Finite Group Databasebeta

← #30648013

Subgroups — #30648013

Order 16256.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861256C211yesmaximal, characteristic, complemented
2C2×#16861233C211yesmaximal, characteristic, complemented
3C2×SD16×D127C211yesmaximal, characteristic, complemented
4#22550442C211yesmaximal, characteristic, complemented
5#22550022C211yesmaximal, characteristic, complemented
6#22550021C211yesmaximal, characteristic, complemented
7#22550018C211yesmaximal, characteristic, complemented
8#22550017C211yesmaximal, characteristic, complemented
9#22550016C211yesmaximal, characteristic, complemented
10#22550014C211yesmaximal, characteristic, complemented
11#22550013C211yesmaximal, characteristic, complemented
12#22549883C211yesmaximal, characteristic, complemented
13#22549869C211yesmaximal, characteristic, complemented
14#22549841C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16#16861256—24no—
17#16861256—24no—
18C2×#14523248—21no—
19C2×#14523248C2211yescharacteristic
20C2×#14523247—21no—
21C2×#14523247—21no—
22C2×#14523247C2211yescharacteristic
23C2×#14523247C2211yescharacteristic, complemented
24C22×Dic254—21no—
25C22×Dic254C2211yescharacteristic, complemented
26C2×Q8×D127C2211yescharacteristic
27C2×Q8×D127C2211yescharacteristic, complemented
28C2×D4×D127C2211yescharacteristic
29#16861233—24no—
30C2×#14523237C2211yescharacteristic
31C2×#14523237C2211yescharacteristic
32C2×#14523236C2211yescharacteristic
33C2×#14523236C2211yescharacteristic
34C2×#14523235C2211yescharacteristic
35SD16×D127—24no—
36C2×#14523233C2211yescharacteristic
37C2×#14523231C2211yescharacteristic
38C2×C8×D127C2211yescharacteristic
39#16861205—21no—
40#16861195—21no—
41#16861184C2211yescharacteristic
42#16861183C2211yescharacteristic
43#16861180C2211yescharacteristic
44#16861179—21no—
45#16861177—21no—
46#16861167C2211yescharacteristic
47#16861134C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861133C2211yescharacteristic, complemented
50#16861132C2211yescharacteristic, complemented
51#16861132C2211yescharacteristic, complemented
52#16861130C2211yescharacteristic, complemented
53C22⋊(C127⋊C8)C2211yescharacteristic, complemented
54(C2×D127)⋊C8C2211yescharacteristic, complemented
55(C2×D127)⋊C8C2211yescharacteristic, complemented
56C2×#41×C127C2211yescharacteristic, complemented
57C2×SD16×C127C2211yescharacteristic
58C2×D8×C127C2211yescharacteristic
59#121×C127C2211yescharacteristic
60#103×C127C2211yescharacteristic, complemented
61#102×C127C2211yescharacteristic, complemented
62C22⋊C8×C127C2211yescharacteristic, complemented
63C23×D127—21no—
64#14523248—28no—
65#14523248—22no—
66#14523248—22no—
67#14523247—28no—
68#14523247—28no—
69#14523247—22no—
70#14523247—22no—
71#14523247—22no—
72#14523247D414yescomplemented
73C2×Dic254—28no—
74C2×Dic254—22no—
75C2×Dic254—22no—
76C2×Dic254—21no—
77C2×Dic254—21no—
78C2×Dic254—21no—
79C2×Dic254D411yescharacteristic
80C2×Dic254C2311yescharacteristic
81C2×Dic254C2311yescharacteristic
82Q8×D127—22no—
83Q8×D127—22no—
84Q8×D127—22no—
85Q8×D127D414yescomplemented
86C2×D508—21no—
87C2×D508D411yescharacteristic
88C2×D508C2311yescharacteristic
89C2×C22⋊D127—21no—
90C2×C22⋊D127—21no—
91C2×C22⋊D127—21no—
92C2×C22⋊D127—21no—
93C2×C22⋊D127—21no—
94C2×C22⋊D127D411yescharacteristic
95C2×C22⋊D127D411yescharacteristic
96D4×D127—22no—
97D4×D127—22no—
98C22×Dic127—21no—
99C22×Dic127—21no—
100C22×Dic127—21no—
101C2×C4×D127—21no—
102C2×C4×D127—21no—
103C2×C4×D127C2311yescharacteristic
104C2×C4×D127C2311yescharacteristic
105#14523237—22no—
106#14523237—22no—
107#14523236—22no—
108#14523236—22no—
109#14523235—22no—
110#14523233—22no—
111#14523231—22no—
112C2×C127⋊C8C2311yescharacteristic
113C2×C127⋊C8C2311yescharacteristic
114C8×D127—22no—
115C4⋊Dic127C2311yescharacteristic
116C22⋊Dic127—21no—
117(C2×D127)⋊C4—21no—
118(C2×D127)⋊C4—21no—
119(C2×D127)⋊C4—21no—
120C4×Dic127—21no—
121#41×C127—22no—
122#41×C127—22no—
123C2×Q8×C127C2311yescharacteristic
124C2×D4×C127—21no—
125C2×D4×C127—21no—
126C2×D4×C127C2311yescharacteristic
127C2×D4×C127C2311yescharacteristic
128C22×C4×C127—21no—
129C22×C4×C127C2311yescharacteristic
130SD16×C127—22no—
131D8×C127—22no—
132C2×C8×C127C2311yescharacteristic
133C2×C8×C127C2311yescharacteristic
134C4⋊C4×C127C2311yescharacteristic
135C22⋊C4×C127—21no—
136C22×D127—21no—
137C22×D127—42no—
138C22×D127—81no—
139C22×D127—81no—
140C22×D127C2×D411yescharacteristic
141C22×D127C2×D411yescharacteristic
142Dic254—24no—
143Dic254—24no—
144Dic254—21no—
145Dic254—21no—
146Dic254—24no—
147Dic254—22no—
148Dic254C2×D412yes—
149Dic254C2×D412yes—
150D508—22no—
151D508—21no—
152D508—21no—
153D508—81no—
154C22⋊D127—24no—
155C22⋊D127—24no—
156C22⋊D127—22no—
157C22⋊D127—22no—
158C22⋊D127—42no—
159C22⋊D127—81no—
160C22⋊D127—81no—
161C2×Dic127—24no—
162C2×Dic127—24no—
163C2×Dic127—21no—
164C2×Dic127—21no—
165C2×Dic127—24no—
166C2×Dic127—21no—
167C2×Dic127C2×D411yescharacteristic
168C2×Dic127C2×D411yescharacteristic
169C4×D127—24no—
170C4×D127—24no—
171C4×D127—21no—
172C4×D127—21no—
173C4×D127C2×D412yes—
174C4×D127C2×D412yes—
175C127⋊C8—21no—
176C127⋊C8—21no—
177C23×C127—21no—
178C23×C127—21no—
179C23×C127C2×D411yescharacteristic
180Q8×C127—21no—
181Q8×C127C2×D412yes—
182D4×C127—24no—
183D4×C127—21no—
184D4×C127—21no—
185D4×C127—21no—
186D4×C127—81no—
187D4×C127C2×D412yes—
188C2×C4×C127—24no—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—21no—
193C2×C4×C127C2×D411yescharacteristic
194C2×C4×C127C2411yescharacteristic
195C8×C127—21no—
196C8×C127—21no—
197C22×C127—21no—
198C22×C127—22no—
199C22×C127—21no—
200C22×C127—21no—
201C22×C127—41no—
202C22×C127—41no—
203C22×C127—41no—
204C22×C127—41no—
205C22×C127—41no—
206C22×C127C22×D411yescharacteristic
207C2×D127—21no—
208C2×D127—21no—
209C2×D127—21no—
210C2×D127—21no—
211C2×D127—21no—
212C2×D127—41no—
213C2×D127—41no—
214C2×D127—41no—
215C2×D127—84no—
216C2×D127#12012yes—
217C4×C127—21no—
218C4×C127—22no—
219C4×C127—41no—
220C4×C127C22×D411yescharacteristic
221C4×C127C22×D411yescharacteristic
222Dic127—22no—
223Dic127—22no—
224Dic127—21no—
225Dic127—41no—
226Dic127#12012yes—
227C2×C127—41no—
228C2×C127—42no—
229C2×C127—81no—
230C2×C127C2×#12011yescharacteristic
231C2×C127#57711yescharacteristic
232C2×C127#57711yescharacteristic
233D127—22no—
234D127—41no—
235D127—81no—
236#3010—1271nomaximal, Sylow
237C127#301011yescharacteristic, complemented, Sylow
238C2×2−1+4—1271no—
239C2×AGL(1,ℤ/8)—1271no—
240C22×SD16—1271no—
241#535—1271no—
242#449—1271no—
243#449—1271no—
244#449—1271no—
245#449D12711yescharacteristic, complemented
246#448—1271no—
247#448—1271no—
248#448—1271no—
249#448—1271no—
250#419—1271no—
251#416—1271no—
252#408—1271no—
2532−1+4—2544no—
2542−1+4—2544no—
255C2×#41—1271no—
256C2×#41—1271no—
257C2×#41—1271no—
258C2×#41—2541no—
259C2×#41—2541no—
260C2×#41—2541no—
261C2×#41C2×D12711yescharacteristic, complemented
262C22×Q8—1271no—
263C22×Q8—1271no—
264C22×Q8—1271no—
265C22×Q8—2541no—
266C22×D4—1271no—
267AGL(1,ℤ/8)—2544no—
268C2×SD16—1271no—
269C2×SD16—1271no—
270C2×SD16—1271no—
271C2×SD16—1271no—
272C2×SD16—1271no—
273C2×SD16—2544no—
274C2×SD16C2×D12711yescharacteristic
275C2×D8—1271no—
276C2×D8C2×D12711yescharacteristic
277C2×M16—1271no—
278C22×C8—1271no—
279#127—2541no—
280#124—2541no—
281#121—1271no—
282#121—1271no—
283#121—1271no—
284#121C2×D12711yescharacteristic
285#120—2541no—
286#120—2541no—
287#117—1271no—
288#103—1271no—
289#103—1271no—
290#103—1271no—
291#103—1271no—
292#103—1271no—
293#103C2×D12711yescharacteristic, complemented
294#102—1271no—
295#102C2×D12711yescharacteristic, complemented
296C22⋊C8—1271no—
297C22⋊C8—1271no—
298C22⋊C8—1271no—
299C22⋊C8C2×D12711yescharacteristic, complemented
300C24—2541no—
301#41—22no—
302#41—22no—
303#41—1274no—
304#41—2542no—
305#41—2542no—
306#41—2548no—
307#41—2542no—
308#41—2542no—
309#41—2548no—
310#41—2548no—
311#41—2542no—
312C2×Q8—1271no—
313C2×Q8—1274no—
314C2×Q8—1271no—
315C2×Q8—1271no—
316C2×Q8—2541no—
317C2×Q8—2541no—
318C2×Q8—2542no—
319C2×Q8—2542no—
320C2×Q8—2548no—
321C2×Q8—2542no—
322C2×Q8—2542no—
323C2×Q8—2542no—
324C2×Q8—2541no—
325C2×Q8C22×D12711yescharacteristic
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—1271no—
329C2×D4—1271no—
330C2×D4—1271no—
331C2×D4—1271no—
332C2×D4—2541no—
333C2×D4—2542no—
334C2×D4—2542no—
335C2×D4—2541no—
336C2×D4—2541no—
337C2×D4—2541no—
338C2×D4—2541no—
339C2×D4—2541no—
340C2×D4C22×D12711yescharacteristic
341C2×D4C22×D12711yescharacteristic
342C22×C4—21no—
343C22×C4—1271no—
344C22×C4—1271no—
345C22×C4—2541no—
346C22×C4—2541no—
347C22×C4—2541no—
348C22×C4—2541no—
349C22×C4—2541no—
350C22×C4C22×D12711yescharacteristic
351SD16—22no—
352SD16—2542no—
353SD16—2542no—
354SD16—2542no—
355SD16—2542no—
356SD16—2542no—
357D8—22no—
358D8—2542no—
359M16—2542no—
360C2×C8—1271no—
361C2×C8—1271no—
362C2×C8—2542no—
363C2×C8C22×D12711yescharacteristic
364C2×C8C22×D12711yescharacteristic
365C4⋊C4—1271no—
366C4⋊C4C22×D12711yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—2541no—
369C22⋊C4—2541no—
370C22⋊C4—2541no—
371C22⋊C4—2541no—
372C42—2541no—
373C23—21no—
374C23—21no—
375C23—1271no—
376C23—1271no—
377C23—10161no—
378C23—10161no—
379C23—5082no—
380C23—2541no—
381C23D4×D12711yescharacteristic
382Q8—21no—
383Q8—1272no—
384Q8—1272no—
385Q8—2542no—
386Q8—2544no—
387Q8—2544no—
388Q8—2544no—
389Q8—2541no—
390Q8—2541no—
391Q8D4×D12712yes—
392D4—24no—
393D4—21no—
394D4—21no—
395D4—21no—
396D4—81no—
397D4—10161no—
398D4—10161no—
399D4—10161no—
400D4—5082no—
401D4—2542no—
402D4—2542no—
403D4—2542no—
404D4—2544no—
405D4—2544no—
406D4—2541no—
407D4—2541no—
408D4D4×D12712yes—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—1271no—
415C2×C4—1272no—
416C2×C4—1272no—
417C2×C4—1271no—
418C2×C4—2544no—
419C2×C4—2544no—
420C2×C4—2544no—
421C2×C4—2541no—
422C2×C4—2544no—
423C2×C4—2544no—
424C2×C4—2541no—
425C2×C4—2541no—
426C2×C4—2541no—
427C2×C4—2541no—
428C2×C4D4×D12711yescharacteristic
429C2×C4C23×D12711yescharacteristic
430C8—21no—
431C8—21no—
432C8—2541no—
433C8—2541no—
434C22—21no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—1272no—
444C22—10164no—
445C22—5081no—
446C22—5081no—
447C22—5081no—
448C22—2541no—
449C22—2541no—
450C22—2541no—
451C22—2541no—
452C22—2541no—
453C22C2×D4×D12711yescharacteristic
454C4—21no—
455C4—22no—
456C4—41no—
457C4—1272no—
458C4—5081no—
459C4—2541no—
460C4—2542no—
461C4—2542no—
462C4C2×D4×D12711yescharacteristic
463C4C2×D4×D12711yescharacteristic
464C2—41no—
465C2—42no—
466C2—81no—
467C2—10161no—
468C2—5081no—
469C2—2542no—
470C2#120×D12711yescharacteristic
471C2#2255068311yescharacteristic
472C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3010
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16861256C2semidirect
2. C2×#16861233C2semidirect
3. C2×SD16×D127C2semidirect
4. #22550442C2semidirect
5. #22550022C2semidirect
6. #22550021C2semidirect
7. #22550018C2semidirect
8. #22550017C2semidirect
9. #22550016C2semidirect
10. #22550014C2semidirect
11. #22550013C2semidirect
12. #22549883C2semidirect
13. #22549869C2semidirect
14. #22549841C2semidirect
15. #449×C127C2semidirect
23. C2×#14523247C22semidirect
25. C22×Dic254C22semidirect
27. C2×Q8×D127C22semidirect
47. #16861134C22semidirect
48. #16861134C22semidirect
49. #16861133C22semidirect
50. #16861132C22semidirect
51. #16861132C22semidirect
52. #16861130C22semidirect
53. C22⋊(C127⋊C8)C22semidirect
54. (C2×D127)⋊C8C22semidirect
55. (C2×D127)⋊C8C22semidirect
56. C2×#41×C127C22semidirect
60. #103×C127C22semidirect
61. #102×C127C22semidirect
62. C22⋊C8×C127C22semidirect
72. #14523247D4semidirect
85. Q8×D127D4semidirect
237. C127#3010semidirect
245. #449D127semidirect
261. C2×#41C2×D127semidirect
293. #103C2×D127semidirect
295. #102C2×D127semidirect
299. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
C2×#168612332
C2×SD16×D1272
#225504422
#225500222
#225500212
#225500182
#225500172
#225500162
#225500142
#225500132
#225498832
#225498692
#225498412
#449×C1272
#3010127