Finite Group Databasebeta

← #30650501

Subgroups — #30650501

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#22550684C211yesmaximal, characteristic, complemented
3#22550683C211yesmaximal, characteristic, complemented
4#22550269C211yesmaximal, characteristic, complemented
5#22550149C211yesmaximal, characteristic, complemented
6#22550148C211yesmaximal, characteristic, complemented
7#22550141C211yesmaximal, characteristic, complemented
8#22550139C211yesmaximal, characteristic, complemented
9#22550059C211yesmaximal, characteristic, complemented
10#22550057C211yesmaximal, characteristic, complemented
11#22550052C211yesmaximal, characteristic, complemented
12#22550049C211yesmaximal, characteristic, complemented
13#22549998C211yesmaximal, characteristic, complemented
14#22549996C211yesmaximal, characteristic, complemented
15#578×C127C211yesmaximal, characteristic, complemented
16#16861256C2211yescharacteristic
17#16861255C2211yescharacteristic
18#16861241—21no—
19#16861240—21no—
20#16861240—21no—
21#16861240C2211yescharacteristic
22#16861240C2211yescharacteristic
23#16861239—21no—
24#16861239C2211yescharacteristic
25#16861239C2211yescharacteristic, complemented
26#16861234—21no—
27#16861234C2211yescharacteristic
28#16861234C2211yescharacteristic
29#16861233—21no—
30#16861230—21no—
31#16861229—21no—
32#16861229—21no—
33#16861229C2211yescharacteristic
34C2×#14523238—21no—
35C2×#14523238C2211yescharacteristic, complemented
36C2×#14523237C2211yescharacteristic
37C2×#14523235—21no—
38C2×#14523235C2211yescharacteristic
39SD16×D127—21no—
40#16861216C2211yescharacteristic
41#16861216C2211yescharacteristic, complemented
42#16861215C2211yescharacteristic
43#16861207C2211yescharacteristic, complemented
44#16861198C2211yescharacteristic
45#16861145C2211yescharacteristic, complemented
46#16861144C2211yescharacteristic, complemented
47#16861136C2211yescharacteristic
48#16861136C2211yescharacteristic, complemented
49#16861136C2211yescharacteristic, complemented
50#16861135C2211yescharacteristic
51#16861128C2211yescharacteristic, complemented
52#16861128C2211yescharacteristic, complemented
53#16861127C2211yescharacteristic, complemented
54#16861125C2211yescharacteristic
55#16861119C2211yescharacteristic, complemented
562−1+4×C127C2211yescharacteristic
572−1+4×C127C2211yescharacteristic, complemented
58#137×C127—21no—
59#137×C127—21no—
60#137×C127C2211yescharacteristic
61#137×C127C2211yescharacteristic, complemented
62#135×C127—21no—
63#135×C127C2211yescharacteristic
64C2×Q16×C127—21no—
65C2×Q16×C127C2211yescharacteristic, complemented
66#131×C127C2211yescharacteristic, complemented
67#14523249—21no—
68#14523248—21no—
69#14523248—21no—
70#14523248C2311yescharacteristic
71#14523247—21no—
72#14523247—41no—
73C2×Dic254—21no—
74C2×Dic254C2311yescharacteristic
75Q8×D127—21no—
76C2×D508—21no—
77C2×D508C2311yescharacteristic
78D4×D127—21no—
79D4×D127—41no—
80#14523238—21no—
81#14523238—21no—
82#14523238—21no—
83#14523238—21no—
84#14523238—21no—
85#14523238—21no—
86#14523238D411yescharacteristic
87#14523237—21no—
88#14523237—21no—
89#14523236—21no—
90#14523236—21no—
91#14523236D411yescharacteristic
92#14523235—21no—
93#14523235—21no—
94#14523235—21no—
95#14523235—21no—
96#14523235—21no—
97#14523235—21no—
98#14523235D411yescharacteristic
99#14523233—21no—
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—
113(C2×D127)⋊C4—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×C127—41no—
121#41×C127C2311yescharacteristic
122#41×C127C2311yescharacteristic
123#41×C127C2311yescharacteristic
124C2×Q8×C127—21no—
125C2×Q8×C127—21no—
126C2×Q8×C127—21no—
127C2×Q8×C127—21no—
128C2×Q8×C127C2311yescharacteristic
129C2×Q8×C127C2311yescharacteristic
130Q16×C127—21no—
131Q16×C127—21no—
132Q16×C127—21no—
133Q16×C127—21no—
134Q16×C127D411yescharacteristic
135SD16×C127—21no—
136SD16×C127—21no—
137SD16×C127D411yescharacteristic
138SD16×C127D411yescharacteristic
139D8×C127D411yescharacteristic
140M16×C127—21no—
141M16×C127C2311yescharacteristic
142C2×C8×C127—21no—
143C2×C8×C127C2311yescharacteristic
144C22×D127—21no—
145C22×D127—41no—
146Dic254—21no—
147Dic254—21no—
148Dic254—41no—
149Dic254C2×D411yescharacteristic
150D508—21no—
151D508—21no—
152D508—41no—
153D508C2×D411yescharacteristic
154C22⋊D127—21no—
155C22⋊D127—41no—
156C22⋊D127—41no—
157C2×Dic127—21no—
158C2×Dic127—21no—
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
168Q8×C127—21no—
169Q8×C127—21no—
170Q8×C127—21no—
171Q8×C127—21no—
172Q8×C127—41no—
173Q8×C127—41no—
174Q8×C127C2×D411yescharacteristic
175Q8×C127C2×D411yescharacteristic
176Q8×C127C2×D411yescharacteristic
177D4×C127—21no—
178D4×C127—21no—
179D4×C127—21no—
180D4×C127—41no—
181D4×C127—41no—
182D4×C127C2×D411yescharacteristic
183D4×C127C2×D411yescharacteristic
184D4×C127C2×D411yescharacteristic
185C2×C4×C127—21no—
186C2×C4×C127—21no—
187C2×C4×C127—21no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—41no—
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×C127H(5,𝔽2)11yescharacteristic
202C2×D127—21no—
203C2×D127—41no—
204C2×D127—81no—
205C2×D127—161no—
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—21no—
209C4×C127—41no—
210C4×C127—41no—
211C4×C127C22×D411yescharacteristic
212C4×C127C22×D411yescharacteristic
213Dic127—21no—
214Dic127—41no—
215Dic127—41no—
216C2×C127—21no—
217C2×C127—41no—
218C2×C127—41no—
219C2×C127—41no—
220C2×C127D4211yescharacteristic
221D127—41no—
222D127—81no—
223#3289—1271nomaximal, Sylow
224C127#328911yescharacteristic, complemented, Sylow
225#578—1271no—
226#578D12711yescharacteristic, complemented
227#577—1271no—
228#577—1271no—
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+4C2×D12711yescharacteristic
2422−1+4C2×D12711yescharacteristic, complemented
243H(5,𝔽2)—1271no—
244#137—21no—
245#137—21no—
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#137C2×D12711yescharacteristic
255#137C2×D12711yescharacteristic, complemented
256AGL(1,ℤ/8)—1271no—
257AGL(1,ℤ/8)—1271no—
258AGL(1,ℤ/8)—2541no—
259AGL(1,ℤ/8)—2541no—
260#135—21no—
261#135—1271no—
262#135—2541no—
263#135—2541no—
264#135—2541no—
265#135C2×D12711yescharacteristic
266C2×Q16—21no—
267C2×Q16—1271no—
268C2×Q16—2541no—
269C2×Q16C2×D12711yescharacteristic, complemented
270C2×SD16—1271no—
271C2×SD16—1271no—
272C2×SD16—2541no—
273C2×SD16—2541no—
274#131—1271no—
275#131—1271no—
276#131—1271no—
277#131C2×D12711yescharacteristic, complemented
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—41no—
297#41—1271no—
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—21no—
309C2×Q8—21no—
310C2×Q8—1271no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8C22×D12711yescharacteristic
314C2×Q8C22×D12711yescharacteristic
315C2×D4—1271no—
316C2×D4—5081no—
317C2×D4—2541no—
318C2×D4—2541no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—1271no—
324Q16—2541no—
325Q16—2541no—
326Q16—2541no—
327Q16—2541no—
328Q16—2541no—
329Q16—2541no—
330Q16C22⋊D12711yescharacteristic
331SD16—21no—
332SD16—21no—
333SD16—1271no—
334SD16—1271no—
335SD16—2541no—
336SD16—2541no—
337SD16—2541no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16C22⋊D12711yescharacteristic
346SD16C22⋊D12711yescharacteristic
347D8—1271no—
348D8—2541no—
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—5081no—
369C23—2541no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—41no—
376Q8—1271no—
377Q8—5081no—
378Q8—2541no—
379Q8—2541no—
380Q8D4×D12711yescharacteristic
381Q8C2×C22⋊D12711yescharacteristic
382Q8C2×C22⋊D12711yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—41no—
387D4—41no—
388D4—1271no—
389D4—5081no—
390D4—5081no—
391D4—5081no—
392D4—2541no—
393D4—2541no—
394D4—2541no—
395D4D4×D12711yescharacteristic
396D4C2×C22⋊D12711yescharacteristic
397D4C2×C22⋊D12711yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—5081no—
408C2×C4—5081no—
409C2×C4—5081no—
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—20321no—
426C22—10161no—
427C22—5081no—
428C22—2541no—
429C22#1686125411yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—5081no—
436C4—5081no—
437C4—2541no—
438C4C2×D4×D12711yescharacteristic
439C4C22×C22⋊D12711yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—10161no—
445C2—5081no—
446C2D4×C22⋊D12711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#578×C127C2
socleC2×C127D42
hypercenter#578D127
hypocenterC127#3289
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #22550691C2semidirect
2. #22550684C2semidirect
3. #22550683C2semidirect
4. #22550269C2semidirect
5. #22550149C2semidirect
6. #22550148C2semidirect
7. #22550141C2semidirect
8. #22550139C2semidirect
9. #22550059C2semidirect
10. #22550057C2semidirect
11. #22550052C2semidirect
12. #22550049C2semidirect
13. #22549998C2semidirect
14. #22549996C2semidirect
15. #578×C127C2semidirect
25. #16861239C22semidirect
35. C2×#14523238C22semidirect
41. #16861216C22semidirect
43. #16861207C22semidirect
45. #16861145C22semidirect
46. #16861144C22semidirect
48. #16861136C22semidirect
49. #16861136C22semidirect
51. #16861128C22semidirect
52. #16861128C22semidirect
53. #16861127C22semidirect
55. #16861119C22semidirect
57. 2−1+4×C127C22semidirect
61. #137×C127C22semidirect
65. C2×Q16×C127C22semidirect
66. #131×C127C22semidirect
224. C127#3289semidirect
226. #578D127semidirect
242. 2−1+4C2×D127semidirect
255. #137C2×D127semidirect
269. C2×Q16C2×D127semidirect
277. #131C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506912
#225506842
#225506832
#225502692
#225501492
#225501482
#225501412
#225501392
#225500592
#225500572
#225500522
#225500492
#225499982
#225499962
#578×C1272
#3289127