Finite Group Databasebeta

← #30650347

Subgroups — #30650347

Order 16256.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861237C211yesmaximal, characteristic, complemented
2C2×#16861233C211yesmaximal, characteristic, complemented
3#22550544C211yesmaximal, characteristic, complemented
4#22550510C211yesmaximal, characteristic, complemented
5#22550251C211yesmaximal, characteristic, complemented
6#22550130C211yesmaximal, characteristic, complemented
7#22550075C211yesmaximal, characteristic, complemented
8#22550070C211yesmaximal, characteristic, complemented
9#22550022C211yesmaximal, characteristic, complemented
10#22550019C211yesmaximal, characteristic, complemented
11#22550011C211yesmaximal, characteristic, complemented
12#22550009C211yesmaximal, characteristic, complemented
13#22549963C211yesmaximal, characteristic, complemented
14#22549952C211yesmaximal, characteristic, complemented
15#469×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic, complemented
17C2×#14523247C2211yescharacteristic, complemented
18C22×D508C2211yescharacteristic
19C2×D4×D127—21no—
20C2×D4×D127C2211yescharacteristic
21#16861237—24no—
22#16861233—24no—
23C2×#14523237C2211yescharacteristic
24C2×#14523237C2211yescharacteristic, complemented
25C2×#14523236C2211yescharacteristic, complemented
26C2×D1016C2211yescharacteristic
27C2×D1016C2211yescharacteristic, complemented
28C2×#14523233C2211yescharacteristic
29C2×#14523233C2211yescharacteristic, complemented
30C2×#14523233C2211yescharacteristic, complemented
31C2×#14523231C2211yescharacteristic, complemented
32#16861205—21no—
33#16861205C2211yescharacteristic
34#16861205C2211yescharacteristic, complemented
35#16861184C2211yescharacteristic
36#16861184C2211yescharacteristic, complemented
37#16861182—21no—
38#16861182—21no—
39#16861181—21no—
40#16861179—21no—
41#16861177—21no—
42#16861174C2211yescharacteristic, complemented
43#16861171C2211yescharacteristic
44C4×C22⋊D127C2211yescharacteristic
45(C22⋊D127)⋊C4—21no—
46Dic127⋊C8C2211yescharacteristic, complemented
47#16861132C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic, complemented
51C22⋊(C127⋊C8)C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53C8×Dic127C2211yescharacteristic, complemented
54C2×D8×C127C2211yescharacteristic, complemented
55C2×M16×C127C2211yescharacteristic, complemented
56#121×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic, complemented
58#106×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63#14523248—22no—
64#14523248—41no—
65#14523247—22no—
66#14523247—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⋊D127—21no—
79C2×C22⋊D127—21no—
80C2×C22⋊D127—21no—
81C2×C22⋊D127—21no—
82C2×C22⋊D127—21no—
83C2×C22⋊D127D411yescharacteristic
84D4×D127—22no—
85D4×D127—41no—
86D4×D127—44no—
87C22×Dic127—21no—
88C2×C4×D127—21no—
89C2×C4×D127—21no—
90C2×C4×D127C2311yescharacteristic
91#14523237—22no—
92#14523237D414yescomplemented
93#14523236—22no—
94D1016—22no—
95D1016—22no—
96#14523233—22no—
97#14523233—41no—
98#14523233—41no—
99#14523231—22no—
100C2×C127⋊C8C2311yescharacteristic
101C2×C127⋊C8C2311yescharacteristic
102Dic127⋊C4—21no—
103Dic127⋊C4D411yescharacteristic
104C22⋊Dic127D411yescharacteristic
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4D411yescharacteristic
109C4×Dic127—21no—
110C4×Dic127C2311yescharacteristic
111C2×D4×C127—21no—
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127C2311yescharacteristic
116D8×C127—22no—
117M16×C127—22no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22⋊C4×C127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—42no—
131C22×D127—81no—
132C22×D127—81no—
133C22×D127C2×D411yescharacteristic
134Dic254—21no—
135Dic254C2×D412yescomplemented
136D508—21no—
137D508—21no—
138D508—21no—
139D508—21no—
140D508—21no—
141D508—42no—
142D508—42no—
143D508—81no—
144D508C2×D412yes—
145C22⋊D127—22no—
146C22⋊D127—42no—
147C22⋊D127—42no—
148C22⋊D127—42no—
149C22⋊D127—42no—
150C22⋊D127—42no—
151C22⋊D127—81no—
152C22⋊D127—81no—
153C2×Dic127—21no—
154C2×Dic127—42no—
155C2×Dic127C2×D411yescharacteristic
156C2×Dic127C2×D411yescharacteristic
157C2×Dic127C2×D411yescharacteristic
158C4×D127—21no—
159C4×D127—21no—
160C4×D127—42no—
161C4×D127—42no—
162C127⋊C8—21no—
163C127⋊C8—21no—
164C23×C127—21no—
165C23×C127—21no—
166C23×C127C2×D411yescharacteristic
167D4×C127—21no—
168D4×C127—21no—
169D4×C127—21no—
170D4×C127—21no—
171D4×C127—42no—
172D4×C127—81no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—21no—
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127C2×D412yescomplemented
181C22×C127—21no—
182C22×C127—21no—
183C22×C127—21no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127C22×D411yescharacteristic
191C2×D127—21no—
192C2×D127—22no—
193C2×D127—21no—
194C2×D127—21no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—41no—
198C2×D127—41no—
199C2×D127—41no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—41no—
203C2×D127—84no—
204C2×D127—84no—
205C4×C127—21no—
206C4×C127—41no—
207C4×C127—41no—
208C4×C127C22×D411yescharacteristic
209C4×C127H(5,𝔽2)11yescharacteristic
210Dic127—21no—
211Dic127—22no—
212Dic127—21no—
213Dic127AGL(1,ℤ/8)12yescomplemented
214C2×C127—41no—
215C2×C127—81no—
216C2×C127—81no—
217C2×C127D4211yescharacteristic
218C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C127#57611yescharacteristic
220D127—41no—
221D127—42no—
222D127—81no—
223D127—81no—
224#3274—1271nomaximal, Sylow
225C127#327411yescharacteristic, complemented, Sylow
226C2×AGL(1,ℤ/8)—1271no—
227C2×AGL(1,ℤ/8)—1271no—
228#550—1271no—
229D42—1271no—
230#493—1271no—
231#469—1271no—
232#469D12711yescharacteristic, complemented
233#460—1271no—
234#459—1271no—
235#449—1271no—
236#449—1271no—
237#447—1271no—
238#447—1271no—
239C4×SD16—1271no—
240#436—1271no—
241C2×#41—1271no—
242C2×#41—1271no—
243C22×D4—1271no—
244C22×D4—1271no—
245C22×D4—2541no—
246AGL(1,ℤ/8)—2544no—
247AGL(1,ℤ/8)—2544no—
248C2×SD16—1271no—
249C2×SD16—1271no—
250C2×SD16—1271no—
251C2×D8—1271no—
252C2×D8—1271no—
253C2×D8—1271no—
254C2×D8—1271no—
255C2×D8—1271no—
256C2×D8C2×D12711yescharacteristic, complemented
257C2×M16—1271no—
258C2×M16C2×D12711yescharacteristic, complemented
259#127—1271no—
260#127—1271no—
261#127—2541no—
262#121—1271no—
263#121—1271no—
264#121—2541no—
265#121—2541no—
266#121—2541no—
267#121C2×D12711yescharacteristic
268#121C2×D12711yescharacteristic, complemented
269#120—2541no—
270#120—2541no—
271C4×Q8—1271no—
272C4×D4—1271no—
273C4×D4—1271no—
274C4×D4—2541no—
275#106C2×D12711yescharacteristic, complemented
276C4⋊C8—1271no—
277#103—1271no—
278#102—1271no—
279#102—1271no—
280#102—1271no—
281#102C2×D12711yescharacteristic, complemented
282#102C2×D12711yescharacteristic, complemented
283C22⋊C8—1271no—
284C22⋊C8—1271no—
285C4×C8—1271no—
286C24—2541no—
287C24—2541no—
288#41—5081no—
289#41—5081no—
290#41—2542no—
291#41—2542no—
292C2×Q8—1271no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1271no—
296C2×D4—1271no—
297C2×D4—1271no—
298C2×D4—1271no—
299C2×D4—5081no—
300C2×D4—5084no—
301C2×D4—5081no—
302C2×D4—2542no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4—2541no—
306C2×D4—2541no—
307C2×D4—2541no—
308C2×D4—2541no—
309C2×D4—2541no—
310C2×D4—2542no—
311C2×D4—2541no—
312C2×D4—2541no—
313C2×D4—2541no—
314C2×D4C22×D12711yescharacteristic
315C2×D4C22×D12711yescharacteristic
316C22×C4—1271no—
317C22×C4—2541no—
318C22×C4—2541no—
319C22×C4—2541no—
320C22×C4C22×D12711yescharacteristic
321SD16—1274no—
322SD16—2542no—
323SD16—2542no—
324D8—22no—
325D8—5081no—
326D8—5081no—
327D8—2542no—
328D8—2542no—
329D8—2542no—
330M16—22no—
331M16—2542no—
332C2×C8—1271no—
333C2×C8—1271no—
334C2×C8C22×D12711yescharacteristic
335C2×C8C22×D12711yescharacteristic
336C4⋊C4—1271no—
337C4⋊C4—2541no—
338C4⋊C4C22×D12711yescharacteristic
339C4⋊C4C22×D12711yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—1271no—
343C22⋊C4—1271no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C42—1271no—
348C42—2541no—
349C23—21no—
350C23—21no—
351C23—1271no—
352C23—10161no—
353C23—10161no—
354C23—5082no—
355C23—5082no—
356C23—5082no—
357C23—5082no—
358C23—2541no—
359C23—2541no—
360C23—2541no—
361C23D4×D12711yescharacteristic
362Q8—1272no—
363Q8—2541no—
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—81no—
370D4—1272no—
371D4—10161no—
372D4—10161no—
373D4—10161no—
374D4—5082no—
375D4—5082no—
376D4—5082no—
377D4—5082no—
378D4—5082no—
379D4—5082no—
380D4—5082no—
381D4—2542no—
382D4—2541no—
383D4—2541no—
384D4—2541no—
385D4—2541no—
386D4—2541no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—1271no—
392C2×C4—1271no—
393C2×C4—1271no—
394C2×C4—5082no—
395C2×C4—5082no—
396C2×C4—5082no—
397C2×C4—2541no—
398C2×C4—2541no—
399C2×C4—2541no—
400C2×C4D4×D12711yescharacteristic
401C2×C4C23×D12711yescharacteristic
402C8—21no—
403C8—2541no—
404C8—2541no—
405C8D4×D12712yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—10164no—
416C22—10164no—
417C22—5081no—
418C22—5081no—
419C22—5081no—
420C22—5081no—
421C22—5081no—
422C22—5081no—
423C22—5081no—
424C22—5081no—
425C22—2541no—
426C22—2541no—
427C22—2541no—
428C22—2542no—
429C22C2×D4×D12711yescharacteristic
430C4—21no—
431C4—41no—
432C4—41no—
433C4—1272no—
434C4—2541no—
435C4—2541no—
436C4—2542no—
437C4C2×D4×D12711yescharacteristic
438C4#1686125511yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—10161no—
443C2—10161no—
444C2—5082no—
445C2—5081no—
446C2#2255051011yescharacteristic
447C2AGL(1,ℤ/8)×D12711yescharacteristic
448C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#469×C127C2
socleC22×C127C22×D4
hypercenter#469D127
hypocenterC127#3274
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. C2×#16861233C2semidirect
3. #22550544C2semidirect
4. #22550510C2semidirect
5. #22550251C2semidirect
6. #22550130C2semidirect
7. #22550075C2semidirect
8. #22550070C2semidirect
9. #22550022C2semidirect
10. #22550019C2semidirect
11. #22550011C2semidirect
12. #22550009C2semidirect
13. #22549963C2semidirect
14. #22549952C2semidirect
15. #469×C127C2semidirect
16. C2×#14523248C22semidirect
17. C2×#14523247C22semidirect
24. C2×#14523237C22semidirect
25. C2×#14523236C22semidirect
27. C2×D1016C22semidirect
29. C2×#14523233C22semidirect
30. C2×#14523233C22semidirect
31. C2×#14523231C22semidirect
34. #16861205C22semidirect
36. #16861184C22semidirect
42. #16861174C22semidirect
46. Dic127⋊C8C22semidirect
47. #16861132C22semidirect
49. #16861129C22semidirect
50. #16861129C22semidirect
51. C22⋊(C127⋊C8)C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. C8×Dic127C22semidirect
54. C2×D8×C127C22semidirect
55. C2×M16×C127C22semidirect
57. #121×C127C22semidirect
58. #106×C127C22semidirect
59. #102×C127C22semidirect
60. #102×C127C22semidirect
92. #14523237D4semidirect
135. Dic254C2×D4semidirect
180. C8×C127C2×D4semidirect
213. Dic127AGL(1,ℤ/8)semidirect
225. C127#3274semidirect
232. #469D127semidirect
256. C2×D8C2×D127semidirect
258. C2×M16C2×D127semidirect
268. #121C2×D127semidirect
275. #106C2×D127semidirect
281. #102C2×D127semidirect
282. #102C2×D127semidirect
405. C8D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612332
#225505442
#225505102
#225502512
#225501302
#225500752
#225500702
#225500222
#225500192
#225500112
#225500092
#225499632
#225499522
#469×C1272
#3274127