Finite Group Databasebeta

← #30648334

Subgroups — #30648334

Order 16256.

500 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861230C211yesmaximal, characteristic, complemented
2C2×M16×D127C211yesmaximal, characteristic, complemented
3#22550389C211yesmaximal, characteristic, complemented
4#22550386C211yesmaximal, characteristic, complemented
5#22550133C211yesmaximal, characteristic, complemented
6#22550132C211yesmaximal, characteristic, complemented
7#22550128C211yesmaximal, characteristic, complemented
8#22550127C211yesmaximal, characteristic, complemented
9#22550123C211yesmaximal, characteristic, complemented
10#22550122C211yesmaximal, characteristic, complemented
11#22550119C211yesmaximal, characteristic, complemented
12#22549913C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14#22549869C211yesmaximal, characteristic, complemented
15#468×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19#16861230—24no—
20C2×#14523238C2211yescharacteristic, complemented
21C2×#14523237C2211yescharacteristic, complemented
22C2×#14523233C2211yescharacteristic, complemented
23C2×#14523232C2211yescharacteristic
24C2×#14523231C2211yescharacteristic
25C2×#14523231C2211yescharacteristic
26M16×D127—24no—
27C2×C8×D127C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861194—21no—
30#16861191—21no—
31#16861189C2211yescharacteristic
32#16861188C2211yescharacteristic
33#16861185C2211yescharacteristic
34#16861184C2211yescharacteristic
35#16861183C2211yescharacteristic
36#16861180C2211yescharacteristic
37#16861179—21no—
38#16861177—21no—
39D4×Dic127—21no—
40#16861171—21no—
41(C22⋊D127)⋊C4—21no—
42(C22⋊D127)⋊C4—21no—
43#16861167C2211yescharacteristic
44#16861167C2211yescharacteristic
45#16861143C2211yescharacteristic, complemented
46#16861140C2211yescharacteristic, complemented
47#16861140C2211yescharacteristic, 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
56C22⋊Q8×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#107×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127D411yescharacteristic
62#14523249—22no—
63#14523249—22no—
64#14523247—22no—
65#14523247—22no—
66C2×Dic254C2311yescharacteristic
67C2×D508—21no—
68C2×D508—21no—
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C22×Dic127—21no—
74C22×Dic127D411yescharacteristic
75C2×C4×D127—22no—
76C2×C4×D127—22no—
77C2×C4×D127—21no—
78C2×C4×D127D411yescharacteristic
79C2×C4×D127D411yescharacteristic
80C2×C4×D127C2311yescharacteristic
81C2×C4×D127C2311yescharacteristic
82#14523238—22no—
83#14523237—22no—
84#14523233—22no—
85#14523232—22no—
86#14523231—22no—
87#14523231—22no—
88C2×C127⋊C8C2311yescharacteristic
89C2×C127⋊C8C2311yescharacteristic
90C8×D127—22no—
91C8×D127D414yescomplemented
92C4⋊Dic127—21no—
93C4⋊Dic127C2311yescharacteristic
94C4⋊Dic127C2311yescharacteristic
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97C22⋊Dic127—21no—
98C22⋊Dic127—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105C4×Dic127—21no—
106C4×Dic127—21no—
107C2×Q8×C127C2311yescharacteristic
108C2×D4×C127—21no—
109C2×D4×C127C2311yescharacteristic
110C22×C4×C127C2311yescharacteristic
111SD16×C127—22no—
112M16×C127—22no—
113C2×C8×C127C2311yescharacteristic
114C2×C8×C127C2311yescharacteristic
115C4⋊C4×C127—21no—
116C4⋊C4×C127C2311yescharacteristic
117C4⋊C4×C127C2311yescharacteristic
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C22×D127—22no—
121C22×D127—21no—
122C22×D127—41no—
123C22×D127—41no—
124C22×D127C2×D411yescharacteristic
125C22×D127C2×D411yescharacteristic
126Dic254—21no—
127Dic254—21no—
128D508—21no—
129D508—21no—
130D508—42no—
131D508—42no—
132C22⋊D127—42no—
133C22⋊D127—42no—
134C22⋊D127—42no—
135C2×Dic127—22no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—42no—
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C4×D127—22no—
143C4×D127—21no—
144C4×D127—21no—
145C4×D127—22no—
146C4×D127—42no—
147C4×D127C2×D412yes—
148C4×D127#4112yes—
149C127⋊C8—21no—
150C127⋊C8C2×D412yes—
151C23×C127—21no—
152C23×C127C2×D411yescharacteristic
153Q8×C127—21no—
154Q8×C127—21no—
155D4×C127—21no—
156D4×C127—21no—
157D4×C127—42no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2411yescharacteristic
165C8×C127—21no—
166C8×C127C2×D412yes—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—41no—
173C22×C127C22×D411yescharacteristic
174C2×D127—21no—
175C2×D127—21no—
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—21no—
179C2×D127—44no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127#12112yes—
184C4×C127—21no—
185C4×C127—41no—
186C4×C127—41no—
187C4×C127—41no—
188C4×C127C22×D411yescharacteristic
189C4×C127C2×#4111yescharacteristic
190Dic127—21no—
191Dic127—41no—
192Dic127—41no—
193Dic127—41no—
194Dic127#12112yes—
195C2×C127—41no—
196C2×C127—81no—
197C2×C127C2×#12111yescharacteristic
198C2×C127#57511yescharacteristic
199C2×C127#57511yescharacteristic
200D127—22no—
201D127—41no—
202D127—81no—
203#3054—1271nomaximal, Sylow
204C127#305411yescharacteristic, complemented, Sylow
205C2×#135—1271no—
206C22×M16—1271no—
207#525—1271no—
208#525—1271no—
209#470—1271no—
210#470—1271no—
211#469—1271no—
212#469—1271no—
213#468—1271no—
214#468—1271no—
215#468—1271no—
216#468D12711yescharacteristic, complemented
217#427—1271no—
218#416—1271no—
219#416—1271no—
220C2×#41—1271no—
221C2×#41—1271no—
222C23×C4—1271no—
223#135—2544no—
224C2×Q16—1271no—
225C2×SD16—1271no—
226C2×SD16C2×D12711yescharacteristic, complemented
227C2×D8—1271no—
228C2×M16—1271no—
229C2×M16—1271no—
230C2×M16—1271no—
231C2×M16—2544no—
232C2×M16C2×D12711yescharacteristic
233C22×C8—1271no—
234C22×C8—1271no—
235#123—2541no—
236#123—2541no—
237C22⋊Q8—1271no—
238C22⋊Q8—1271no—
239C22⋊Q8—1271no—
240C22⋊Q8C2×D12711yescharacteristic
241#121—1271no—
242#121—1271no—
243#121—1271no—
244#121C2×D12711yescharacteristic
245#120—2541no—
246#120—2541no—
247C4×D4—2541no—
248C4×D4—2541no—
249C4×D4—2541no—
250C4×D4—2541no—
251#117—1271no—
252#117—1271no—
253#107—1271no—
254#107C2×D12711yescharacteristic, complemented
255#106—1271no—
256#106—1271no—
257#103—1271no—
258#103—1271no—
259#103—1271no—
260#103C2×D12711yescharacteristic, complemented
261#102—1271no—
262#102—1271no—
263#102—1271no—
264#102C2×D12711yescharacteristic, complemented
265C24—1271no—
266#41—2542no—
267#41—2542no—
268#41—2542no—
269#41—2542no—
270C2×Q8—1271no—
271C2×Q8C22×D12711yescharacteristic
272C2×D4—21no—
273C2×D4—1271no—
274C2×D4—2541no—
275C2×D4—2541no—
276C2×D4—2541no—
277C2×D4—2541no—
278C2×D4—2541no—
279C2×D4C22×D12711yescharacteristic
280C22×C4—1271no—
281C22×C4—1271no—
282C22×C4—1271no—
283C22×C4—1271no—
284C22×C4—1271no—
285C22×C4—2541no—
286C22×C4—2542no—
287C22×C4—2542no—
288C22×C4—2541no—
289C22×C4C22×D12711yescharacteristic
290Q16—2542no—
291SD16—22no—
292SD16—2542no—
293D8—2542no—
294M16—22no—
295M16—2542no—
296M16—2542no—
297M16—2542no—
298C2×C8—1271no—
299C2×C8—1274no—
300C2×C8—1271no—
301C2×C8—2542no—
302C2×C8C22×D12711yescharacteristic
303C2×C8C22×D12711yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—1271no—
306C4⋊C4—1271no—
307C4⋊C4—2541no—
308C4⋊C4—2541no—
309C4⋊C4—2541no—
310C4⋊C4C22×D12711yescharacteristic
311C4⋊C4C22×D12711yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4—2541no—
317C22⋊C4—2541no—
318C22⋊C4—2541no—
319C22⋊C4—2541no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C42—2541no—
323C42—2541no—
324C23—21no—
325C23—1271no—
326C23—1271no—
327C23—5081no—
328C23—5081no—
329C23—2542no—
330C23—2541no—
331C23D4×D12711yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—2541no—
335Q8—2541no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—5082no—
340D4—5082no—
341D4—5082no—
342D4—5082no—
343D4—5082no—
344D4—2541no—
345D4—2541no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—1271no—
352C2×C4—1272no—
353C2×C4—1272no—
354C2×C4—1271no—
355C2×C4—5082no—
356C2×C4—5082no—
357C2×C4—2542no—
358C2×C4—2542no—
359C2×C4—2542no—
360C2×C4—2541no—
361C2×C4—2541no—
362C2×C4—2541no—
363C2×C4—2541no—
364C2×C4—2541no—
365C2×C4D4×D12711yescharacteristic
366C2×C4C23×D12711yescharacteristic
367C8—21no—
368C8—1272no—
369C8—2541no—
370C8D4×D12712yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—1272no—
378C22—5084no—
379C22—5081no—
380C22—5081no—
381C22—5081no—
382C22—2541no—
383C22—2541no—
384C22—2541no—
385C22—2541no—
386C22—2541no—
387C22C2×D4×D12711yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—1272no—
393C4—5081no—
394C4—5081no—
395C4—5081no—
396C4—2541no—
397C4C2×D4×D12711yescharacteristic
398C4#41×D12711yescharacteristic
399C2—41no—
400C2—81no—
401C2—10161no—
402C2—5081no—
403C2—2542no—
404C2#121×D12711yescharacteristic
405C2#2255067211yescharacteristic
406C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#468×C127C2
socleC22×C127C22×D4
hypercenter#468D127
hypocenterC127#3054
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×#16861230C2semidirect
2. C2×M16×D127C2semidirect
3. #22550389C2semidirect
4. #22550386C2semidirect
5. #22550133C2semidirect
6. #22550132C2semidirect
7. #22550128C2semidirect
8. #22550127C2semidirect
9. #22550123C2semidirect
10. #22550122C2semidirect
11. #22550119C2semidirect
12. #22549913C2semidirect
13. #22549871C2semidirect
14. #22549869C2semidirect
15. #468×C127C2semidirect
20. C2×#14523238C22semidirect
21. C2×#14523237C22semidirect
22. C2×#14523233C22semidirect
45. #16861143C22semidirect
46. #16861140C22semidirect
47. #16861140C22semidirect
48. #16861134C22semidirect
49. #16861133C22semidirect
50. #16861132C22semidirect
51. #16861131C22semidirect
52. #16861130C22semidirect
53. #16861129C22semidirect
54. C2×SD16×C127C22semidirect
58. #107×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
91. C8×D127D4semidirect
204. C127#3054semidirect
216. #468D127semidirect
226. C2×SD16C2×D127semidirect
254. #107C2×D127semidirect
260. #103C2×D127semidirect
264. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612302
C2×M16×D1272
#225503892
#225503862
#225501332
#225501322
#225501282
#225501272
#225501232
#225501222
#225501192
#225499132
#225498712
#225498692
#468×C1272
#3054127