Finite Group Databasebeta

← #30648409

Subgroups — #30648409

Order 16256.

504 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×#16861233C211yesmaximal, characteristic, complemented
2C2×#16861215C211yesmaximal, characteristic, complemented
3#22550455C211yesmaximal, characteristic, complemented
4#22550442C211yesmaximal, characteristic, complemented
5#22550127C211yesmaximal, characteristic, complemented
6#22550126C211yesmaximal, characteristic, complemented
7#22550123C211yesmaximal, characteristic, complemented
8#22550120C211yesmaximal, characteristic, complemented
9#22550111C211yesmaximal, characteristic, complemented
10#22550105C211yesmaximal, characteristic, complemented
11#22550104C211yesmaximal, characteristic, complemented
12#22549917C211yesmaximal, characteristic, complemented
13#22549869C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#466×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861233—24no—
20C2×#14523237C2211yescharacteristic
21C2×#14523236C2211yescharacteristic
22C2×#14523233C2211yescharacteristic
23#16861215—24no—
24C2×#14523232C2211yescharacteristic, complemented
25C2×#14523231C2211yescharacteristic
26C2×#14523231C2211yescharacteristic, complemented
27C2×C8×D127C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic, complemented
29#16861205—21no—
30#16861195—21no—
31#16861189C2211yescharacteristic, complemented
32#16861188C2211yescharacteristic, complemented
33#16861186—21no—
34#16861184C2211yescharacteristic
35#16861183C2211yescharacteristic
36#16861182—21no—
37#16861181—21no—
38#16861180—21no—
39#16861180C2211yescharacteristic
40#16861180C2211yescharacteristic, complemented
41#16861179—21no—
42#16861177—21no—
43#16861167C2211yescharacteristic
44C4⋊C4×D127C2211yescharacteristic
45#16861142C2211yescharacteristic, complemented
46#16861141C2211yescharacteristic, complemented
47#16861140C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861132C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic
51#16861130C2211yescharacteristic
52#16861130C2211yescharacteristic, complemented
53#16861129C2211yescharacteristic
54C2×D8×C127C2211yescharacteristic
55C22×C8×C127C2211yescharacteristic, complemented
56#121×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic, complemented
58#107×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523248—22no—
63#14523248—22no—
64#14523247—22no—
65#14523247—22no—
66C2×Dic254D411yescharacteristic
67C2×Dic254C2311yescharacteristic
68C2×D508—21no—
69C2×D508D411yescharacteristic
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127D411yescharacteristic
77C2×C22⋊D127D411yescharacteristic
78D4×D127—22no—
79D4×D127—22no—
80C22×Dic127—21no—
81C2×C4×D127—21no—
82C2×C4×D127C2311yescharacteristic
83C2×C4×D127C2311yescharacteristic
84#14523237—22no—
85#14523236—22no—
86#14523233—22no—
87#14523232—22no—
88#14523231—22no—
89#14523231D414yescomplemented
90C2×C127⋊C8C2311yescharacteristic
91C2×C127⋊C8C2311yescharacteristic
92C8×D127—22no—
93C8×D127—22no—
94C4⋊Dic127—21no—
95C4⋊Dic127C2311yescharacteristic
96C4⋊Dic127C2311yescharacteristic
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106C4×Dic127—21no—
107C2×D4×C127—21no—
108C2×D4×C127—21no—
109C2×D4×C127C2311yescharacteristic
110C2×D4×C127C2311yescharacteristic
111C22×C4×C127C2311yescharacteristic
112D8×C127—22no—
113C2×C8×C127—22no—
114C2×C8×C127C2311yescharacteristic
115C2×C8×C127C2311yescharacteristic
116C4⋊C4×C127C2311yescharacteristic
117C4⋊C4×C127C2311yescharacteristic
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C22×D127—21no—
121C22×D127—42no—
122C22×D127—81no—
123C22×D127—81no—
124C22×D127C2×D411yescharacteristic
125C22×D127C2×D411yescharacteristic
126Dic254—22no—
127Dic254—21no—
128Dic254—21no—
129D508—22no—
130D508—21no—
131D508—21no—
132D508—81no—
133C22⋊D127—22no—
134C22⋊D127—22no—
135C22⋊D127—42no—
136C22⋊D127—42no—
137C22⋊D127—81no—
138C22⋊D127—81no—
139C22⋊D127—81no—
140C2×Dic127—21no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—42no—
144C2×Dic127C2×D411yescharacteristic
145C2×Dic127C2×D411yescharacteristic
146C4×D127—21no—
147C4×D127—21no—
148C4×D127—81no—
149C4×D127C2×D412yes—
150C4×D127#4112yes—
151C127⋊C8—21no—
152C127⋊C8C2×D412yes—
153C23×C127—21no—
154C23×C127—21no—
155C23×C127C2×D411yescharacteristic
156D4×C127—21no—
157D4×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—81no—
161D4×C127—81no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127C2×D411yescharacteristic
167C2×C4×C127C2411yescharacteristic
168C8×C127—21no—
169C8×C127C2×D412yes—
170C22×C127—21no—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127C22×D411yescharacteristic
180C2×D127—21no—
181C2×D127—21no—
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C2×D127—84no—
189C2×D127#12112yes—
190C4×C127—21no—
191C4×C127—41no—
192C4×C127—41no—
193C4×C127C22×D411yescharacteristic
194C4×C127C2×#4111yescharacteristic
195Dic127—21no—
196Dic127—41no—
197Dic127—41no—
198Dic127—41no—
199Dic127#12112yes—
200C2×C127—41no—
201C2×C127—81no—
202C2×C127—81no—
203C2×C127C2×#12111yescharacteristic
204C2×C127#57611yescharacteristic
205C2×C127#57711yescharacteristic
206D127—22no—
207D127—41no—
208D127—81no—
209#3062—1271nomaximal, Sylow
210C127#306211yescharacteristic, complemented, Sylow
211C2×AGL(1,ℤ/8)—1271no—
212C2×#131—1271no—
213#537—1271no—
214#535—1271no—
215#469—1271no—
216#469—1271no—
217#468—1271no—
218#468—1271no—
219#466—1271no—
220#466D12711yescharacteristic, complemented
221#465—1271no—
222#465—1271no—
223#428—1271no—
224#416—1271no—
225C2×#102—1271no—
226C2×#41—1271no—
227C2×#41—1271no—
228C22×D4—1271no—
229AGL(1,ℤ/8)—2544no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×D8—1271no—
233C2×D8C2×D12711yescharacteristic
234#131—2544no—
235C2×M16—1271no—
236C2×M16—1271no—
237C2×M16—1271no—
238C22×C8—1271no—
239C22×C8—1271no—
240C22×C8C2×D12711yescharacteristic, complemented
241#127—2541no—
242#124—2541no—
243C22⋊Q8—1271no—
244C22⋊Q8—1271no—
245C22⋊Q8—2541no—
246#121—1271no—
247#121—1271no—
248#121—1271no—
249#121—1271no—
250#121—2541no—
251#121—2541no—
252#121—2541no—
253#121C2×D12711yescharacteristic
254#121C2×D12711yescharacteristic, complemented
255#120—2541no—
256#120—2541no—
257#117—1271no—
258C2×C4⋊C4—1271no—
259#107—1271no—
260#107C2×D12711yescharacteristic, complemented
261#106—1271no—
262#106—1271no—
263#103—1271no—
264#103—1271no—
265#102—1271no—
266#102—1271no—
267#102—1271no—
268#102—1271no—
269#102C2×D12711yescharacteristic
270#102C2×D12711yescharacteristic, complemented
271C24—2541no—
272#41—2542no—
273#41—2542no—
274#41—2542no—
275#41—2542no—
276C2×Q8—1271no—
277C2×Q8—1271no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—1271no—
281C2×D4—1271no—
282C2×D4—1271no—
283C2×D4—1271no—
284C2×D4—2541no—
285C2×D4—2542no—
286C2×D4—2542no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4—2541no—
291C2×D4—2541no—
292C2×D4C22×D12711yescharacteristic
293C2×D4C22×D12711yescharacteristic
294C22×C4—1271no—
295C22×C4—1271no—
296C22×C4—2541no—
297C22×C4—2541no—
298C22×C4C22×D12711yescharacteristic
299SD16—2542no—
300SD16—2542no—
301D8—22no—
302D8—2542no—
303M16—1274no—
304M16—2542no—
305M16—2542no—
306C2×C8—22no—
307C2×C8—1271no—
308C2×C8—1271no—
309C2×C8—2542no—
310C2×C8—2542no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—1271no—
314C4⋊C4—1271no—
315C4⋊C4—2541no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4C22×D12711yescharacteristic
319C4⋊C4C22×D12711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C42—2541no—
330C23—21no—
331C23—21no—
332C23—1271no—
333C23—1271no—
334C23—10161no—
335C23—10161no—
336C23—5082no—
337C23—2541no—
338C23D4×D12711yescharacteristic
339Q8—2542no—
340Q8—2541no—
341Q8—2541no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—81no—
347D4—81no—
348D4—10161no—
349D4—10161no—
350D4—10161no—
351D4—10161no—
352D4—5082no—
353D4—5082no—
354D4—2542no—
355D4—2542no—
356D4—2542no—
357D4—2541no—
358D4—2541no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—1271no—
364C2×C4—1272no—
365C2×C4—1272no—
366C2×C4—1271no—
367C2×C4—10161no—
368C2×C4—5082no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4D4×D12711yescharacteristic
375C2×C4C23×D12711yescharacteristic
376C8—21no—
377C8—1272no—
378C8—2541no—
379C8D4×D12712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—1272no—
390C22—10164no—
391C22—5081no—
392C22—5081no—
393C22—5081no—
394C22—2541no—
395C22—2541no—
396C22—2541no—
397C22—2541no—
398C22—2541no—
399C22C2×D4×D12711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—1272no—
404C4—5081no—
405C4—5081no—
406C4—5081no—
407C4—2541no—
408C4C2×D4×D12711yescharacteristic
409C4#41×D12711yescharacteristic
410C2—41no—
411C2—81no—
412C2—81no—
413C2—10161no—
414C2—5081no—
415C2—2542no—
416C2#121×D12711yescharacteristic
417C2#2255067711yescharacteristic
418C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#466×C127C2
socleC22×C127C22×D4
hypercenter#466D127
hypocenterC127#3062
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. C2×#16861215C2semidirect
3. #22550455C2semidirect
4. #22550442C2semidirect
5. #22550127C2semidirect
6. #22550126C2semidirect
7. #22550123C2semidirect
8. #22550120C2semidirect
9. #22550111C2semidirect
10. #22550105C2semidirect
11. #22550104C2semidirect
12. #22549917C2semidirect
13. #22549869C2semidirect
14. #102×D127C2semidirect
15. #466×C127C2semidirect
24. C2×#14523232C22semidirect
26. C2×#14523231C22semidirect
28. C2×C8×D127C22semidirect
31. #16861189C22semidirect
32. #16861188C22semidirect
40. #16861180C22semidirect
45. #16861142C22semidirect
46. #16861141C22semidirect
47. #16861140C22semidirect
48. #16861134C22semidirect
49. #16861132C22semidirect
52. #16861130C22semidirect
55. C22×C8×C127C22semidirect
57. #121×C127C22semidirect
58. #107×C127C22semidirect
60. #102×C127C22semidirect
89. #14523231D4semidirect
210. C127#3062semidirect
220. #466D127semidirect
240. C22×C8C2×D127semidirect
254. #121C2×D127semidirect
260. #107C2×D127semidirect
270. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
C2×#168612152
#225504552
#225504422
#225501272
#225501262
#225501232
#225501202
#225501112
#225501052
#225501042
#225499172
#225498692
#102×D1272
#466×C1272
#3062127