—2541no—
380C8C2×D50812yes—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—1272no—
385C22—10164no—
386C22—5081no—
387C22—5081no—
388C22—5081no—
389C22—5081no—
390C22—5081no—
391C22—5081no—
392C22—2541no—
393C22—2541no—
394C22—2541no—
395C22—2541no—
396C22—2541no—
397C22C2×D4×D12711yescharacteristic
398C4—21no—
399C4—21no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—5081no—
404C4—5081no—
405C4—2541no—
406C4—2541no—
407C4C22×D50811yescharacteristic
408C4#1686125511yescharacteristic
409C2—42no—
410C2—10161no—
411C2—10161no—
412C2—5081no—
413C2—2542no—
414C2D4×D50811yescharacteristic
415C2#135×D12711yescharacteristic
416C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×SD16×C127C2
socleC22×C127C22×D4
hypercenterC4×SD16D127
hypocenterC127#3282
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16861230C2semidirect
2. C2×SD16×D127C2semidirect
3. #22550538C2semidirect
4. #22550521C2semidirect
5. #22550260C2semidirect
6. #22550106C2semidirect
7. #22550106C2semidirect
8. #22550098C2semidirect
9. #22550091C2semidirect
10. #22550039C2semidirect
11. #22550033C2semidirect
12. #22550024C2semidirect
13. #22550011C2semidirect
14. C8×D508C2semidirect
15. C4×SD16×C127C2semidirect
16. C2×#14523249C22semidirect
21. C2×Dic508C22semidirect
22. C2×#14523238C22semidirect
24. C2×#14523237C22semidirect
31. C2×C8×D127C22semidirect
34. #16861197C22semidirect
38. #16861189C22semidirect
47. C4⋊(C127⋊C8)C22semidirect
48. #16861132C22semidirect
49. #16861132C22semidirect
51. #16861129C22semidirect
53. (C2×D127)⋊C8C22semidirect
55. C4×Q8×C127C22semidirect
57. #106×C127C22semidirect
58. #103×C127C22semidirect
60. C4×C8×C127C22semidirect
114. SD16×C127D4semidirect
188. C2×D127#135semidirect
209. C127#3282semidirect
223. C4×SD16D127semidirect
254. C4×Q8C2×D127semidirect
259. #106C2×D127semidirect
263. #103C2×D127semidirect
269. C4×C8C2×D127semidirect
304. SD16D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612302
C2×SD16×D1272
#225505382
#225505212
#225502602
#225501062
#225500982
#225500912
#225500392
#225500332
#225500242
#225500112
C8×D5082
C4×SD16×C1272
#3282127