Finite Group Databasebeta

← #30647973

Subgroups — #30647973

Order 16256.

758 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D127C211yesmaximal, characteristic, complemented
2C2×#16861241C211yesmaximal, characteristic, complemented
3C2×#16861236C211yesmaximal, characteristic, complemented
4#22550389C211yesmaximal, characteristic, complemented
5#22550037C211yesmaximal, characteristic, complemented
6#22550036C211yesmaximal, characteristic, complemented
7#22550035C211yesmaximal, characteristic, complemented
8#22550024C211yesmaximal, characteristic, complemented
9#22550023C211yesmaximal, characteristic, complemented
10#22550022C211yesmaximal, characteristic, complemented
11#22550021C211yesmaximal, characteristic, complemented
12#22549883C211yesmaximal, characteristic, complemented
13#22549879C211yesmaximal, characteristic, complemented
14#22549838C211yesmaximal, characteristic, complemented
15#452×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523249C2211yescharacteristic
18C2×#14523248—21no—
19C2×#14523248C2211yescharacteristic
20C2×#14523247—21no—
21C2×#14523247C2211yescharacteristic
22#41×D127—24no—
23#41×D127—24no—
24C2×Q8×D127C2211yescharacteristic
25C2×D4×D127—21no—
26C2×D4×D127C2211yescharacteristic
27C22×C4×D127—21no—
28C22×C4×D127C2211yescharacteristic
29#16861241—24no—
30#16861236—24no—
31C2×#14523238C2211yescharacteristic, complemented
32C2×#14523237C2211yescharacteristic, complemented
33C2×#14523235C2211yescharacteristic, complemented
34C2×#14523235C2211yescharacteristic, complemented
35C2×D1016C2211yescharacteristic, complemented
36C2×#14523233C2211yescharacteristic, complemented
37C2×#14523231C2211yescharacteristic
38C2×#14523231C2211yescharacteristic
39#16861194—21no—
40#16861185C2211yescharacteristic, complemented
41#16861184C2211yescharacteristic, complemented
42#16861183C2211yescharacteristic, complemented
43#16861179—21no—
44#16861171—21no—
45(C22⋊D127)⋊C4—21no—
46#16861167C2211yescharacteristic
47#16861134C2211yescharacteristic, complemented
48#16861133C2211yescharacteristic, complemented
49#16861132C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic, complemented
52#16861129C2211yescharacteristic, complemented
53C22⋊(C127⋊C8)C2211yescharacteristic, complemented
54(C2×D127)⋊C8C2211yescharacteristic, complemented
55(C2×D127)⋊C8C2211yescharacteristic, complemented
56C2×#41×C127C2211yescharacteristic
57C2×Q16×C127C2211yescharacteristic, complemented
58C2×SD16×C127C2211yescharacteristic, complemented
59C22⋊Q8×C127C2211yescharacteristic, complemented
60#103×C127C2211yescharacteristic, complemented
61#102×C127C2211yescharacteristic, complemented
62C22⋊C8×C127C2211yescharacteristic, complemented
63C23×D127—21no—
64C23×D127D411yescharacteristic
65#14523249—22no—
66#14523249—22no—
67#14523249—22no—
68#14523249—22no—
69#14523248—28no—
70#14523248—22no—
71#14523248—22no—
72#14523247—28no—
73#14523247—22no—
74#14523247—22no—
75C2×Dic254—21no—
76C2×Dic254C2311yescharacteristic
77Q8×D127—22no—
78Q8×D127D414yescomplemented
79C2×D508—21no—
80C2×D508—21no—
81C2×D508—21no—
82C2×D508C2311yescharacteristic
83C2×D508C2311yescharacteristic
84C2×C22⋊D127—21no—
85C2×C22⋊D127—21no—
86C2×C22⋊D127—21no—
87C2×C22⋊D127—21no—
88D4×D127—28no—
89D4×D127—22no—
90D4×D127D414yescomplemented
91C22×Dic127—21no—
92C22×Dic127D411yescharacteristic
93C2×C4×D127—28no—
94C2×C4×D127—21no—
95C2×C4×D127—21no—
96C2×C4×D127—22no—
97C2×C4×D127—22no—
98C2×C4×D127—21no—
99C2×C4×D127—21no—
100C2×C4×D127D411yescharacteristic
101C2×C4×D127D411yescharacteristic
102C2×C4×D127C2311yescharacteristic
103C2×C4×D127C2311yescharacteristic
104#14523238—22no—
105#14523237—22no—
106#14523235—22no—
107#14523235—22no—
108D1016—22no—
109#14523233—22no—
110#14523231—22no—
111#14523231—22no—
112C2×C127⋊C8C2311yescharacteristic
113C2×C127⋊C8C2311yescharacteristic
114C4⋊Dic127C2311yescharacteristic
115Dic127⋊C4—21no—
116(C2×D127)⋊C4—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×Q8×C127C2311yescharacteristic
125C2×D4×C127—21no—
126C2×D4×C127C2311yescharacteristic
127C22×C4×C127—21no—
128C22×C4×C127C2311yescharacteristic
129Q16×C127—22no—
130SD16×C127—22no—
131C2×C8×C127C2311yescharacteristic
132C2×C8×C127C2311yescharacteristic
133C4⋊C4×C127—21no—
134C4⋊C4×C127C2311yescharacteristic
135C22⋊C4×C127—21no—
136C22×D127—21no—
137C22×D127—22no—
138C22×D127—24no—
139C22×D127—21no—
140C22×D127—41no—
141C22×D127—41no—
142C22×D127—42no—
143C22×D127—42no—
144C22×D127C2×D411yescharacteristic
145C22×D127C2×D411yescharacteristic
146Dic254—21no—
147Dic254—24no—
148Dic254C2×D412yes—
149D508—24no—
150D508—21no—
151D508—21no—
152D508—21no—
153D508—42no—
154D508—42no—
155D508C2×D412yes—
156C22⋊D127—24no—
157C22⋊D127—24no—
158C22⋊D127—24no—
159C22⋊D127—42no—
160C2×Dic127—24no—
161C2×Dic127—21no—
162C2×Dic127—22no—
163C2×Dic127—21no—
164C2×Dic127C2×D411yescharacteristic
165C2×Dic127C2×D411yescharacteristic
166C4×D127—24no—
167C4×D127—24no—
168C4×D127—22no—
169C4×D127—24no—
170C4×D127—21no—
171C4×D127—21no—
172C4×D127—22no—
173C4×D127—42no—
174C4×D127C2×D412yes—
175C4×D127C2×D412yes—
176C127⋊C8—21no—
177C127⋊C8—21no—
178C23×C127—21no—
179C23×C127C2×D411yescharacteristic
180Q8×C127—21no—
181Q8×C127—21no—
182Q8×C127—21no—
183Q8×C127C2×D412yes—
184D4×C127—24no—
185D4×C127—21no—
186D4×C127C2×D412yes—
187C2×C4×C127—24no—
188C2×C4×C127—21no—
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×C127C22×D411yescharacteristic
204C2×D127—21no—
205C2×D127—21no—
206C2×D127—21no—
207C2×D127—22no—
208C2×D127—21no—
209C2×D127—21no—
210C2×D127—44no—
211C2×D127—48no—
212C2×D127—41no—
213C2×D127—41no—
214C2×D127—41no—
215C2×D127—41no—
216C2×D127—41no—
217C2×D127#12012yes—
218C4×C127—21no—
219C4×C127—22no—
220C4×C127—41no—
221C4×C127—41no—
222C4×C127C22×D411yescharacteristic
223C4×C127C22×D411yescharacteristic
224Dic127—21no—
225Dic127—22no—
226Dic127—41no—
227Dic127#12012yes—
228C2×C127—41no—
229C2×C127—42no—
230C2×C127C2×#12011yescharacteristic
231C2×C127#57511yescharacteristic
232C2×C127#57511yescharacteristic
233D127—22no—
234D127—41no—
235D127—42no—
236D127—81no—
237#3005—1271nomaximal, Sylow
238C127#300511yescharacteristic, complemented, Sylow
239C22×#41—1271no—
240C2×#137—1271no—
241C2×AGL(1,ℤ/8)—1271no—
242#525—1271no—
243#452—1271no—
244#452—1271no—
245#452—1271no—
246#452D12711yescharacteristic, complemented
247#449—1271no—
248#449—1271no—
249#449—1271no—
250#449—1271no—
251#419—1271no—
252#418—1271no—
253#407—1271no—
254C2×#41—1271no—
255C2×#41—1271no—
256C2×#41—1271no—
257C2×#41—1271no—
258C2×#41—2541no—
259C2×#41—2541no—
260C2×#41—2544no—
261C2×#41—2544no—
262C2×#41C2×D12711yescharacteristic
263C22×Q8—1271no—
264C22×D4—1271no—
265C22×D4—2541no—
266C23×C4—1271no—
267C23×C4—2541no—
268#137—2544no—
269AGL(1,ℤ/8)—2544no—
270C2×Q16—1271no—
271C2×Q16C2×D12711yescharacteristic, complemented
272C2×SD16—1271no—
273C2×SD16—1271no—
274C2×SD16—1271no—
275C2×SD16C2×D12711yescharacteristic, complemented
276C2×D8—1271no—
277C2×D8—1271no—
278C2×M16—1271no—
279C2×M16—1271no—
280#123—2541no—
281C22⋊Q8—1271no—
282C22⋊Q8C2×D12711yescharacteristic, complemented
283#121—1271no—
284#121—1271no—
285#120—2541no—
286C4×D4—2541no—
287C4×D4—2541no—
288#117—1271no—
289#103—1271no—
290#103—1271no—
291#103—1271no—
292#103C2×D12711yescharacteristic, complemented
293#102—1271no—
294#102—1271no—
295#102—1271no—
296#102C2×D12711yescharacteristic, complemented
297C22⋊C8—1271no—
298C22⋊C8—1271no—
299C22⋊C8—1271no—
300C22⋊C8C2×D12711yescharacteristic, complemented
301C24—1271no—
302C24—2541no—
303#41—22no—
304#41—22no—
305#41—2542no—
306#41—2542no—
307#41—2548no—
308#41—2542no—
309#41—2542no—
310#41—2542no—
311#41—2548no—
312#41—2542no—
313#41—2542no—
314#41—2542no—
315C2×Q8—1274no—
316C2×Q8—1271no—
317C2×Q8—2541no—
318C2×Q8—2542no—
319C2×Q8C22×D12711yescharacteristic
320C2×Q8C22×D12711yescharacteristic
321C2×D4—21no—
322C2×D4—1274no—
323C2×D4—1271no—
324C2×D4—1271no—
325C2×D4—2541no—
326C2×D4—2541no—
327C2×D4—2541no—
328C2×D4—2542no—
329C2×D4—2541no—
330C2×D4—2548no—
331C2×D4—2541no—
332C2×D4—2541no—
333C2×D4—2541no—
334C2×D4C22×D12711yescharacteristic
335C22×C4—21no—
336C22×C4—1271no—
337C22×C4—1271no—
338C22×C4—1271no—
339C22×C4—1271no—
340C22×C4—1271no—
341C22×C4—2541no—
342C22×C4—2542no—
343C22×C4—2542no—
344C22×C4—2541no—
345C22×C4—2541no—
346C22×C4—2548no—
347C22×C4—2541no—
348C22×C4—2541no—
349C22×C4C22×D12711yescharacteristic
350Q16—22no—
351Q16—2542no—
352SD16—22no—
353SD16—2542no—
354SD16—2542no—
355SD16—2542no—
356D8—2542no—
357D8—2542no—
358M16—2542no—
359M16—2542no—
360C2×C8—1271no—
361C2×C8—1271no—
362C2×C8C22×D12711yescharacteristic
363C2×C8C22×D12711yescharacteristic
364C4⋊C4—21no—
365C4⋊C4—1271no—
366C4⋊C4—2541no—
367C4⋊C4C22×D12711yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—2541no—
370C22⋊C4—2541no—
371C22⋊C4—2541no—
372C22⋊C4—2541no—
373C42—2541no—
374C23—21no—
375C23—1271no—
376C23—1271no—
377C23—5081no—
378C23—5081no—
379C23—5082no—
380C23—5082no—
381C23—2544no—
382C23—2542no—
383C23—2541no—
384C23—2541no—
385C23D4×D12711yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—1272no—
390Q8—2544no—
391Q8—2541no—
392Q8D4×D12712yes—
393D4—24no—
394D4—21no—
395D4—1272no—
396D4—5082no—
397D4—5082no—
398D4—5082no—
399D4—2544no—
400D4—2544no—
401D4—2544no—
402D4—2544no—
403D4—2541no—
404D4—2541no—
405D4—2541no—
406D4D4×D12712yes—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—1271no—
414C2×C4—1272no—
415C2×C4—1272no—
416C2×C4—1271no—
417C2×C4—5082no—
418C2×C4—2542no—
419C2×C4—2542no—
420C2×C4—2544no—
421C2×C4—2544no—
422C2×C4—2544no—
423C2×C4—2542no—
424C2×C4—2541no—
425C2×C4—2544no—
426C2×C4—2541no—
427C2×C4—2541no—
428C2×C4—2541no—
429C2×C4D4×D12711yescharacteristic
430C2×C4C23×D12711yescharacteristic
431C8—21no—
432C8—21no—
433C8—2541no—
434C8—2541no—
435C22—21no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—41no—
440C22—41no—
441C22—1272no—
442C22—5084no—
443C22—5088no—
444C22—5081no—
445C22—5081no—
446C22—5081no—
447C22—5081no—
448C22—5081no—
449C22—2542no—
450C22—2541no—
451C22—2541no—
452C22—2541no—
453C22—2541no—
454C22—2541no—
455C22C2×D4×D12711yescharacteristic
456C4—21no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—1272no—
461C4—5081no—
462C4—2541no—
463C4—2542no—
464C4C2×D4×D12711yescharacteristic
465C4C2×D4×D12711yescharacteristic
466C2—41no—
467C2—42no—
468C2—10161no—
469C2—5082no—
470C2—5081no—
471C2—2542no—
472C2#120×D12711yescharacteristic
473C2#2255067311yescharacteristic
474C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#452×C127C2
socleC22×C127C22×D4
hypercenter#452D127
hypocenterC127#3005
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#41×D127C2semidirect
2. C2×#16861241C2semidirect
3. C2×#16861236C2semidirect
4. #22550389C2semidirect
5. #22550037C2semidirect
6. #22550036C2semidirect
7. #22550035C2semidirect
8. #22550024C2semidirect
9. #22550023C2semidirect
10. #22550022C2semidirect
11. #22550021C2semidirect
12. #22549883C2semidirect
13. #22549879C2semidirect
14. #22549838C2semidirect
15. #452×C127C2semidirect
31. C2×#14523238C22semidirect
32. C2×#14523237C22semidirect
33. C2×#14523235C22semidirect
34. C2×#14523235C22semidirect
35. C2×D1016C22semidirect
36. C2×#14523233C22semidirect
40. #16861185C22semidirect
41. #16861184C22semidirect
42. #16861183C22semidirect
47. #16861134C22semidirect
48. #16861133C22semidirect
49. #16861132C22semidirect
50. #16861131C22semidirect
51. #16861130C22semidirect
52. #16861129C22semidirect
53. C22⋊(C127⋊C8)C22semidirect
54. (C2×D127)⋊C8C22semidirect
55. (C2×D127)⋊C8C22semidirect
57. C2×Q16×C127C22semidirect
58. C2×SD16×C127C22semidirect
59. C22⋊Q8×C127C22semidirect
60. #103×C127C22semidirect
61. #102×C127C22semidirect
62. C22⋊C8×C127C22semidirect
78. Q8×D127D4semidirect
90. D4×D127D4semidirect
238. C127#3005semidirect
246. #452D127semidirect
271. C2×Q16C2×D127semidirect
275. C2×SD16C2×D127semidirect
282. C22⋊Q8C2×D127semidirect
292. #103C2×D127semidirect
296. #102C2×D127semidirect
300. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1272
C2×#168612412
C2×#168612362
#225503892
#225500372
#225500362
#225500352
#225500242
#225500232
#225500222
#225500212
#225498832
#225498792
#225498382
#452×C1272
#3005127