Finite Group Databasebeta

← #468×D127 #30648314

Subgroups — #468×D127 #30648314

Order 16256.

544 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D127C211yesmaximal, characteristic, complemented
2C2×M16×D127C211yesmaximal, characteristic, complemented
3C22⋊Q8×D127C211yesmaximal, characteristic, complemented
4#121×D127C211yesmaximal, characteristic
5#22550125C211yesmaximal, characteristic, complemented
6#22550124C211yesmaximal, characteristic, complemented
7#22550123C211yesmaximal, characteristic, complemented
8#22550122C211yesmaximal, characteristic, complemented
9#22550121C211yesmaximal, characteristic, complemented
10#22550120C211yesmaximal, characteristic, complemented
11#22550119C211yesmaximal, characteristic, complemented
12#107×D127C211yesmaximal, characteristic, complemented
13#103×D127C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#468×C127C211yesmaximal, characteristic, complemented
16C2×Q8×D127C2211yescharacteristic
17C2×D4×D127—21no—
18C2×D4×D127C2211yescharacteristic
19C22×C4×D127C2211yescharacteristic
20C2×#14523237C2211yescharacteristic, complemented
21C2×#14523236C2211yescharacteristic, complemented
22C2×#14523235C2211yescharacteristic, complemented
23SD16×D127—24no—
24C2×#14523232C2211yescharacteristic, complemented
25C2×#14523231C2211yescharacteristic, complemented
26C2×#14523231C2211yescharacteristic, complemented
27M16×D127—24no—
28C2×C8×D127C2211yescharacteristic
29C2×C8×D127C2211yescharacteristic
30#16861189C2211yescharacteristic, complemented
31#16861188C2211yescharacteristic, complemented
32#16861187—21no—
33#16861186—21no—
34#16861185C2211yescharacteristic, complemented
35#16861184C2211yescharacteristic
36#16861183C2211yescharacteristic
37#16861182—21no—
38#16861181—21no—
39#16861180C2211yescharacteristic
40C4⋊C4×D127—21no—
41C4⋊C4×D127C2211yescharacteristic
42C4⋊C4×D127C2211yescharacteristic
43C22⋊C4×D127—21no—
44C22⋊C4×D127—21no—
45#16861143C2211yescharacteristic, complemented
46#16861142C2211yescharacteristic, complemented
47#16861142C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861133C2211yescharacteristic, complemented
50#16861132C2211yescharacteristic, complemented
51#16861131C2211yescharacteristic, complemented
52#16861130C2211yescharacteristic, complemented
53#16861129C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×M16×C127C2211yescharacteristic, complemented
56C22⋊Q8×C127C2211yescharacteristic, complemented
57#121×C127C2211yescharacteristic
58#107×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127D411yescharacteristic
63C2×Dic254—21no—
64C2×Dic254C2311yescharacteristic
65Q8×D127—22no—
66Q8×D127—22no—
67C2×D508—21no—
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72D4×D127—22no—
73D4×D127—22no—
74D4×D127—44no—
75C22×Dic127D411yescharacteristic
76C2×C4×D127—22no—
77C2×C4×D127—22no—
78C2×C4×D127—21no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127D411yescharacteristic
82C2×C4×D127D411yescharacteristic
83C2×C4×D127C2311yescharacteristic
84C2×C4×D127C2311yescharacteristic
85#14523237—22no—
86#14523236—22no—
87#14523235—22no—
88#14523232—22no—
89#14523231—22no—
90#14523231—22no—
91C2×C127⋊C8C2311yescharacteristic
92C2×C127⋊C8C2311yescharacteristic
93C8×D127—22no—
94C8×D127D414yescomplemented
95C4⋊Dic127—21no—
96C4⋊Dic127C2311yescharacteristic
97C4⋊Dic127C2311yescharacteristic
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101Dic127⋊C4—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127—21no—
110C2×D4×C127C2311yescharacteristic
111C22×C4×C127C2311yescharacteristic
112SD16×C127—22no—
113M16×C127—22no—
114C2×C8×C127C2311yescharacteristic
115C2×C8×C127C2311yescharacteristic
116C4⋊C4×C127—21no—
117C4⋊C4×C127C2311yescharacteristic
118C4⋊C4×C127C2311yescharacteristic
119C22⋊C4×C127—21no—
120C22⋊C4×C127—21no—
121C22×D127—22no—
122C22×D127—22no—
123C22×D127—22no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127C2×D411yescharacteristic
129C22×D127C2×D411yescharacteristic
130Dic254—21no—
131Dic254—21no—
132Dic254—42no—
133D508—21no—
134D508—21no—
135D508—42no—
136C22⋊D127—42no—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C2×Dic127—22no—
140C2×Dic127—21no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C4×D127—22no—
146C4×D127—21no—
147C4×D127—21no—
148C4×D127—22no—
149C4×D127—42no—
150C4×D127—42no—
151C4×D127—42no—
152C4×D127C2×D412yes—
153C4×D127#4112yes—
154C127⋊C8—21no—
155C127⋊C8C2×D412yescomplemented
156C23×C127—21no—
157C23×C127C2×D411yescharacteristic
158Q8×C127—21no—
159Q8×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127—42no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127C2×D412yescomplemented
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127C22×D411yescharacteristic
179C2×D127—21no—
180C2×D127—21no—
181C2×D127—21no—
182C2×D127—44no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—84no—
187C2×D127#12112yes—
188C2×D127AGL(1,ℤ/8)12yes—
189C2×D127#13712yes—
190C4×C127—21no—
191C4×C127—41no—
192C4×C127—41no—
193C4×C127—41no—
194C4×C127C22×D411yescharacteristic
195C4×C127C2×#4111yescharacteristic
196Dic127—21no—
197Dic127—41no—
198Dic127—41no—
199Dic127—41no—
200Dic127#12112yes—
201C2×C127—41no—
202C2×C127—81no—
203C2×C127C2×#12111yescharacteristic
204C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C127C2×#13711yescharacteristic
206D127—41no—
207D127—81no—
208D127#46814yesdirect product factor
209C2×#468—1271nomaximal, Sylow
210C127C2×#46811yescharacteristic, complemented, Sylow
211C22×SD16—1271no—
212C22×M16—1271no—
213C2×C22⋊Q8—1271no—
214C2×#121—1271no—
215#468—1271no—
216#468—1271no—
217#468—1271no—
218#468—1271no—
219#468—1271no—
220#468—1271no—
221#468—1271no—
222#468D12711yescharacteristic, direct product factor
223C2×#107—1271no—
224C2×#103—1271no—
225C2×#102—1271no—
226C22×Q8—1271no—
227C22×D4—1271no—
228C22×D4—2541no—
229C23×C4—1271no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×SD16—1271no—
233C2×SD16—2544no—
234C2×SD16C2×D12711yescharacteristic, complemented
235C2×M16—1271no—
236C2×M16—1271no—
237C2×M16—1271no—
238C2×M16—2544no—
239C2×M16C2×D12711yescharacteristic, complemented
240C22×C8—1271no—
241C22×C8—1271no—
242C22⋊Q8—1271no—
243C22⋊Q8—1271no—
244C22⋊Q8—1271no—
245C22⋊Q8—2541no—
246C22⋊Q8—2541no—
247C22⋊Q8C2×D12711yescharacteristic, complemented
248#121—1271no—
249#121—1271no—
250#121—1271no—
251#121—2541no—
252#121—2541no—
253#121C2×D12711yescharacteristic
254C2×C4⋊C4—1271no—
255C2×C4⋊C4—1271no—
256C2×C4⋊C4—2541no—
257C2×C22⋊C4—2541no—
258C2×C22⋊C4—2541no—
259#107—1271no—
260#107—1271no—
261#107—1271no—
262#107C2×D12711yescharacteristic, complemented
263#103—1271no—
264#103—1271no—
265#103—1271no—
266#103C2×D12711yescharacteristic, complemented
267#102—1271no—
268#102—1271no—
269#102—1271no—
270#102C2×D12711yescharacteristic, complemented
271C24—1271no—
272C24—2541no—
273C2×Q8—1271no—
274C2×Q8—2541no—
275C2×Q8—2542no—
276C2×Q8—2542no—
277C2×Q8C22×D12711yescharacteristic
278C2×D4—21no—
279C2×D4—1271no—
280C2×D4—5084no—
281C2×D4—2541no—
282C2×D4—2542no—
283C2×D4—2542no—
284C2×D4—2541no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4C22×D12711yescharacteristic
288C22×C4—1271no—
289C22×C4—1271no—
290C22×C4—1271no—
291C22×C4—1271no—
292C22×C4—1271no—
293C22×C4—2542no—
294C22×C4—2542no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4—2541no—
298C22×C4C22×D12711yescharacteristic
299SD16—22no—
300SD16—2542no—
301SD16—2542no—
302SD16—2542no—
303M16—22no—
304M16—2542no—
305M16—2542no—
306M16—2542no—
307C2×C8—1271no—
308C2×C8—1274no—
309C2×C8—1271no—
310C2×C8—2542no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—1271no—
315C4⋊C4—1271no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4—2541no—
320C4⋊C4—2541no—
321C4⋊C4C22×D12711yescharacteristic
322C4⋊C4C22×D12711yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C22⋊C4—2541no—
331C23—21no—
332C23—1271no—
333C23—1271no—
334C23—5082no—
335C23—5082no—
336C23—5082no—
337C23—2542no—
338C23—2542no—
339C23—2542no—
340C23—2541no—
341C23D4×D12711yescharacteristic
342Q8—21no—
343Q8—21no—
344Q8—5082no—
345Q8—2541no—
346Q8—2541no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—5082no—
351D4—5082no—
352D4—5082no—
353D4—5082no—
354D4—2541no—
355D4—2541no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—1271no—
362C2×C4—1272no—
363C2×C4—1272no—
364C2×C4—1271no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—2542no—
369C2×C4—2542no—
370C2×C4—2542no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4—2541no—
375C2×C4—2541no—
376C2×C4D4×D12711yescharacteristic
377C2×C4C23×D12711yescharacteristic
378C8—21no—
379C8—1272no—
380C8—2541no—
381C8D4×D12712yescomplemented
382C22—21no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—1272no—
389C22—1272no—
390C22—1272no—
391C22—10164no—
392C22—5084no—
393C22—5081no—
394C22—5081no—
395C22—5081no—
396C22—2541no—
397C22—2541no—
398C22—2541no—
399C22C2×D4×D12711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—1272no—
405C4—5081no—
406C4—5081no—
407C4—5081no—
408C4—2541no—
409C4C2×D4×D12711yescharacteristic
410C4#41×D12711yescharacteristic
411C2—41no—
412C2—81no—
413C2—1274no—
414C2—10161no—
415C2—5081no—
416C2#121×D12711yescharacteristic
417C2AGL(1,ℤ/8)×D12711yescharacteristic
418C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#468×C127C2
socleC22×C127C22×D4
prime radical#121×D127C2
hypercenter#468D127
hypocenterC127C2×#468
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D127C2semidirect
2. C2×M16×D127C2semidirect
3. C22⋊Q8×D127C2semidirect
5. #22550125C2semidirect
6. #22550124C2semidirect
7. #22550123C2semidirect
8. #22550122C2semidirect
9. #22550121C2semidirect
10. #22550120C2semidirect
11. #22550119C2semidirect
12. #107×D127C2semidirect
13. #103×D127C2semidirect
14. #102×D127C2semidirect
15. #468×C127C2semidirect
20. C2×#14523237C22semidirect
21. C2×#14523236C22semidirect
22. C2×#14523235C22semidirect
24. C2×#14523232C22semidirect
25. C2×#14523231C22semidirect
26. C2×#14523231C22semidirect
30. #16861189C22semidirect
31. #16861188C22semidirect
34. #16861185C22semidirect
45. #16861143C22semidirect
46. #16861142C22semidirect
47. #16861142C22semidirect
48. #16861134C22semidirect
49. #16861133C22semidirect
50. #16861132C22semidirect
51. #16861131C22semidirect
52. #16861130C22semidirect
53. #16861129C22semidirect
54. C2×SD16×C127C22semidirect
55. C2×M16×C127C22semidirect
56. C22⋊Q8×C127C22semidirect
58. #107×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
94. C8×D127D4semidirect
155. C127⋊C8C2×D4semidirect
171. C8×C127C2×D4semidirect
208. D127#468direct (N × H)
210. C127C2×#468semidirect
222. #468D127direct (N × H)
234. C2×SD16C2×D127semidirect
239. C2×M16C2×D127semidirect
247. C22⋊Q8C2×D127semidirect
262. #107C2×D127semidirect
266. #103C2×D127semidirect
270. #102C2×D127semidirect
381. C8D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1272
C2×M16×D1272
C22⋊Q8×D1272
#121×D1272
#225501252
#225501242
#225501232
#225501222
#225501212
#225501202
#225501192
#107×D1272
#103×D1272
#102×D1272
#468×C1272
C2×#468127