Finite Group Databasebeta

← #30647987

Subgroups — #30647987

Order 16256.

754 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D127C211yesmaximal, characteristic
2C2×#16861241C211yesmaximal, characteristic, complemented
3C2×#16861238C211yesmaximal, characteristic, complemented
4C22⋊Q8×D127C211yesmaximal, characteristic, complemented
5#22550040C211yesmaximal, characteristic, complemented
6#22550038C211yesmaximal, characteristic, complemented
7#22550036C211yesmaximal, characteristic, complemented
8#22550033C211yesmaximal, characteristic, complemented
9#22550030C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22550025C211yesmaximal, characteristic, complemented
12#22549876C211yesmaximal, characteristic, complemented
13#22549874C211yesmaximal, characteristic, complemented
14#22549838C211yesmaximal, characteristic, complemented
15#452×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248—21no—
18C2×#14523248C2211yescharacteristic
19C2×#14523247—21no—
20C2×#14523247C2211yescharacteristic
21#41×D127—24no—
22#41×D127—24no—
23C2×Q8×D127C2211yescharacteristic
24C2×Q8×D127C2211yescharacteristic, complemented
25C2×D4×D127—21no—
26C2×D4×D127C2211yescharacteristic
27C22×C4×D127—21no—
28C22×C4×D127C2211yescharacteristic
29#16861241—24no—
30#16861238—24no—
31C2×Dic508C2211yescharacteristic, complemented
32C2×#14523238C2211yescharacteristic, complemented
33C2×#14523238C2211yescharacteristic, complemented
34C2×#14523237C2211yescharacteristic, complemented
35C2×#14523236C2211yescharacteristic, complemented
36C2×#14523235C2211yescharacteristic, complemented
37C2×#14523231C2211yescharacteristic, complemented
38C2×#14523231C2211yescharacteristic, complemented
39#16861189C2211yescharacteristic
40#16861188C2211yescharacteristic, complemented
41#16861187—21no—
42#16861186—21no—
43#16861185C2211yescharacteristic
44C4⋊C4×D127—21no—
45C4⋊C4×D127C2211yescharacteristic, complemented
46C22⋊C4×D127—21no—
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
55(C2×D127)⋊C8C2211yescharacteristic
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#14523249D414yes—
67#14523248—28no—
68#14523248—22no—
69#14523248—22no—
70#14523247—28no—
71#14523247—22no—
72#14523247D414yes—
73C2×Dic254—21no—
74C2×Dic254—21no—
75C2×Dic254C2311yescharacteristic
76C2×Dic254C2311yescharacteristic
77Q8×D127—22no—
78Q8×D127—22no—
79Q8×D127—22no—
80Q8×D127—22no—
81C2×D508—21no—
82C2×D508C2311yescharacteristic
83C2×C22⋊D127—21no—
84C2×C22⋊D127—21no—
85C2×C22⋊D127—21no—
86D4×D127—28no—
87D4×D127—22no—
88D4×D127—22no—
89C22×Dic127—21no—
90C22×Dic127D411yescharacteristic
91C2×C4×D127—28no—
92C2×C4×D127—21no—
93C2×C4×D127—21no—
94C2×C4×D127—22no—
95C2×C4×D127—22no—
96C2×C4×D127—21no—
97C2×C4×D127—21no—
98C2×C4×D127—21no—
99C2×C4×D127D411yescharacteristic
100C2×C4×D127D411yescharacteristic
101C2×C4×D127C2311yescharacteristic
102C2×C4×D127C2311yescharacteristic
103Dic508—22no—
104#14523238—22no—
105#14523238—22no—
106#14523237—22no—
107#14523236—22no—
108#14523235—22no—
109#14523231—22no—
110#14523231—22no—
111C2×C127⋊C8C2311yescharacteristic
112C2×C127⋊C8C2311yescharacteristic
113C4⋊Dic127—21no—
114C4⋊Dic127C2311yescharacteristic
115Dic127⋊C4—21no—
116Dic127⋊C4—21no—
117Dic127⋊C4—21no—
118C22⋊Dic127—21no—
119(C2×D127)⋊C4—21no—
120(C2×D127)⋊C4—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—24no—
137C22×D127—21no—
138C22×D127—22no—
139C22×D127—22no—
140C22×D127—41no—
141C22×D127—42no—
142C22×D127—42no—
143C22×D127C2×D411yescharacteristic
144C22×D127C2×D411yescharacteristic
145Dic254—24no—
146Dic254—21no—
147Dic254—21no—
148Dic254—21no—
149Dic254—42no—
150Dic254C2×D412yes—
151D508—24no—
152D508—21no—
153D508C2×D412yes—
154C22⋊D127—24no—
155C22⋊D127—24no—
156C22⋊D127—24no—
157C2×Dic127—21no—
158C2×Dic127—24no—
159C2×Dic127—22no—
160C2×Dic127—21no—
161C2×Dic127—21no—
162C2×Dic127C2×D411yescharacteristic
163C2×Dic127C2×D411yescharacteristic
164C4×D127—24no—
165C4×D127—24no—
166C4×D127—22no—
167C4×D127—21no—
168C4×D127—21no—
169C4×D127—22no—
170C4×D127—24no—
171C4×D127—42no—
172C4×D127—42no—
173C4×D127C2×D412yes—
174C4×D127C2×D412yes—
175C127⋊C8—21no—
176C127⋊C8—21no—
177C23×C127—21no—
178C23×C127C2×D411yescharacteristic
179Q8×C127—21no—
180Q8×C127—21no—
181Q8×C127—21no—
182Q8×C127C2×D412yes—
183D4×C127—24no—
184D4×C127—21no—
185D4×C127C2×D412yes—
186C2×C4×C127—24no—
187C2×C4×C127—21no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127C2×D411yescharacteristic
193C2×C4×C127C2411yescharacteristic
194C8×C127—21no—
195C8×C127—21no—
196C22×C127—21no—
197C22×C127—22no—
198C22×C127—21no—
199C22×C127—21no—
200C22×C127—41no—
201C22×C127—41no—
202C22×C127C22×D411yescharacteristic
203C2×D127—21no—
204C2×D127—21no—
205C2×D127—21no—
206C2×D127—21no—
207C2×D127—22no—
208C2×D127—44no—
209C2×D127—48no—
210C2×D127—41no—
211C2×D127—41no—
212C2×D127#12012yes—
213C2×D127#13712yescomplemented
214C4×C127—21no—
215C4×C127—22no—
216C4×C127—41no—
217C4×C127—41no—
218C4×C127C22×D411yescharacteristic
219C4×C127C22×D411yescharacteristic
220Dic127—22no—
221Dic127—21no—
222Dic127—41no—
223Dic127—41no—
224Dic127#12012yes—
225C2×C127—41no—
226C2×C127—42no—
227C2×C127C2×#12011yescharacteristic
228C2×C127C2×#13711yescharacteristic
229C2×C127#57511yescharacteristic
230D127—22no—
231D127—22no—
232D127—42no—
233#3007—1271nomaximal, Sylow
234C127#300711yescharacteristic, complemented, Sylow
235C22×#41—1271no—
236C2×#137—1271no—
237C2×#137—1271no—
238C2×C22⋊Q8—1271no—
239#452—1271no—
240#452—1271no—
241#452—1271no—
242#452D12711yescharacteristic, complemented
243#451—1271no—
244#451—1271no—
245#450—1271no—
246#450—1271no—
247#417—1271no—
248#417—1271no—
249#407—1271no—
250C2×#41—1271no—
251C2×#41—1271no—
252C2×#41—1271no—
253C2×#41—2541no—
254C2×#41—2541no—
255C2×#41—2544no—
256C2×#41—2544no—
257C2×#41C2×D12711yescharacteristic
258C22×Q8—1271no—
259C22×Q8—1271no—
260C22×D4—1271no—
261C22×D4—2541no—
262C23×C4—1271no—
263C23×C4—2541no—
264#137—2544no—
265#137—2544no—
266C2×Q16—1271no—
267C2×Q16—1271no—
268C2×Q16—1271no—
269C2×Q16C2×D12711yescharacteristic, complemented
270C2×SD16—1271no—
271C2×SD16—1271no—
272C2×SD16—1271no—
273C2×SD16C2×D12711yescharacteristic, complemented
274C2×M16—1271no—
275C2×M16—1271no—
276C22⋊Q8—1271no—
277C22⋊Q8—1271no—
278C22⋊Q8—1271no—
279C22⋊Q8—2541no—
280C22⋊Q8—2541no—
281C22⋊Q8C2×D12711yescharacteristic, complemented
282C2×C4⋊C4—1271no—
283C2×C4⋊C4—2541no—
284C2×C22⋊C4—2541no—
285#103—1271no—
286#103—1271no—
287#103—1271no—
288#103C2×D12711yescharacteristic, complemented
289#102—1271no—
290#102—1271no—
291#102—1271no—
292#102C2×D12711yescharacteristic, complemented
293C22⋊C8—1271no—
294C22⋊C8—1271no—
295C22⋊C8—1271no—
296C22⋊C8C2×D12711yescharacteristic, complemented
297C24—1271no—
298C24—2541no—
299#41—22no—
300#41—22no—
301#41—1274no—
302#41—1274no—
303#41—2542no—
304#41—2542no—
305#41—2548no—
306#41—2542no—
307#41—2542no—
308#41—2548no—
309C2×Q8—1271no—
310C2×Q8—1271no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8—2542no—
314C2×Q8—2542no—
315C2×Q8—2542no—
316C2×Q8—2542no—
317C2×Q8C22×D12711yescharacteristic
318C2×Q8C22×D12711yescharacteristic
319C2×D4—21no—
320C2×D4—1271no—
321C2×D4—2541no—
322C2×D4—2541no—
323C2×D4—2542no—
324C2×D4—2548no—
325C2×D4—2541no—
326C2×D4—2541no—
327C2×D4—2542no—
328C2×D4C22×D12711yescharacteristic
329C22×C4—21no—
330C22×C4—1271no—
331C22×C4—1271no—
332C22×C4—1271no—
333C22×C4—1271no—
334C22×C4—1271no—
335C22×C4—2542no—
336C22×C4—2542no—
337C22×C4—2541no—
338C22×C4—2541no—
339C22×C4—2541no—
340C22×C4—2541no—
341C22×C4—2548no—
342C22×C4—2541no—
343C22×C4—2541no—
344C22×C4C22×D12711yescharacteristic
345Q16—22no—
346Q16—2542no—
347Q16—2542no—
348Q16—2542no—
349SD16—22no—
350SD16—2542no—
351SD16—2542no—
352SD16—2542no—
353M16—2542no—
354M16—2542no—
355C2×C8—1271no—
356C2×C8—1271no—
357C2×C8C22×D12711yescharacteristic
358C2×C8C22×D12711yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—1271no—
361C4⋊C4—2541no—
362C4⋊C4—2541no—
363C4⋊C4—2541no—
364C4⋊C4—2541no—
365C4⋊C4C22×D12711yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—2541no—
368C22⋊C4—2541no—
369C22⋊C4—2541no—
370C23—21no—
371C23—1271no—
372C23—1271no—
373C23—5081no—
374C23—5082no—
375C23—5082no—
376C23—2542no—
377C23—2542no—
378C23—2544no—
379C23—2541no—
380C23D4×D12711yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—1272no—
385Q8—5082no—
386Q8—2544no—
387Q8—2541no—
388Q8—2541no—
389Q8—2541no—
390Q8D4×D12712yes—
391D4—24no—
392D4—21no—
393D4—1272no—
394D4—2544no—
395D4—2544no—
396D4—2544no—
397D4—2544no—
398D4—2541no—
399D4D4×D12712yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—1271no—
407C2×C4—1272no—
408C2×C4—1272no—
409C2×C4—1271no—
410C2×C4—5082no—
411C2×C4—5082no—
412C2×C4—2542no—
413C2×C4—2544no—
414C2×C4—2542no—
415C2×C4—2544no—
416C2×C4—2544no—
417C2×C4—2542no—
418C2×C4—2544no—
419C2×C4—2541no—
420C2×C4—2541no—
421C2×C4—2541no—
422C2×C4—2541no—
423C2×C4—2541no—
424C2×C4D4×D12711yescharacteristic
425C2×C4C23×D12711yescharacteristic
426C8—21no—
427C8—21no—
428C8—2541no—
429C8—2541no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—1272no—
437C22—1272no—
438C22—5088no—
439C22—5084no—
440C22—5081no—
441C22—5081no—
442C22—2542no—
443C22—2541no—
444C22—2541no—
445C22—2541no—
446C22—2541no—
447C22C2×D4×D12711yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—1272no—
453C4—5081no—
454C4—5081no—
455C4—2541no—
456C4—2542no—
457C4C2×D4×D12711yescharacteristic
458C4C2×D4×D12711yescharacteristic
459C2—41no—
460C2—42no—
461C2—5082no—
462C2—2542no—
463C2—2542no—
464C2#120×D12711yescharacteristic
465C2#137×D12711yescharacteristic
466C2#2255067311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#452×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×D127C2
hypercenter#452D127
hypocenterC127#3007
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
2. C2×#16861241C2semidirect
3. C2×#16861238C2semidirect
4. C22⋊Q8×D127C2semidirect
5. #22550040C2semidirect
6. #22550038C2semidirect
7. #22550036C2semidirect
8. #22550033C2semidirect
9. #22550030C2semidirect
10. #22550028C2semidirect
11. #22550025C2semidirect
12. #22549876C2semidirect
13. #22549874C2semidirect
14. #22549838C2semidirect
15. #452×C127C2semidirect
24. C2×Q8×D127C22semidirect
31. C2×Dic508C22semidirect
32. C2×#14523238C22semidirect
33. C2×#14523238C22semidirect
34. C2×#14523237C22semidirect
35. C2×#14523236C22semidirect
36. C2×#14523235C22semidirect
37. C2×#14523231C22semidirect
38. C2×#14523231C22semidirect
40. #16861188C22semidirect
45. C4⋊C4×D127C22semidirect
47. #16861134C22semidirect
48. #16861133C22semidirect
49. #16861132C22semidirect
50. #16861131C22semidirect
51. #16861130C22semidirect
52. #16861129C22semidirect
53. C22⋊(C127⋊C8)C22semidirect
57. C2×Q16×C127C22semidirect
58. C2×SD16×C127C22semidirect
59. C22⋊Q8×C127C22semidirect
60. #103×C127C22semidirect
61. #102×C127C22semidirect
62. C22⋊C8×C127C22semidirect
213. C2×D127#137semidirect
234. C127#3007semidirect
242. #452D127semidirect
269. C2×Q16C2×D127semidirect
273. C2×SD16C2×D127semidirect
281. C22⋊Q8C2×D127semidirect
288. #103C2×D127semidirect
292. #102C2×D127semidirect
296. 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×#168612382
C22⋊Q8×D1272
#225500402
#225500382
#225500362
#225500332
#225500302
#225500282
#225500252
#225498762
#225498742
#225498382
#452×C1272
#3007127