Finite Group Databasebeta

← #30650338

Subgroups — #30650338

Order 16256.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861237C211yesmaximal, characteristic, complemented
2C2×#16861236C211yesmaximal, characteristic, complemented
3#22550523C211yesmaximal, characteristic, complemented
4#22550517C211yesmaximal, characteristic, complemented
5#22550257C211yesmaximal, characteristic, complemented
6#22550123C211yesmaximal, characteristic, complemented
7#22550094C211yesmaximal, characteristic, complemented
8#22550090C211yesmaximal, characteristic, complemented
9#22550041C211yesmaximal, characteristic, complemented
10#22550036C211yesmaximal, characteristic, complemented
11#22550011C211yesmaximal, characteristic, complemented
12#22550009C211yesmaximal, characteristic, complemented
13#22549958C211yesmaximal, characteristic, complemented
14#22549952C211yesmaximal, characteristic, complemented
15#468×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C22×D508C2211yescharacteristic, complemented
19C2×D4×D127C2211yescharacteristic, complemented
20#16861237—24no—
21#16861236—24no—
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic, complemented
24C2×#14523237C2211yescharacteristic, complemented
25C2×#14523235C2211yescharacteristic, complemented
26C2×D1016C2211yescharacteristic
27C2×D1016C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523233C2211yescharacteristic
30C2×#14523231C2211yescharacteristic, complemented
31#16861198C2211yescharacteristic, complemented
32#16861196C2211yescharacteristic, complemented
33#16861194—21no—
34#16861191—21no—
35#16861189C2211yescharacteristic, complemented
36#16861184—21no—
37#16861184C2211yescharacteristic
38#16861180—21no—
39#16861179—21no—
40#16861177—21no—
41#16861171C2211yescharacteristic, complemented
42#16861171C2211yescharacteristic, complemented
43C4×C22⋊D127C2211yescharacteristic
44C22⋊C4×D127—21no—
45C22⋊C4×D127—21no—
46Dic127⋊C8C2211yescharacteristic, complemented
47#16861132C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic, complemented
51C22⋊(C127⋊C8)C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53C8×Dic127C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×M16×C127C2211yescharacteristic, complemented
56C22⋊Q8×C127C2211yescharacteristic, complemented
57#121×C127C2211yescharacteristic
58#107×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63#14523249—22no—
64#14523249—41no—
65#14523248—22no—
66#14523248—41no—
67C2×Dic254C2311yescharacteristic
68C2×D508—22no—
69C2×D508—21no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508—41no—
73C2×D508C2311yescharacteristic
74C2×D508C2311yescharacteristic
75C2×D508C2311yescharacteristic
76C2×C22⋊D127—21no—
77C2×C22⋊D127—21no—
78C2×C22⋊D127D411yescharacteristic
79D4×D127—22no—
80D4×D127—41no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127—21no—
84C2×C4×D127—21no—
85C2×C4×D127C2311yescharacteristic
86#14523238—41no—
87#14523237—22no—
88#14523237—22no—
89#14523235—22no—
90D1016—22no—
91D1016D414yescomplemented
92#14523233—22no—
93#14523233—41no—
94#14523231—22no—
95C2×C127⋊C8C2311yescharacteristic
96C2×C127⋊C8C2311yescharacteristic
97C4⋊Dic127—21no—
98Dic127⋊C4D411yescharacteristic
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127D411yescharacteristic
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4D411yescharacteristic
109C4×Dic127C2311yescharacteristic
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127—21no—
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127C2311yescharacteristic
114SD16×C127—22no—
115M16×C127—22no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C4⋊C4×C127—21no—
119C4⋊C4×C127C2311yescharacteristic
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—81no—
131C22×D127—81no—
132C22×D127C2×D411yescharacteristic
133Dic254—21no—
134Dic254—21no—
135D508—21no—
136D508—21no—
137D508—21no—
138D508—21no—
139D508—42no—
140D508—42no—
141D508—81no—
142D508C2×D412yes—
143D508C2×D412yes—
144C22⋊D127—41no—
145C22⋊D127—42no—
146C22⋊D127—42no—
147C2×Dic127—21no—
148C2×Dic127C2×D411yescharacteristic
149C2×Dic127C2×D411yescharacteristic
150C2×Dic127C2×D411yescharacteristic
151C4×D127—21no—
152C4×D127—21no—
153C4×D127—42no—
154C4×D127—42no—
155C4×D127—81no—
156C4×D127—81no—
157C127⋊C8—21no—
158C127⋊C8—21no—
159C23×C127—21no—
160C23×C127C2×D411yescharacteristic
161Q8×C127—21no—
162Q8×C127—21no—
163D4×C127—21no—
164D4×C127—21no—
165D4×C127—81no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127C2×D411yescharacteristic
172C2×C4×C127C2411yescharacteristic
173C8×C127—21no—
174C8×C127C2×D412yes—
175C22×C127—21no—
176C22×C127—21no—
177C22×C127—21no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127C22×D411yescharacteristic
182C2×D127—21no—
183C2×D127—22no—
184C2×D127—22no—
185C2×D127—21no—
186C2×D127—21no—
187C2×D127—41no—
188C2×D127—41no—
189C2×D127—41no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—84no—
195C2×D127—84no—
196C4×C127—21no—
197C4×C127—41no—
198C4×C127—41no—
199C4×C127—41no—
200C4×C127C22×D411yescharacteristic
201C4×C127H(5,𝔽2)11yescharacteristic
202Dic127—21no—
203Dic127—21no—
204Dic127—21no—
205Dic127—41no—
206C2×C127—41no—
207C2×C127—81no—
208C2×C127D4211yescharacteristic
209C2×C127#57511yescharacteristic
210C2×C127#57611yescharacteristic
211D127—41no—
212D127—42no—
213D127—42no—
214D127—81no—
215#3273—1271nomaximal, Sylow
216C127#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—1271no—
218C2×AGL(1,ℤ/8)—1271no—
219#546—1271no—
220#546—1271no—
221#495—1271no—
222#468—1271no—
223#468D12711yescharacteristic, complemented
224#463—1271no—
225#463—1271no—
226#452—1271no—
227#452—1271no—
228#447—1271no—
229#447—1271no—
230C4×D8—1271no—
231#436—1271no—
232C2×#41—1271no—
233C2×#41—1271no—
234C22×D4—1271no—
235C22×D4—1271no—
236AGL(1,ℤ/8)—2544no—
237AGL(1,ℤ/8)—2544no—
238C2×Q16—1271no—
239C2×SD16—1271no—
240C2×SD16—1271no—
241C2×SD16—1271no—
242C2×SD16C2×D12711yescharacteristic, complemented
243C2×D8—1271no—
244C2×D8—1271no—
245C2×D8—1271no—
246C2×D8—1271no—
247C2×M16—1271no—
248C2×M16C2×D12711yescharacteristic, complemented
249#124—1271no—
250#124—1271no—
251#123—2541no—
252#123—2541no—
253C22⋊Q8—1271no—
254C22⋊Q8C2×D12711yescharacteristic, complemented
255#121—1271no—
256#121—2541no—
257#121—2541no—
258#121C2×D12711yescharacteristic
259#120—2541no—
260#120—2541no—
261C4×D4—1271no—
262C4×D4—1271no—
263C4×D4—1271no—
264C2×C22⋊C4—2541no—
265C2×C22⋊C4—2541no—
266#107C2×D12711yescharacteristic, complemented
267C4⋊C8—1271no—
268#103—1271no—
269#103C2×D12711yescharacteristic, complemented
270#102—1271no—
271#102—1271no—
272#102—1271no—
273#102C2×D12711yescharacteristic, complemented
274C22⋊C8—1271no—
275C22⋊C8—1271no—
276C4×C8—1271no—
277C24—2541no—
278C24—2541no—
279#41—5081no—
280#41—5081no—
281#41—2542no—
282#41—2542no—
283C2×Q8—1271no—
284C2×Q8C22×D12711yescharacteristic
285C2×D4—21no—
286C2×D4—1271no—
287C2×D4—1271no—
288C2×D4—1271no—
289C2×D4—1271no—
290C2×D4—5081no—
291C2×D4—5081no—
292C2×D4—2542no—
293C2×D4—2541no—
294C2×D4—2541no—
295C2×D4—2541no—
296C2×D4—2542no—
297C2×D4—2541no—
298C2×D4—2541no—
299C2×D4C22×D12711yescharacteristic
300C22×C4—1271no—
301C22×C4—2541no—
302C22×C4—2541no—
303C22×C4—2541no—
304C22×C4—2541no—
305C22×C4C22×D12711yescharacteristic
306Q16—5081no—
307SD16—22no—
308SD16—2542no—
309SD16—2542no—
310SD16—2542no—
311D8—1274no—
312D8—5081no—
313D8—2542no—
314D8—2542no—
315M16—22no—
316M16—2542no—
317C2×C8—1271no—
318C2×C8—1271no—
319C2×C8C22×D12711yescharacteristic
320C2×C8C22×D12711yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—1271no—
323C4⋊C4—2541no—
324C4⋊C4C22×D12711yescharacteristic
325C4⋊C4C22×D12711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—1271no—
329C22⋊C4—1271no—
330C22⋊C4—2541no—
331C22⋊C4—2541no—
332C22⋊C4—2541no—
333C22⋊C4—2541no—
334C22⋊C4—2541no—
335C22⋊C4—2541no—
336C22⋊C4—2541no—
337C22⋊C4—2541no—
338C42—1271no—
339C23—21no—
340C23—1271no—
341C23—10161no—
342C23—10161no—
343C23—5082no—
344C23—5082no—
345C23—5082no—
346C23—5082no—
347C23—2541no—
348C23—2541no—
349C23—2541no—
350C23D4×D12711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2541no—
354Q8—2541no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—1272no—
359D4—1272no—
360D4—10161no—
361D4—5082no—
362D4—5082no—
363D4—5082no—
364D4—5081no—
365D4—5082no—
366D4—2541no—
367D4—2541no—
368D4—2541no—
369D4—2541no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—1271no—
376C2×C4—1271no—
377C2×C4—1271no—
378C2×C4—10161no—
379C2×C4—10161no—
380C2×C4—5082no—
381C2×C4—5082no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4—2541no—
385C2×C4D4×D12711yescharacteristic
386C2×C4C23×D12711yescharacteristic
387C8—21no—
388C8—2541no—
389C8—2541no—
390C8D4×D12712yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—10164no—
398C22—10164no—
399C22—5081no—
400C22—5081no—
401C22—5081no—
402C22—5081no—
403C22—5081no—
404C22—5081no—
405C22—5081no—
406C22—2541no—
407C22—2541no—
408C22—2542no—
409C22—2542no—
410C22—2541no—
411C22C2×D4×D12711yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—5081no—
417C4—2541no—
418C4—2541no—
419C4—2541no—
420C4C2×D4×D12711yescharacteristic
421C4#1686125511yescharacteristic
422C2—41no—
423C2—81no—
424C2—10161no—
425C2—5081no—
426C2—5082no—
427C2—5082no—
428C2#2255051011yescharacteristic
429C2#2255067511yescharacteristic
430C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#468×C127C2
socleC22×C127C22×D4
hypercenter#468D127
hypocenterC127#3273
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. C2×#16861236C2semidirect
3. #22550523C2semidirect
4. #22550517C2semidirect
5. #22550257C2semidirect
6. #22550123C2semidirect
7. #22550094C2semidirect
8. #22550090C2semidirect
9. #22550041C2semidirect
10. #22550036C2semidirect
11. #22550011C2semidirect
12. #22550009C2semidirect
13. #22549958C2semidirect
14. #22549952C2semidirect
15. #468×C127C2semidirect
18. C22×D508C22semidirect
19. C2×D4×D127C22semidirect
22. C2×#14523238C22semidirect
23. C2×#14523237C22semidirect
24. C2×#14523237C22semidirect
25. C2×#14523235C22semidirect
30. C2×#14523231C22semidirect
31. #16861198C22semidirect
32. #16861196C22semidirect
35. #16861189C22semidirect
41. #16861171C22semidirect
42. #16861171C22semidirect
46. Dic127⋊C8C22semidirect
47. #16861132C22semidirect
48. #16861129C22semidirect
49. #16861129C22semidirect
50. #16861129C22semidirect
51. C22⋊(C127⋊C8)C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. C8×Dic127C22semidirect
54. C2×SD16×C127C22semidirect
55. C2×M16×C127C22semidirect
56. C22⋊Q8×C127C22semidirect
58. #107×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
91. D1016D4semidirect
216. C127#3273semidirect
223. #468D127semidirect
242. C2×SD16C2×D127semidirect
248. C2×M16C2×D127semidirect
254. C22⋊Q8C2×D127semidirect
266. #107C2×D127semidirect
269. #103C2×D127semidirect
273. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612362
#225505232
#225505172
#225502572
#225501232
#225500942
#225500902
#225500412
#225500362
#225500112
#225500092
#225499582
#225499522
#468×C1272
#3273127