Finite Group Databasebeta

← #30650503

Subgroups — #30650503

Order 16256.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550691C211yesmaximal, characteristic, complemented
2#22550690C211yesmaximal, characteristic, complemented
3#22550687C211yesmaximal, characteristic, complemented
4#22550267C211yesmaximal, characteristic, complemented
5#22550151C211yesmaximal, characteristic, complemented
6#22550149C211yesmaximal, characteristic, complemented
7#22550137C211yesmaximal, characteristic, complemented
8#22550136C211yesmaximal, characteristic, complemented
9#22550059C211yesmaximal, characteristic, complemented
10#22550057C211yesmaximal, characteristic, complemented
11#22550054C211yesmaximal, characteristic, complemented
12#22550047C211yesmaximal, characteristic, complemented
13#22549998C211yesmaximal, characteristic, complemented
14#22549996C211yesmaximal, characteristic, complemented
15#577×C127C211yesmaximal, characteristic, complemented
16#16861256C2211yescharacteristic
17#16861256C2211yescharacteristic, complemented
18#16861240—21no—
19#16861240C2211yescharacteristic, complemented
20#16861239—21no—
21#16861239—21no—
22#16861239C2211yescharacteristic
23#16861239C2211yescharacteristic
24#16861239C2211yescharacteristic
25#16861239C2211yescharacteristic, complemented
26#16861238—21no—
27#16861238—21no—
28#16861235—21no—
29#16861235C2211yescharacteristic
30#16861229—21no—
31#16861229—21no—
32#16861229C2211yescharacteristic
33#16861228—21no—
34C2×Dic508C2211yescharacteristic, complemented
35C2×#14523238—21no—
36C2×#14523238C2211yescharacteristic, complemented
37Q16×D127—21no—
38C2×#14523236—21no—
39C2×#14523236C2211yescharacteristic
40#16861216C2211yescharacteristic
41#16861216C2211yescharacteristic, complemented
42#16861215C2211yescharacteristic, complemented
43#16861207C2211yescharacteristic, complemented
44#16861196C2211yescharacteristic
45#16861145C2211yescharacteristic, complemented
46#16861144C2211yescharacteristic, complemented
47#16861136C2211yescharacteristic
48#16861136C2211yescharacteristic
49#16861136C2211yescharacteristic, complemented
50#16861135C2211yescharacteristic, complemented
51#16861128C2211yescharacteristic, complemented
52#16861127C2211yescharacteristic, complemented
53#16861127C2211yescharacteristic, complemented
54#16861126C2211yescharacteristic
55#16861119C2211yescharacteristic, complemented
562−1+4×C127C2211yescharacteristic
57H(5,𝔽2)×C127C2211yescharacteristic
58#137×C127—21no—
59#137×C127C2211yescharacteristic
60AGL(1,ℤ/8)×C127—21no—
61AGL(1,ℤ/8)×C127C2211yescharacteristic
62#135×C127—21no—
63#135×C127C2211yescharacteristic
64C2×SD16×C127—21no—
65C2×SD16×C127C2211yescharacteristic
66#131×C127C2211yescharacteristic
67#14523248—21no—
68#14523248—21no—
69#14523248C2311yescharacteristic
70#14523247—21no—
71#14523247—21no—
72#14523247—41no—
73#14523247—41no—
74C2×Dic254—21no—
75C2×Dic254—21no—
76C2×Dic254C2311yescharacteristic
77C2×Dic254C2311yescharacteristic
78Q8×D127—21no—
79Q8×D127—21no—
80Dic508—21no—
81Dic508—21no—
82#14523238—21no—
83#14523238—21no—
84#14523238—21no—
85#14523238—21no—
86#14523238—21no—
87#14523238—21no—
88#14523238—21no—
89#14523238D411yescharacteristic
90#14523236—21no—
91#14523236—21no—
92#14523236—21no—
93#14523236—21no—
94#14523236—21no—
95#14523236—21no—
96#14523236—21no—
97#14523236D411yescharacteristic
98#14523235—21no—
99#14523235D411yescharacteristic
100#14523233—21no—
101#14523233D411yescharacteristic
102#14523232—21no—
103#14523232—21no—
104#14523232C2311yescharacteristic
105#14523232C2311yescharacteristic
106#14523232C2311yescharacteristic
107#14523231—21no—
108C2×C127⋊C8—21no—
109C2×C127⋊C8—21no—
110C2×C127⋊C8C2311yescharacteristic
111C8×D127—21no—
112Dic127⋊C4—41no—
113C22⋊Dic127—41no—
114C4×Dic127C2311yescharacteristic
115#41×C127—21no—
116#41×C127—21no—
117#41×C127—21no—
118#41×C127—21no—
119#41×C127—41no—
120#41×C127C2311yescharacteristic
121#41×C127C2311yescharacteristic
122#41×C127C2311yescharacteristic
123C2×Q8×C127—21no—
124C2×Q8×C127—21no—
125C2×Q8×C127C2311yescharacteristic
126C2×D4×C127—21no—
127C2×D4×C127—21no—
128C2×D4×C127—41no—
129C2×D4×C127C2311yescharacteristic
130Q16×C127—21no—
131Q16×C127D411yescharacteristic
132SD16×C127—21no—
133SD16×C127—21no—
134SD16×C127—21no—
135SD16×C127—21no—
136SD16×C127D411yescharacteristic
137SD16×C127D411yescharacteristic
138D8×C127—21no—
139D8×C127D411yescharacteristic
140M16×C127—21no—
141M16×C127C2311yescharacteristic
142C2×C8×C127—21no—
143C2×C8×C127C2311yescharacteristic
144Dic254—21no—
145Dic254—21no—
146Dic254—21no—
147Dic254—21no—
148Dic254—41no—
149Dic254—41no—
150Dic254C2×D411yescharacteristic
151D508C2×D411yescharacteristic
152C22⋊D127—21no—
153C22⋊D127—41no—
154C22⋊D127—41no—
155C2×Dic127—21no—
156C2×Dic127—21no—
157C2×Dic127—21no—
158C2×Dic127—41no—
159C2×Dic127—41no—
160C4×D127—21no—
161C4×D127—41no—
162C4×D127—41no—
163C127⋊C8—21no—
164C127⋊C8—21no—
165C127⋊C8—21no—
166C127⋊C8C2×D411yescharacteristic
167C127⋊C8C2×D411yescharacteristic
168C23×C127—21no—
169C23×C127—41no—
170Q8×C127—21no—
171Q8×C127—21no—
172Q8×C127—41no—
173Q8×C127C2×D411yescharacteristic
174Q8×C127C2×D411yescharacteristic
175Q8×C127C2×D411yescharacteristic
176D4×C127—21no—
177D4×C127—21no—
178D4×C127—21no—
179D4×C127—21no—
180D4×C127—21no—
181D4×C127—41no—
182D4×C127—41no—
183D4×C127—41no—
184D4×C127C2×D411yescharacteristic
185D4×C127C2×D411yescharacteristic
186D4×C127C2×D411yescharacteristic
187C2×C4×C127—21no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—41no—
192C2×C4×C127—41no—
193C2×C4×C127—41no—
194C2×C4×C127C2411yescharacteristic
195C8×C127—21no—
196C8×C127C2×D411yescharacteristic
197C8×C127C2×D411yescharacteristic
198C22×C127—21no—
199C22×C127—21no—
200C22×C127—21no—
201C22×C127—41no—
202C22×C127—81no—
203C22×C127—161no—
204C22×C127H(5,𝔽2)11yescharacteristic
205C2×D127—21no—
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—21no—
209C4×C127—41no—
210C4×C127C22×D411yescharacteristic
211C4×C127C22×D411yescharacteristic
212Dic127—21no—
213Dic127—41no—
214Dic127—41no—
215Dic127—41no—
216C2×C127—21no—
217C2×C127—41no—
218C2×C127—41no—
219C2×C127—41no—
220C2×C127—81no—
221C2×C127D4211yescharacteristic
222D127—41no—
223#3289—1271nomaximal, Sylow
224C127#328911yescharacteristic, complemented, Sylow
225#578—1271no—
226#578—1271no—
227#577—1271no—
228#577D12711yescharacteristic, complemented
229#497—1271no—
230#473—1271no—
231#473—1271no—
232#471—1271no—
233#471—1271no—
234#456—1271no—
235#456—1271no—
236#455—1271no—
237#454—1271no—
238#444—1271no—
239#444—1271no—
2402−1+4—1271no—
2412−1+4—1271no—
2422−1+4C2×D12711yescharacteristic
243H(5,𝔽2)C2×D12711yescharacteristic
244#137—21no—
245#137—1271no—
246#137—1271no—
247#137—1271no—
248#137—1271no—
249#137—1271no—
250#137—2541no—
251#137—2541no—
252#137—2541no—
253#137—2541no—
254#137—2541no—
255#137C2×D12711yescharacteristic
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—1271no—
258AGL(1,ℤ/8)—2541no—
259AGL(1,ℤ/8)C2×D12711yescharacteristic
260#135—21no—
261#135—1271no—
262#135—2541no—
263#135—2541no—
264#135—2541no—
265#135C2×D12711yescharacteristic
266C2×Q16—1271no—
267C2×Q16—1271no—
268C2×Q16—2541no—
269C2×Q16—2541no—
270C2×SD16—21no—
271C2×SD16—1271no—
272C2×SD16—2541no—
273C2×SD16C2×D12711yescharacteristic
274#131—1271no—
275#131—1271no—
276#131—1271no—
277#131C2×D12711yescharacteristic
278C4⋊Q8—1271no—
279#124—1271no—
280#108—1271no—
281#108—1271no—
282#104—1271no—
283#104—1271no—
284#104—1271no—
285#104—1271no—
286#101—1271no—
287#101—1271no—
288#101—1271no—
289#100—1271no—
290#97—1271no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—41no—
296#41—1271no—
297#41—5081no—
298#41—5081no—
299#41—2541no—
300#41—2541no—
301#41—2541no—
302#41—2541no—
303#41C22×D12711yescharacteristic
304#41C22×D12711yescharacteristic
305#41C22×D12711yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—1271no—
309C2×Q8—1271no—
310C2×Q8—2541no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8—2541no—
314C2×Q8C22×D12711yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4C22×D12711yescharacteristic
319Q16—21no—
320Q16—1271no—
321Q16—2541no—
322Q16—2541no—
323Q16—2541no—
324Q16—2541no—
325Q16—2541no—
326Q16—2541no—
327Q16—2541no—
328Q16—2541no—
329Q16—2541no—
330Q16C22⋊D12711yescharacteristic
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—1271no—
336SD16—1271no—
337SD16—2541no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16C22⋊D12711yescharacteristic
346SD16C22⋊D12711yescharacteristic
347D8—21no—
348D8—1271no—
349D8—2541no—
350D8C22⋊D12711yescharacteristic
351M16—21no—
352M16—1271no—
353M16—1271no—
354M16—1271no—
355M16—2541no—
356M16—2541no—
357M16—2541no—
358M16C22×D12711yescharacteristic
359C2×C8—21no—
360C2×C8—1271no—
361C2×C8—2541no—
362C2×C8—2541no—
363C2×C8—2541no—
364C2×C8C22×D12711yescharacteristic
365C4⋊C4—5081no—
366C22⋊C4—5081no—
367C42—1271no—
368C23—21no—
369C23—41no—
370Q8—21no—
371Q8—21no—
372Q8—41no—
373Q8—1271no—
374Q8—5081no—
375Q8—5081no—
376Q8—2541no—
377Q8—2541no—
378Q8—2541no—
379Q8—2541no—
380Q8D4×D12711yescharacteristic
381Q8C2×C22⋊D12711yescharacteristic
382Q8C2×C22⋊D12711yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—21no—
387D4—21no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—1271no—
392D4—5081no—
393D4—5081no—
394D4—2541no—
395D4D4×D12711yescharacteristic
396D4C2×C22⋊D12711yescharacteristic
397D4C2×C22⋊D12711yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—5081no—
406C2×C4—5081no—
407C2×C4—5081no—
408C2×C4—5081no—
409C2×C4—2541no—
410C2×C4—2541no—
411C2×C4—2541no—
412C2×C4—2541no—
413C2×C4C23×D12711yescharacteristic
414C8—21no—
415C8—1271no—
416C8—1271no—
417C8—2541no—
418C8—2541no—
419C8—2541no—
420C8C2×C22⋊D12711yescharacteristic
421C8C2×C22⋊D12711yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—41no—
426C22—81no—
427C22—161no—
428C22—2541no—
429C22#1686125411yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—5081no—
435C4—5081no—
436C4—5081no—
437C4—2541no—
438C4C2×D4×D12711yescharacteristic
439C4C22×C22⋊D12711yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—81no—
445C2—5081no—
446C2D4×C22⋊D12711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#577×C127C2
socleC2×C127D42
hypercenter#577D127
hypocenterC127#3289
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #22550691C2semidirect
2. #22550690C2semidirect
3. #22550687C2semidirect
4. #22550267C2semidirect
5. #22550151C2semidirect
6. #22550149C2semidirect
7. #22550137C2semidirect
8. #22550136C2semidirect
9. #22550059C2semidirect
10. #22550057C2semidirect
11. #22550054C2semidirect
12. #22550047C2semidirect
13. #22549998C2semidirect
14. #22549996C2semidirect
15. #577×C127C2semidirect
17. #16861256C22semidirect
19. #16861240C22semidirect
25. #16861239C22semidirect
34. C2×Dic508C22semidirect
36. C2×#14523238C22semidirect
41. #16861216C22semidirect
42. #16861215C22semidirect
43. #16861207C22semidirect
45. #16861145C22semidirect
46. #16861144C22semidirect
49. #16861136C22semidirect
50. #16861135C22semidirect
51. #16861128C22semidirect
52. #16861127C22semidirect
53. #16861127C22semidirect
55. #16861119C22semidirect
224. C127#3289semidirect
228. #577D127semidirect

Maximal subgroups

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

subgroupindex
#225506912
#225506902
#225506872
#225502672
#225501512
#225501492
#225501372
#225501362
#225500592
#225500572
#225500542
#225500472
#225499982
#225499962
#577×C1272
#3289127