Finite Group Databasebeta

← #30652383

Subgroups — #30652383

Order 16256.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861256C211yesmaximal, characteristic, complemented
2C2×#41×D127C211yesmaximal, characteristic, complemented
3#22550546C213yesmaximal, complemented
4#22550545C211yesmaximal, characteristic, complemented
5#22550544C213yesmaximal, complemented
6#22550538C211yesmaximal, characteristic, complemented
7#22550535C213yesmaximal, complemented
8#22550533C211yesmaximal, characteristic, complemented
9#22550532C213yesmaximal, complemented
10#22550442C213yesmaximal, complemented
11#22550439C213yesmaximal, complemented
12#22550386C213yesmaximal, complemented
13#22550385C213yesmaximal, complemented
14#22550352C211yesmaximal, characteristic, complemented
15#550×C127C211yesmaximal, characteristic, complemented
16#16861256—28no—
17C2×#14523249C2211yescharacteristic
18C2×#14523248C2213yes—
19C2×#14523248C2213yes—
20C2×#14523247C2213yes—
21C2×#14523247C2213yes—
22C2×#14523247C2213yescomplemented
23#41×D127—28no—
24C22×Dic254C2213yescomplemented
25C2×Q8×D127C2211yescharacteristic
26C2×Q8×D127C2211yescharacteristic, complemented
27C2×D4×D127C2213yes—
28C22×C4×D127C2213yes—
29#16861205C2213yescomplemented
30#16861205C2216yescomplemented
31#16861197C2213yescomplemented
32#16861196C2213yescomplemented
33#16861195C2216yescomplemented
34#16861194C2211yescharacteristic
35#16861191C2213yes—
36#16861189C2213yescomplemented
37#16861188C2213yescomplemented
38#16861187C2213yescomplemented
39#16861186C2213yescomplemented
40#16861184C2213yescomplemented
41#16861183C2213yescomplemented
42#16861182C2213yescomplemented
43#16861181C2216yescomplemented
44#16861181C2216yescomplemented
45#16861181C2213yescomplemented
46#16861180C2213yescomplemented
47#16861180C2213yescomplemented
48#16861179C2213yes—
49#16861177C2213yes—
50#16861177C2216yes—
51Q8×Dic127C2211yescharacteristic, complemented
52C4×Dic254C2213yescomplemented
53#16861174C2213yescomplemented
54C4×D508C2213yescomplemented
55D4×Dic127C2216yescomplemented
56C4×C22⋊D127C2213yescomplemented
57C4×C22⋊D127C2213yescomplemented
58(C22⋊D127)⋊C4C2216yescomplemented
59#16861167C2213yes—
60#16861166C2213yes—
61C2×#41×C127C2211yescharacteristic
62C2×#41×C127C2211yescharacteristic, complemented
63#127×C127C2213yescomplemented
64#121×C127C2213yescomplemented
65#121×C127C2213yescomplemented
66C4×Q8×C127C2211yescharacteristic, complemented
67C4×D4×C127C2213yescomplemented
68C23×D127C2313yes—
69#14523249—24no—
70#14523248—212no—
71#14523248—212no—
72#14523247—212no—
73#14523247—212no—
74#14523247—212no—
75C2×Dic254—212no—
76C2×Dic254C2313yes—
77C2×Dic254C2313yes—
78C2×Dic254C2313yes—
79Q8×D127—24no—
80Q8×D127D418yescomplemented
81C2×D508C2313yes—
82C2×D508C2311yescharacteristic
83C2×D508C2313yes—
84C2×C22⋊D127C2313yes—
85C2×C22⋊D127C2316yes—
86C2×C22⋊D127C2313yes—
87C2×C22⋊D127C2316yes—
88C2×C22⋊D127C2313yes—
89C2×C22⋊D127C2313yes—
90D4×D127—212no—
91C22×Dic127C2313yes—
92C22×Dic127C2316yes—
93C2×C4×D127—212no—
94C2×C4×D127C2313yes—
95C2×C4×D127C2313yes—
96C2×C4×D127C2313yes—
97C2×C4×D127C2311yescharacteristic
98C2×C4×D127C2313yes—
99C4⋊Dic127C2313yes—
100C4⋊Dic127C2313yes—
101Dic127⋊C4C2311yescharacteristic
102Dic127⋊C4C2313yes—
103Dic127⋊C4C2313yes—
104Dic127⋊C4C2313yes—
105C22⋊Dic127C2313yes—
106C22⋊Dic127C2316yes—
107C22⋊Dic127C2313yes—
108(C2×D127)⋊C4C2316yes—
109(C2×D127)⋊C4C2313yes—
110(C2×D127)⋊C4C2313yes—
111(C2×D127)⋊C4C2313yes—
112(C2×D127)⋊C4C2311yescharacteristic
113(C2×D127)⋊C4C2311yescharacteristic
114(C2×D127)⋊C4C2311yescharacteristic
115C4×Dic127C2316yes—
116C4×Dic127C2313yes—
117#41×C127—24no—
118#41×C127—24no—
119C2×Q8×C127C2311yescharacteristic
120C2×D4×C127C2313yes—
121C2×D4×C127C2313yes—
122C2×D4×C127C2316yes—
123C2×D4×C127C2313yes—
124C22×C4×C127C2313yes—
125C22×C4×C127C2313yes—
126C4⋊C4×C127C2313yes—
127C4⋊C4×C127C2311yescharacteristic
128C22⋊C4×C127C2313yes—
129C22⋊C4×C127C2313yes—
130C42×C127C2313yes—
131C22×D127—26no—
132C22×D127—43no—
133C22×D127—43no—
134C22×D127C2413yes—
135C22×D127C2411yescharacteristic
136C22×D127C2411yescharacteristic
137Dic254—26no—
138Dic254—26no—
139Dic254C2×D4112yes—
140D508—22no—
141D508—26no—
142D508—43no—
143C22⋊D127—26no—
144C22⋊D127—212no—
145C22⋊D127—26no—
146C22⋊D127—43no—
147C22⋊D127—46no—
148C22⋊D127—43no—
149C2×Dic127—26no—
150C2×Dic127—212no—
151C2×Dic127C2411yescharacteristic
152C2×Dic127C2413yes—
153C2×Dic127C2413yes—
154C2×Dic127C2411yescharacteristic
155C2×Dic127C2413yes—
156C4×D127—22no—
157C4×D127—26no—
158C4×D127—26no—
159C4×D127—26no—
160C4×D127C2×D4112yes—
161C23×C127C2413yes—
162C23×C127C2413yes—
163Q8×C127C2×D414yes—
164D4×C127—26no—
165D4×C127—26no—
166D4×C127—43no—
167D4×C127—46no—
168C2×C4×C127—26no—
169C2×C4×C127—26no—
170C2×C4×C127C2411yescharacteristic
171C2×C4×C127C2413yes—
172C2×C4×C127C2411yescharacteristic
173C2×C4×C127C2411yescharacteristic
174C2×C4×C127C2413yes—
175C22×C127—23no—
176C22×C127—23no—
177C22×C127—23no—
178C22×C127—23no—
179C22×C127—23no—
180C22×C127—23no—
181C22×C127C2511yescharacteristic
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C2×D127—23no—
186C2×D127—21no—
187C2×D127—23no—
188C2×D127—23no—
189C2×D127—21no—
190C2×D127—412no—
191C2×D127C22×D412yes—
192C4×C127—21no—
193C4×C127—21no—
194C4×C127—21no—
195C4×C127—23no—
196C4×C127C22×D416yes—
197Dic127—21no—
198Dic127—23no—
199Dic127—21no—
200Dic127—23no—
201Dic127C22×D416yes—
202C2×C127—43no—
203C2×C127—43no—
204C2×C127C23×D411yescharacteristic
205C2×C127#58511yescharacteristic
206C2×C127#58511yescharacteristic
207D127—22no—
208D127—41no—
209D127—43no—
210#3487—1271nomaximal, Sylow
211C127#348711yescharacteristic, complemented, Sylow
212C2×2−1+4—1271no—
213C22×#41—1271no—
214#550—1271no—
215#550—1273no—
216#550—1273no—
217#550D12711yescharacteristic, complemented
218#548—1271no—
219#548—1273no—
220#548—1271no—
221#548—1273no—
222#535—1273no—
223#535—1273no—
224#525—1273no—
225#525—1273no—
226#519—1271no—
2272−1+4—2548no—
228C2×#41—1271no—
229C2×#41—1273no—
230C2×#41—1273no—
231C2×#41—1273no—
232C2×#41—1273no—
233C2×#41—1273no—
234C2×#41—2548no—
235C2×#41C2×D12711yescharacteristic
236C2×#41C2×D12711yescharacteristic, complemented
237C22×Q8—1271no—
238C22×Q8—1271no—
239C22×Q8—1273no—
240C22×D4—1273no—
241C23×C4—1273no—
242#127—1273no—
243#127—1276no—
244#127C2×D12713yescomplemented
245#124—1273no—
246#124—1276no—
247#124—1273no—
248#123—1271no—
249#123—1273no—
250C22⋊Q8—1273no—
251C22⋊Q8—1273no—
252C22⋊Q8—1273no—
253C22⋊Q8—1273no—
254#121—1273no—
255#121—1273no—
256#121—1273no—
257#121—1276no—
258#121—1276no—
259#121—1273no—
260#121—1273no—
261#121—1273no—
262#121C2×D12713yescomplemented
263#121C2×D12713yescomplemented
264#120—1276no—
265#120—1273no—
266#120—1273no—
267C4×Q8—1271no—
268C4×Q8—1273no—
269C4×Q8—1273no—
270C4×Q8C2×D12711yescharacteristic, complemented
271C4×D4—1276no—
272C4×D4—1273no—
273C4×D4—1273no—
274C4×D4—1276no—
275C4×D4—1273no—
276C4×D4C2×D12713yescomplemented
277#117—1273no—
278#117—1273no—
279C24—1273no—
280#41—24no—
281#41—24no—
282#41—25412no—
283#41—25412no—
284#41—25412no—
285#41—25412no—
286#41—25412no—
287#41—2544no—
288C2×Q8—1278no—
289C2×Q8—1273no—
290C2×Q8—1273no—
291C2×Q8—1273no—
292C2×Q8—25412no—
293C2×Q8—2544no—
294C2×Q8C22×D12711yescharacteristic
295C2×D4—1276no—
296C2×D4—1273no—
297C2×D4—1273no—
298C2×D4—1273no—
299C2×D4—1276no—
300C2×D4—1273no—
301C2×D4—1273no—
302C2×D4—1271no—
303C2×D4—1273no—
304C2×D4—25412no—
305C2×D4C22×D12713yes—
306C2×D4C22×D12713yes—
307C2×D4C22×D12716yes—
308C2×D4C22×D12713yes—
309C22×C4—1273no—
310C22×C4—1276no—
311C22×C4—1273no—
312C22×C4—1273no—
313C22×C4—1273no—
314C22×C4—1273no—
315C22×C4—1271no—
316C22×C4—25412no—
317C22×C4C22×D12713yes—
318C22×C4C22×D12713yes—
319C4⋊C4—1273no—
320C4⋊C4—1273no—
321C4⋊C4—1273no—
322C4⋊C4—1271no—
323C4⋊C4—1273no—
324C4⋊C4—1273no—
325C4⋊C4C22×D12713yes—
326C4⋊C4C22×D12711yescharacteristic
327C22⋊C4—1273no—
328C22⋊C4—1273no—
329C22⋊C4—1276no—
330C22⋊C4—1273no—
331C22⋊C4—1271no—
332C22⋊C4—1271no—
333C22⋊C4—1271no—
334C22⋊C4—1273no—
335C22⋊C4—1276no—
336C22⋊C4—1273no—
337C22⋊C4C22×D12713yes—
338C22⋊C4C22×D12713yes—
339C42—1276no—
340C42—1273no—
341C42C22×D12713yes—
342C23—1271no—
343C23—1273no—
344C23—1271no—
345C23—5083no—
346C23—5083no—
347C23—2546no—
348C23C23×D12713yes—
349C23C23×D12713yes—
350Q8—12712no—
351Q8—2546no—
352Q8—2546no—
353Q8D4×D12714yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—5083no—
359D4—5083no—
360D4—5086no—
361D4—5083no—
362D4—2542no—
363D4—25412no—
364D4—2546no—
365D4—2546no—
366D4—2546no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—1273no—
370C2×C4—12712no—
371C2×C4—1271no—
372C2×C4—1273no—
373C2×C4—1273no—
374C2×C4—1271no—
375C2×C4—2546no—
376C2×C4—2542no—
377C2×C4—2546no—
378C2×C4—25412no—
379C2×C4—2546no—
380C2×C4—2546no—
381C2×C4C23×D12713yes—
382C2×C4C23×D12711yescharacteristic
383C2×C4C23×D12711yescharacteristic
384C2×C4C23×D12711yescharacteristic
385C2×C4C23×D12713yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—1272no—
393C22—50812no—
394C22—2541no—
395C22—2543no—
396C22—2541no—
397C22—2543no—
398C22—2541no—
399C22—2543no—
400C22—2541no—
401C22—2541no—
402C22C24×D12711yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—1276no—
408C4—2541no—
409C4—2543no—
410C4—2541no—
411C4—2543no—
412C4C2×D4×D12716yes—
413C2—43no—
414C2—43no—
415C2—5083no—
416C2—5081no—
417C2—2542no—
418C2C22×D4×D12711yescharacteristic
419C2#2255071111yescharacteristic
420C2#2255071211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
Fitting#550×C127C2
hypercenter#550D127
hypocenterC127#3487
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#16861256C2semidirect
2. C2×#41×D127C2semidirect
3. #22550546C2semidirect
4. #22550545C2semidirect
5. #22550544C2semidirect
6. #22550538C2semidirect
7. #22550535C2semidirect
8. #22550533C2semidirect
9. #22550532C2semidirect
10. #22550442C2semidirect
11. #22550439C2semidirect
12. #22550386C2semidirect
13. #22550385C2semidirect
14. #22550352C2semidirect
15. #550×C127C2semidirect
22. C2×#14523247C22semidirect
24. C22×Dic254C22semidirect
26. C2×Q8×D127C22semidirect
29. #16861205C22semidirect
30. #16861205C22semidirect
31. #16861197C22semidirect
32. #16861196C22semidirect
33. #16861195C22semidirect
36. #16861189C22semidirect
37. #16861188C22semidirect
38. #16861187C22semidirect
39. #16861186C22semidirect
40. #16861184C22semidirect
41. #16861183C22semidirect
42. #16861182C22semidirect
43. #16861181C22semidirect
44. #16861181C22semidirect
45. #16861181C22semidirect
46. #16861180C22semidirect
47. #16861180C22semidirect
51. Q8×Dic127C22semidirect
52. C4×Dic254C22semidirect
53. #16861174C22semidirect
54. C4×D508C22semidirect
55. D4×Dic127C22semidirect
56. C4×C22⋊D127C22semidirect
57. C4×C22⋊D127C22semidirect
58. (C22⋊D127)⋊C4C22semidirect
62. C2×#41×C127C22semidirect
63. #127×C127C22semidirect
64. #121×C127C22semidirect
65. #121×C127C22semidirect
66. C4×Q8×C127C22semidirect
67. C4×D4×C127C22semidirect
80. Q8×D127D4semidirect
211. C127#3487semidirect
217. #550D127semidirect
236. C2×#41C2×D127semidirect
244. #127C2×D127semidirect
262. #121C2×D127semidirect
263. #121C2×D127semidirect
270. C4×Q8C2×D127semidirect
276. C4×D4C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
C2×#41×D1272
#225505462
#225505452
#225505442
#225505382
#225505352
#225505332
#225505322
#225504422
#225504392
#225503862
#225503852
#225503522
#550×C1272
#3487127