Finite Group Databasebeta

← #30650343

Subgroups — #30650343

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×#16861236C211yesmaximal, characteristic, complemented
2C2×#16861234C211yesmaximal, characteristic, complemented
3#22550545C211yesmaximal, characteristic, complemented
4#22550510C211yesmaximal, characteristic, complemented
5#22550251C211yesmaximal, characteristic, complemented
6#22550128C211yesmaximal, characteristic, complemented
7#22550127C211yesmaximal, characteristic, complemented
8#22550074C211yesmaximal, characteristic, complemented
9#22550070C211yesmaximal, characteristic, complemented
10#22550024C211yesmaximal, characteristic, complemented
11#22550012C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549961C211yesmaximal, characteristic, complemented
14#22549951C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic, complemented
17C22×D508C2211yescharacteristic
18C2×D4×D127—21no—
19C2×D4×D127C2211yescharacteristic
20#16861236—24no—
21#16861234—24no—
22C2×#14523235C2211yescharacteristic
23C2×#14523235C2211yescharacteristic, complemented
24C2×D1016C2211yescharacteristic
25C2×D1016C2211yescharacteristic, complemented
26C2×#14523233C2211yescharacteristic
27C2×#14523233C2211yescharacteristic, complemented
28C2×#14523233C2211yescharacteristic, complemented
29C2×#14523232C2211yescharacteristic, complemented
30C2×#14523231C2211yescharacteristic, complemented
31#16861205—21no—
32#16861205C2211yescharacteristic
33#16861205C2211yescharacteristic, complemented
34#16861184C2211yescharacteristic
35#16861183—21no—
36#16861183C2211yescharacteristic, complemented
37#16861182—21no—
38#16861180—21no—
39#16861180C2211yescharacteristic, complemented
40#16861179—21no—
41#16861177—21no—
42Q8×Dic127C2211yescharacteristic, complemented
43#16861171C2211yescharacteristic
44C4×C22⋊D127—21no—
45C4×C22⋊D127C2211yescharacteristic
46#16861140C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic
49#16861131C2211yescharacteristic, complemented
50#16861130C2211yescharacteristic, complemented
51#16861129C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53C8×Dic127C2211yescharacteristic, complemented
54C2×#41×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic, complemented
56C2×D8×C127C2211yescharacteristic, complemented
57#121×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic, complemented
60C22⋊C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63#14523249—22no—
64#14523249—41no—
65C2×D508—22no—
66C2×D508—21no—
67C2×D508—21no—
68C2×D508—21no—
69C2×D508—41no—
70C2×D508C2311yescharacteristic
71C2×D508C2311yescharacteristic
72C2×D508C2311yescharacteristic
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77C2×C22⋊D127—21no—
78C2×C22⋊D127—21no—
79C2×C22⋊D127—21no—
80C2×C22⋊D127D411yescharacteristic
81D4×D127—22no—
82D4×D127—41no—
83D4×D127—44no—
84C2×C4×D127—21no—
85C2×C4×D127—21no—
86C2×C4×D127C2311yescharacteristic
87#14523235—22no—
88#14523235D414yescomplemented
89D1016—22no—
90D1016—41no—
91#14523233—22no—
92#14523233—22no—
93#14523233—22no—
94#14523232—22no—
95#14523231—22no—
96C2×C127⋊C8C2311yescharacteristic
97C2×C127⋊C8C2311yescharacteristic
98C4⋊Dic127—21no—
99C4⋊Dic127C2311yescharacteristic
100Dic127⋊C4D411yescharacteristic
101C22⋊Dic127—21no—
102C22⋊Dic127D411yescharacteristic
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4D411yescharacteristic
107C4×Dic127—21no—
108C4×Dic127C2311yescharacteristic
109#41×C127—22no—
110#41×C127—41no—
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127—21no—
114C2×D4×C127C2311yescharacteristic
115C2×D4×C127C2311yescharacteristic
116C22×C4×C127—21no—
117C22×C4×C127C2311yescharacteristic
118SD16×C127—22no—
119D8×C127—41no—
120C2×C8×C127C2311yescharacteristic
121C2×C8×C127C2311yescharacteristic
122C4⋊C4×C127C2311yescharacteristic
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
134D508—21no—
135D508—21no—
136D508—21no—
137D508—21no—
138D508—21no—
139D508—42no—
140D508—42no—
141D508—42no—
142D508C2×D412yes—
143C22⋊D127—22no—
144C22⋊D127—42no—
145C22⋊D127—42no—
146C22⋊D127—42no—
147C22⋊D127—81no—
148C22⋊D127—81no—
149C22⋊D127—81no—
150C22⋊D127—81no—
151C2×Dic127—21no—
152C2×Dic127C2×D411yescharacteristic
153C2×Dic127C2×D411yescharacteristic
154C2×Dic127C2×D411yescharacteristic
155C4×D127—21no—
156C4×D127—21no—
157C4×D127—42no—
158C4×D127—42no—
159C127⋊C8—21no—
160C127⋊C8C2×D412yescomplemented
161C23×C127—21no—
162C23×C127—21no—
163C23×C127C2×D411yescharacteristic
164Q8×C127—21no—
165Q8×C127C2×D412yescomplemented
166D4×C127—21no—
167D4×C127—21no—
168D4×C127—21no—
169D4×C127—21no—
170D4×C127—42no—
171D4×C127—42no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—42no—
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127—21no—
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—22no—
207C4×C127—41no—
208C4×C127C22×D411yescharacteristic
209C4×C127H(5,𝔽2)11yescharacteristic
210Dic127—21no—
211Dic127—21no—
212Dic127—41no—
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#469—1271no—
233#460—1271no—
234#459—1271no—
235#449—1271no—
236#449D12711yescharacteristic, complemented
237#447—1271no—
238#447—1271no—
239C4×SD16—1271no—
240#436—1271no—
241C2×#41—1271no—
242C2×#41C2×D12711yescharacteristic, complemented
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×SD16C2×D12711yescharacteristic, complemented
251C2×D8—1271no—
252C2×D8—1271no—
253C2×D8—1271no—
254C2×D8—1271no—
255C2×D8—1271no—
256C2×D8C2×D12711yescharacteristic, complemented
257C2×M16—1271no—
258C2×M16—1271no—
259#127—1271no—
260#127—1271no—
261#127—2541no—
262#121—1271no—
263#121—1271no—
264#121—1271no—
265#121—2541no—
266#121—2541no—
267#121—2541no—
268#121C2×D12711yescharacteristic
269#120—2541no—
270#120—2541no—
271C4×Q8—1271no—
272C4×D4—1271no—
273C4×D4—1271no—
274C4×D4—2541no—
275#106—1271no—
276C4⋊C8—1271no—
277#103C2×D12711yescharacteristic, complemented
278#102—1271no—
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102C2×D12711yescharacteristic, complemented
283C22⋊C8—1271no—
284C22⋊C8C2×D12711yescharacteristic, complemented
285C4×C8—1271no—
286C24—2541no—
287C24—2541no—
288#41—22no—
289#41—41no—
290#41—5081no—
291#41—2542no—
292C2×Q8C22×D12711yescharacteristic
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—21no—
317C22×C4—1271no—
318C22×C4—2541no—
319C22×C4—2541no—
320C22×C4C22×D12711yescharacteristic
321SD16—22no—
322SD16—1274no—
323SD16—2542no—
324D8—41no—
325D8—5081no—
326D8—2542no—
327D8—2542no—
328D8—2542no—
329D8—2542no—
330M16—2542no—
331M16—2542no—
332C2×C8—1271no—
333C2×C8—1271no—
334C2×C8C22×D12711yescharacteristic
335C2×C8C22×D12711yescharacteristic
336C4⋊C4—1271no—
337C4⋊C4—1271no—
338C4⋊C4—2541no—
339C4⋊C4C22×D12711yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—1271no—
342C22⋊C4—1271no—
343C22⋊C4—2541no—
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—21no—
363Q8D4×D12712yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—42no—
370D4—1272no—
371D4—10161no—
372D4—10161no—
373D4—10161no—
374D4—10161no—
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—42no—
392C2×C4—1271no—
393C2×C4—1271no—
394C2×C4—1271no—
395C2×C4—5082no—
396C2×C4—5082no—
397C2×C4—2541no—
398C2×C4—2541no—
399C2×C4—2541no—
400C2×C4D4×D12711yescharacteristic
401C2×C4C23×D12711yescharacteristic
402C8—21no—
403C8—21no—
404C8—1272no—
405C8—2541no—
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—22no—
432C4—41no—
433C4—1272no—
434C4—5081no—
435C4—2541no—
436C4—2541no—
437C4C2×D4×D12711yescharacteristic
438C4#1686125411yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—10161no—
443C2—10161no—
444C2—5082no—
445C2—5081no—
446C2#2255050811yescharacteristic
447C2AGL(1,ℤ/8)×D12711yescharacteristic
448C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3274
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×#16861234C2semidirect
3. #22550545C2semidirect
4. #22550510C2semidirect
5. #22550251C2semidirect
6. #22550128C2semidirect
7. #22550127C2semidirect
8. #22550074C2semidirect
9. #22550070C2semidirect
10. #22550024C2semidirect
11. #22550012C2semidirect
12. #22550008C2semidirect
13. #22549961C2semidirect
14. #22549951C2semidirect
15. #449×C127C2semidirect
16. C2×#14523249C22semidirect
23. C2×#14523235C22semidirect
25. C2×D1016C22semidirect
27. C2×#14523233C22semidirect
28. C2×#14523233C22semidirect
29. C2×#14523232C22semidirect
30. C2×#14523231C22semidirect
33. #16861205C22semidirect
36. #16861183C22semidirect
39. #16861180C22semidirect
42. Q8×Dic127C22semidirect
46. #16861140C22semidirect
47. Dic127⋊C8C22semidirect
49. #16861131C22semidirect
50. #16861130C22semidirect
51. #16861129C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. C8×Dic127C22semidirect
54. C2×#41×C127C22semidirect
55. C2×SD16×C127C22semidirect
56. C2×D8×C127C22semidirect
58. #103×C127C22semidirect
59. #102×C127C22semidirect
60. C22⋊C8×C127C22semidirect
88. #14523235D4semidirect
160. C127⋊C8C2×D4semidirect
165. Q8×C127C2×D4semidirect
213. Dic127AGL(1,ℤ/8)semidirect
225. C127#3274semidirect
236. #449D127semidirect
242. C2×#41C2×D127semidirect
250. C2×SD16C2×D127semidirect
256. C2×D8C2×D127semidirect
277. #103C2×D127semidirect
282. #102C2×D127semidirect
284. C22⋊C8C2×D127semidirect
363. Q8D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×#168612342
#225505452
#225505102
#225502512
#225501282
#225501272
#225500742
#225500702
#225500242
#225500122
#225500082
#225499612
#225499512
#449×C1272
#3274127