Finite Group Databasebeta

← #30650500

Subgroups — #30650500

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#22550692C211yesmaximal, characteristic, complemented
2#22550691C211yesmaximal, characteristic, complemented
3#22550685C211yesmaximal, characteristic, complemented
4#22550684C211yesmaximal, characteristic, complemented
5#22550268C211yesmaximal, characteristic, complemented
6#22550150C211yesmaximal, characteristic, complemented
7#22550149C211yesmaximal, characteristic, complemented
8#22550142C211yesmaximal, characteristic, complemented
9#22550141C211yesmaximal, characteristic, complemented
10#22550058C211yesmaximal, characteristic, complemented
11#22550055C211yesmaximal, characteristic, complemented
12#22550050C211yesmaximal, characteristic, complemented
13#22549995C211yesmaximal, characteristic, complemented
14#22549995C211yesmaximal, characteristic, complemented
15#456×C127C211yesmaximal, characteristic, complemented
16#16861257C2211yescharacteristic
17#16861256C2211yescharacteristic, complemented
18#16861255C2211yescharacteristic
19#16861242C2211yescharacteristic
20#16861241—21no—
21#16861241—21no—
22#16861240—21no—
23#16861240—21no—
24#16861240C2211yescharacteristic
25#16861240C2211yescharacteristic, complemented
26#16861239—21no—
27#16861239C2211yescharacteristic
28#16861238—21no—
29#16861237C2211yescharacteristic
30#16861236—21no—
31#16861234—21no—
32#16861234C2211yescharacteristic
33#16861232—21no—
34#16861230—21no—
35#16861229—21no—
36#16861229—21no—
37#16861229C2211yescharacteristic
38#16861229C2211yescharacteristic
39C2×#14523238—21no—
40C2×#14523238C2211yescharacteristic, complemented
41C2×#14523238C2211yescharacteristic, complemented
42Q16×D127—21no—
43C2×#14523235—21no—
44C2×#14523235C2211yescharacteristic
45C2×#14523235C2211yescharacteristic
46SD16×D127—21no—
47#16861217C2211yescharacteristic
48#16861217C2211yescharacteristic, complemented
49#16861216C2211yescharacteristic
50#16861216C2211yescharacteristic, complemented
51#16861197C2211yescharacteristic
52#16861144C2211yescharacteristic, complemented
53#16861144C2211yescharacteristic, complemented
54#16861137C2211yescharacteristic
55#16861137C2211yescharacteristic, complemented
56#16861128C2211yescharacteristic, complemented
57#16861127C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic
59#16861120C2211yescharacteristic, complemented
602−1+4×C127C2211yescharacteristic
61#137×C127C2211yescharacteristic
62#137×C127C2211yescharacteristic, complemented
63C4⋊Q8×C127C2211yescharacteristic, complemented
64#104×C127C2211yescharacteristic
65#104×C127C2211yescharacteristic, complemented
66#101×C127C2211yescharacteristic, complemented
67#14523249—21no—
68#14523249—21no—
69#14523249—21no—
70#14523248—21no—
71#14523248—21no—
72#14523248—41no—
73#14523248C2311yescharacteristic
74#14523248C2311yescharacteristic
75#14523247—21no—
76#14523247—41no—
77C2×Dic254—21no—
78C2×Dic254C2311yescharacteristic
79Q8×D127—21no—
80Q8×D127—21no—
81Q8×D127—21no—
82C2×D508—21no—
83C2×D508C2311yescharacteristic
84D4×D127—21no—
85D4×D127—41no—
86Dic508—21no—
87#14523238—21no—
88#14523238—21no—
89#14523238—21no—
90#14523238—21no—
91#14523238—21no—
92#14523238—21no—
93#14523238—21no—
94#14523238D411yescharacteristic
95#14523238D411yescharacteristic
96#14523237—21no—
97#14523237—21no—
98#14523236—21no—
99#14523236D411yescharacteristic
100#14523236D411yescharacteristic
101#14523235—21no—
102#14523235—21no—
103#14523235—21no—
104#14523235—21no—
105#14523235—21no—
106#14523235—21no—
107#14523235—21no—
108#14523235D411yescharacteristic
109#14523235D411yescharacteristic
110D1016—21no—
111#14523233—21no—
112#14523233D411yescharacteristic
113#14523233D411yescharacteristic
114#14523232—21no—
115#14523232—21no—
116#14523232C2311yescharacteristic
117#14523232C2311yescharacteristic
118#14523231—21no—
119#14523231—21no—
120C2×C127⋊C8—21no—
121C2×C127⋊C8—21no—
122C2×C127⋊C8C2311yescharacteristic
123C2×C127⋊C8C2311yescharacteristic
124C8×D127—21no—
125C8×D127—21no—
126(C2×D127)⋊C4—41no—
127#41×C127—21no—
128#41×C127—21no—
129#41×C127—41no—
130#41×C127C2311yescharacteristic
131#41×C127C2311yescharacteristic
132C2×Q8×C127—21no—
133C2×Q8×C127—21no—
134C2×Q8×C127C2311yescharacteristic
135C2×Q8×C127C2311yescharacteristic
136Q16×C127—21no—
137Q16×C127—21no—
138SD16×C127—21no—
139SD16×C127—21no—
140M16×C127C2311yescharacteristic
141M16×C127C2311yescharacteristic
142C4⋊C4×C127—41no—
143C42×C127C2311yescharacteristic
144C22×D127—21no—
145C22×D127—41no—
146Dic254—21no—
147Dic254—21no—
148Dic254—41no—
149Dic254—41no—
150Dic254C2×D411yescharacteristic
151Dic254C2×D411yescharacteristic
152D508—21no—
153D508—21no—
154D508—41no—
155D508—41no—
156D508C2×D411yescharacteristic
157D508C2×D411yescharacteristic
158C22⋊D127—21no—
159C22⋊D127—21no—
160C22⋊D127—41no—
161C22⋊D127—41no—
162C2×Dic127—21no—
163C2×Dic127—41no—
164C4×D127—21no—
165C4×D127—21no—
166C4×D127—41no—
167C4×D127—41no—
168C4×D127—41no—
169C4×D127—41no—
170C127⋊C8—21no—
171C127⋊C8—21no—
172C127⋊C8C2×D411yescharacteristic
173C127⋊C8C2×D411yescharacteristic
174C127⋊C8C2×D411yescharacteristic
175C127⋊C8C2×D411yescharacteristic
176Q8×C127—21no—
177Q8×C127—21no—
178Q8×C127—21no—
179Q8×C127—21no—
180Q8×C127—41no—
181Q8×C127C2×D411yescharacteristic
182Q8×C127C2×D411yescharacteristic
183D4×C127—21no—
184D4×C127—21no—
185D4×C127—41no—
186D4×C127C2×D411yescharacteristic
187D4×C127C2×D411yescharacteristic
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—21no—
193C2×C4×C127—41no—
194C2×C4×C127—41no—
195C2×C4×C127C2411yescharacteristic
196C8×C127—21no—
197C8×C127—21no—
198C22×C127—21no—
199C22×C127—21no—
200C22×C127H(5,𝔽2)11yescharacteristic
201C2×D127—21no—
202C2×D127—21no—
203C2×D127—41no—
204C2×D127—81no—
205C2×D127—161no—
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—41no—
209C4×C127—41no—
210C4×C127—41no—
211C4×C127C22×D411yescharacteristic
212C4×C127C22×D411yescharacteristic
213Dic127—21no—
214Dic127—21no—
215Dic127—41no—
216C2×C127—21no—
217C2×C127—41no—
218C2×C127—41no—
219C2×C127D4211yescharacteristic
220D127—41no—
221D127—41no—
222D127—81no—
223#3289—1271nomaximal, Sylow
224C127#328911yescharacteristic, complemented, Sylow
225#578—1271no—
226#578—1271no—
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#456D12711yescharacteristic, complemented
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)—1271no—
244#137—1271no—
245#137—1271no—
246#137—1271no—
247#137—1271no—
248#137—2541no—
249#137—2541no—
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—1271no—
261#135—1271no—
262#135—2541no—
263#135—2541no—
264#135—2541no—
265#135—2541no—
266C2×Q16—1271no—
267C2×Q16—1271no—
268C2×Q16—2541no—
269C2×Q16—2541no—
270C2×SD16—1271no—
271C2×SD16—1271no—
272C2×SD16—2541no—
273C2×SD16—2541no—
274#131—1271no—
275#131—1271no—
276#131—1271no—
277#131—1271no—
278C4⋊Q8C2×D12711yescharacteristic, complemented
279#124—1271no—
280#108—1271no—
281#108—1271no—
282#104—1271no—
283#104—1271no—
284#104C2×D12711yescharacteristic
285#104C2×D12711yescharacteristic, complemented
286#101—1271no—
287#101—1271no—
288#101C2×D12711yescharacteristic, complemented
289#100—1271no—
290#97—1271no—
291#41—21no—
292#41—21no—
293#41—41no—
294#41—1271no—
295#41—1271no—
296#41—5081no—
297#41—5081no—
298#41—2541no—
299#41—2541no—
300#41—2541no—
301#41—2541no—
302#41—2541no—
303#41—2541no—
304#41C22×D12711yescharacteristic
305#41C22×D12711yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—1271no—
309C2×Q8—2541no—
310C2×Q8—2541no—
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—1271no—
322Q16—1271no—
323Q16—2541no—
324Q16—2541no—
325Q16—2541no—
326Q16—2541no—
327Q16—2541no—
328Q16—2541no—
329Q16—2541no—
330Q16—2541no—
331SD16—21no—
332SD16—21no—
333SD16—1271no—
334SD16—1271no—
335SD16—1271no—
336SD16—1271no—
337SD16—2541no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16—2541no—
346SD16—2541no—
347D8—1271no—
348D8—1271no—
349D8—2541no—
350D8—2541no—
351M16—1271no—
352M16—1271no—
353M16—2541no—
354M16—2541no—
355M16—2541no—
356M16—2541no—
357M16C22×D12711yescharacteristic
358M16C22×D12711yescharacteristic
359C2×C8—1271no—
360C2×C8—1271no—
361C2×C8—2541no—
362C2×C8—2541no—
363C2×C8—2541no—
364C2×C8—2541no—
365C4⋊C4—41no—
366C22⋊C4—5081no—
367C42C22×D12711yescharacteristic
368C23—5081no—
369C23—2541no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—1271no—
376Q8—1271no—
377Q8—5081no—
378Q8—5081no—
379Q8—2541no—
380Q8—2541no—
381Q8D4×D12711yescharacteristic
382Q8D4×D12711yescharacteristic
383D4—21no—
384D4—21no—
385D4—41no—
386D4—1271no—
387D4—1271no—
388D4—5081no—
389D4—5081no—
390D4—5081no—
391D4—5081no—
392D4—2541no—
393D4—2541no—
394D4—2541no—
395D4—2541no—
396D4D4×D12711yescharacteristic
397D4D4×D12711yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—5081no—
406C2×C4—5081no—
407C2×C4—5081no—
408C2×C4—5081no—
409C2×C4—5081no—
410C2×C4—2541no—
411C2×C4—2541no—
412C2×C4—2541no—
413C2×C4C23×D12711yescharacteristic
414C8—21no—
415C8—21no—
416C8—1271no—
417C8—1271no—
418C8—1271no—
419C8—1271no—
420C8—2541no—
421C8—2541no—
422C22—21no—
423C22—21no—
424C22—20321no—
425C22—10161no—
426C22—5081no—
427C22—2541no—
428C22—2541no—
429C22#1686125411yescharacteristic
430C4—21no—
431C4—21no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—5081no—
436C4—2541no—
437C4—2541no—
438C4C2×D4×D12711yescharacteristic
439C4C2×D4×D12711yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—10161no—
444C2—5081no—
445C2—5081no—
446C2#2255050811yescharacteristic
subgroupG/H
centerC2#22550508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#456×C127C2
socleC2×C127D42
hypercenter#456D127
hypocenterC127#3289
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #22550692C2semidirect
2. #22550691C2semidirect
3. #22550685C2semidirect
4. #22550684C2semidirect
5. #22550268C2semidirect
6. #22550150C2semidirect
7. #22550149C2semidirect
8. #22550142C2semidirect
9. #22550141C2semidirect
10. #22550058C2semidirect
11. #22550055C2semidirect
12. #22550050C2semidirect
13. #22549995C2semidirect
14. #22549995C2semidirect
15. #456×C127C2semidirect
17. #16861256C22semidirect
25. #16861240C22semidirect
40. C2×#14523238C22semidirect
41. C2×#14523238C22semidirect
48. #16861217C22semidirect
50. #16861216C22semidirect
52. #16861144C22semidirect
53. #16861144C22semidirect
55. #16861137C22semidirect
56. #16861128C22semidirect
57. #16861127C22semidirect
59. #16861120C22semidirect
62. #137×C127C22semidirect
63. C4⋊Q8×C127C22semidirect
65. #104×C127C22semidirect
66. #101×C127C22semidirect
224. C127#3289semidirect
235. #456D127semidirect
255. #137C2×D127semidirect
278. C4⋊Q8C2×D127semidirect
285. #104C2×D127semidirect
288. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506922
#225506912
#225506852
#225506842
#225502682
#225501502
#225501492
#225501422
#225501412
#225500582
#225500552
#225500502
#225499952
#456×C1272
#3289127