Finite Group Databasebeta

← #30650336

Subgroups — #30650336

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×#16861236C211yesmaximal, characteristic, complemented
2C2×#16861233C211yesmaximal, characteristic, complemented
3#22550521C211yesmaximal, characteristic, complemented
4#22550519C211yesmaximal, characteristic, complemented
5#22550258C211yesmaximal, characteristic, complemented
6#22550125C211yesmaximal, characteristic, complemented
7#22550121C211yesmaximal, characteristic, complemented
8#22550092C211yesmaximal, characteristic, complemented
9#22550041C211yesmaximal, characteristic, complemented
10#22550035C211yesmaximal, characteristic, complemented
11#22550011C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549953C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic, complemented
19C2×D4×D127C2211yescharacteristic, complemented
20#16861236—24no—
21#16861233—24no—
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic, complemented
24C2×#14523236C2211yescharacteristic, complemented
25C2×#14523235C2211yescharacteristic, complemented
26C2×D1016C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523231C2211yescharacteristic, complemented
30C2×#14523231C2211yescharacteristic, complemented
31#16861197C2211yescharacteristic, complemented
32#16861193—21no—
33#16861191—21no—
34#16861189C2211yescharacteristic, complemented
35#16861185C2211yescharacteristic, complemented
36#16861184C2211yescharacteristic
37#16861183—21no—
38#16861182—21no—
39#16861180C2211yescharacteristic
40#16861179—21no—
41#16861179—21no—
42C4×D508C2211yescharacteristic
43C4×D508C2211yescharacteristic, complemented
44C2×(C2×D127)⋊C4—21no—
45C22⋊C4×D127—21no—
46#16861142C2211yescharacteristic, complemented
47#16861134C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53C4×C127⋊C8C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×D8×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic, complemented
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63#14523249—22no—
64#14523249—41no—
65#14523247—22no—
66#14523247—41no—
67C2×Dic254C2311yescharacteristic
68C2×D508—21no—
69C2×D508—21no—
70C2×D508D411yescharacteristic
71C2×D508C2311yescharacteristic
72C2×D508C2311yescharacteristic
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77D4×D127—22no—
78D4×D127—22no—
79D4×D127—41no—
80D4×D127—41no—
81C22×Dic127—21no—
82C2×C4×D127—21no—
83C2×C4×D127—21no—
84C2×C4×D127C2311yescharacteristic
85C2×C4×D127C2311yescharacteristic
86#14523238—41no—
87#14523237—22no—
88#14523236—22no—
89#14523235—22no—
90D1016—22no—
91#14523233—41no—
92#14523233D414yescomplemented
93#14523231—22no—
94#14523231—22no—
95C2×C127⋊C8C2311yescharacteristic
96C2×C127⋊C8C2311yescharacteristic
97C4⋊Dic127—21no—
98C4⋊Dic127D411yescharacteristic
99C4⋊Dic127C2311yescharacteristic
100Dic127⋊C4—21no—
101C22⋊Dic127—21no—
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
109(C2×D127)⋊C4D411yescharacteristic
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127—21no—
114SD16×C127—22no—
115D8×C127—22no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C4⋊C4×C127C2311yescharacteristic
119C22⋊C4×C127—21no—
120C22⋊C4×C127—21no—
121C22⋊C4×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—81no—
130C22×D127—81no—
131C22×D127C2×D411yescharacteristic
132C22×D127C2×D411yescharacteristic
133Dic254—21no—
134Dic254—21no—
135D508—21no—
136D508—21no—
137D508—21no—
138D508—41no—
139D508—42no—
140D508—81no—
141D508C2×D412yes—
142C22⋊D127—42no—
143C22⋊D127—42no—
144C22⋊D127—42no—
145C22⋊D127—81no—
146C2×Dic127—21no—
147C2×Dic127—21no—
148C2×Dic127—42no—
149C2×Dic127C2×D411yescharacteristic
150C2×Dic127C2×D411yescharacteristic
151C4×D127—21no—
152C4×D127—21no—
153C4×D127—21no—
154C4×D127—21no—
155C4×D127—42no—
156C4×D127—81no—
157C127⋊C8—21no—
158C127⋊C8C2×D412yes—
159C23×C127—21no—
160C23×C127—21no—
161Q8×C127—21no—
162Q8×C127—21no—
163D4×C127—21no—
164D4×C127—21no—
165D4×C127—21no—
166D4×C127C2×D412yes—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—81no—
170C2×C4×C127C2×D411yescharacteristic
171C2×C4×C127C2×D411yescharacteristic
172C2×C4×C127C2411yescharacteristic
173C8×C127—21no—
174C8×C127—21no—
175C22×C127—22no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127C22×D411yescharacteristic
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—22no—
185C2×D127—21no—
186C2×D127—21no—
187C2×D127—21no—
188C2×D127—21no—
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—21no—
198C4×C127—41no—
199C4×C127—41no—
200C4×C127C22×D411yescharacteristic
201C4×C127H(5,𝔽2)11yescharacteristic
202Dic127—21no—
203Dic127—21no—
204Dic127—41no—
205Dic127—41no—
206C2×C127—42no—
207C2×C127—81no—
208C2×C127D4211yescharacteristic
209C2×C127#57511yescharacteristic
210C2×C127#57611yescharacteristic
211D127—41no—
212D127—41no—
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#468—1271no—
224#463—1271no—
225#463D12711yescharacteristic, complemented
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×D8C2×D12711yescharacteristic
247C2×M16—1271no—
248C2×M16—1271no—
249#124—1271no—
250#124C2×D12711yescharacteristic, complemented
251#123—2541no—
252#123—2541no—
253C22⋊Q8—1271no—
254C22⋊Q8—1271no—
255#121—1271no—
256#121—1271no—
257#121—2541no—
258#121—2541no—
259#120—2541no—
260#120—2541no—
261C4×D4—1271no—
262C4×D4—1271no—
263C4×D4C2×D12711yescharacteristic, complemented
264C2×C22⋊C4—2541no—
265C2×C22⋊C4—2541no—
266#107—1271no—
267C4⋊C8C2×D12711yescharacteristic, complemented
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—1271no—
286C2×D4—1271no—
287C2×D4—1271no—
288C2×D4—5081no—
289C2×D4—5081no—
290C2×D4—2541no—
291C2×D4—2541no—
292C2×D4—2541no—
293C2×D4—2541no—
294C2×D4—2542no—
295C2×D4—2541no—
296C2×D4—2541no—
297C2×D4—2542no—
298C2×D4C22×D12711yescharacteristic
299C2×D4C22×D12711yescharacteristic
300C22×C4—21no—
301C22×C4—1271no—
302C22×C4—1271no—
303C22×C4—2541no—
304C22×C4—2541no—
305C22×C4—2541no—
306Q16—5081no—
307SD16—22no—
308SD16—2542no—
309SD16—2542no—
310SD16—2542no—
311D8—22no—
312D8—1274no—
313D8—5081no—
314D8—2542no—
315M16—2542no—
316M16—2542no—
317C2×C8—1271no—
318C2×C8—1271no—
319C2×C8C22×D12711yescharacteristic
320C2×C8C22×D12711yescharacteristic
321C4⋊C4—1271no—
322C4⋊C4—1271no—
323C4⋊C4—2541no—
324C4⋊C4—2541no—
325C4⋊C4C22×D12711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—1271no—
330C22⋊C4—1271no—
331C22⋊C4—2541no—
332C22⋊C4—2541no—
333C22⋊C4—2541no—
334C22⋊C4—2541no—
335C22⋊C4—2541no—
336C22⋊C4—2541no—
337C22⋊C4—2541no—
338C42C22×D12711yescharacteristic
339C23—21no—
340C23—21no—
341C23—1271no—
342C23—1271no—
343C23—10161no—
344C23—10161no—
345C23—5082no—
346C23—5082no—
347C23—5082no—
348C23—5082no—
349C23—2541no—
350C23—2541no—
351Q8—21no—
352Q8—21no—
353Q8—2541no—
354Q8—2541no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—1272no—
359D4—10161no—
360D4—10161no—
361D4—5081no—
362D4—5082no—
363D4—5082no—
364D4—5082no—
365D4—5082no—
366D4—2541no—
367D4—2541no—
368D4—2541no—
369D4D4×D12712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—1271no—
374C2×C4—1271no—
375C2×C4—10161no—
376C2×C4—5082no—
377C2×C4—5082no—
378C2×C4—2541no—
379C2×C4—2541no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4D4×D12711yescharacteristic
385C2×C4D4×D12711yescharacteristic
386C2×C4C23×D12711yescharacteristic
387C8—21no—
388C8—21no—
389C8—1272no—
390C8—2541no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—10164no—
398C22—10164no—
399C22—5081no—
400C22—5081no—
401C22—5081no—
402C22—5081no—
403C22—5081no—
404C22—2541no—
405C22—2541no—
406C22—2541no—
407C22—2541no—
408C22—2542no—
409C22—2541no—
410C22—2541no—
411C22C2×D4×D12711yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—5081no—
417C4—5081no—
418C4—2541no—
419C4—2541no—
420C4C2×D4×D12711yescharacteristic
421C4#1686125511yescharacteristic
422C2—42no—
423C2—81no—
424C2—10161no—
425C2—5081no—
426C2—5081no—
427C2—5082no—
428C2#2255051011yescharacteristic
429C2#2255067311yescharacteristic
430C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
hypercenter#463D127
hypocenterC127#3273
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×#16861233C2semidirect
3. #22550521C2semidirect
4. #22550519C2semidirect
5. #22550258C2semidirect
6. #22550125C2semidirect
7. #22550121C2semidirect
8. #22550092C2semidirect
9. #22550041C2semidirect
10. #22550035C2semidirect
11. #22550011C2semidirect
12. #22550008C2semidirect
13. C4×#14523233C2semidirect
14. #22549953C2semidirect
15. #463×C127C2semidirect
18. C2×D4×D127C22semidirect
19. C2×D4×D127C22semidirect
22. C2×#14523238C22semidirect
23. C2×#14523237C22semidirect
24. C2×#14523236C22semidirect
25. C2×#14523235C22semidirect
29. C2×#14523231C22semidirect
30. C2×#14523231C22semidirect
31. #16861197C22semidirect
34. #16861189C22semidirect
35. #16861185C22semidirect
43. C4×D508C22semidirect
46. #16861142C22semidirect
47. #16861134C22semidirect
48. #16861131C22semidirect
49. #16861130C22semidirect
50. #16861129C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. C4×C127⋊C8C22semidirect
54. C2×SD16×C127C22semidirect
56. #124×C127C22semidirect
57. C4×D4×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
92. #14523233D4semidirect
216. C127#3273semidirect
225. #463D127semidirect
242. C2×SD16C2×D127semidirect
250. #124C2×D127semidirect
263. C4×D4C2×D127semidirect
267. C4⋊C8C2×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×#168612362
C2×#168612332
#225505212
#225505192
#225502582
#225501252
#225501212
#225500922
#225500412
#225500352
#225500112
#225500082
C4×#145232332
#225499532
#463×C1272
#3273127