Finite Group Databasebeta

← #30650276

Subgroups — #30650276

Order 16256.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861233C211yesmaximal, characteristic, complemented
2C2×#16861228C211yesmaximal, characteristic, complemented
3#22550546C211yesmaximal, characteristic, complemented
4#22550521C211yesmaximal, characteristic, complemented
5#22550263C211yesmaximal, characteristic, complemented
6#22550121C211yesmaximal, characteristic, complemented
7#22550111C211yesmaximal, characteristic, complemented
8#22550099C211yesmaximal, characteristic, complemented
9#22550039C211yesmaximal, characteristic, complemented
10#22550022C211yesmaximal, characteristic, complemented
11#22550018C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549976C211yesmaximal, characteristic, complemented
14#22549946C211yesmaximal, characteristic, complemented
15#459×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×#14523247C2211yescharacteristic, complemented
19C2×D4×D127C2211yescharacteristic
20#16861233—24no—
21#16861228—24no—
22C2×Dic508C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic, complemented
26C2×#14523236C2211yescharacteristic, complemented
27C2×#14523233C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523231C2211yescharacteristic
30C2×C8×D127C2211yescharacteristic, complemented
31#16861205—21no—
32#16861197C2211yescharacteristic
33#16861193—21no—
34#16861189C2211yescharacteristic
35#16861184C2211yescharacteristic
36#16861183—21no—
37#16861181—21no—
38#16861181—21no—
39#16861180C2211yescharacteristic
40#16861180C2211yescharacteristic, complemented
41#16861179—21no—
42C4×Dic254C2211yescharacteristic, complemented
43C4×D508C2211yescharacteristic
44D4×Dic127—21no—
45C2×(C2×D127)⋊C4—21no—
46#16861142C2211yescharacteristic, complemented
47#16861134C2211yescharacteristic, complemented
48#16861132C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic
50#16861130C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861120C2211yescharacteristic, complemented
54C2×D8×C127C2211yescharacteristic
55C2×D8×C127C2211yescharacteristic
56#127×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523247—22no—
63#14523247—22no—
64#14523247—22no—
65#14523247—41no—
66#14523247—41no—
67#14523247—41no—
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70C2×D508—21no—
71C2×D508D411yescharacteristic
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—
79D4×D127—22no—
80D4×D127—41no—
81C22×Dic127—21no—
82C22×Dic127—21no—
83C22×Dic127—21no—
84C2×C4×D127C2311yescharacteristic
85C2×C4×D127C2311yescharacteristic
86Dic508—22no—
87#14523237—22no—
88#14523236—22no—
89#14523236—41no—
90#14523236D414yescomplemented
91#14523233—22no—
92#14523233—41no—
93#14523231—22no—
94C2×C127⋊C8C2311yescharacteristic
95C2×C127⋊C8C2311yescharacteristic
96C8×D127—22no—
97C4⋊Dic127—21no—
98C4⋊Dic127D411yescharacteristic
99C4⋊Dic127C2311yescharacteristic
100Dic127⋊C4—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108(C2×D127)⋊C4D411yescharacteristic
109C4×Dic127—21no—
110C2×D4×C127—21no—
111C2×D4×C127—21no—
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127—21no—
116D8×C127—22no—
117D8×C127—22no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—42no—
125C22×D127—42no—
126C22×D127—81no—
127C22×D127C2×D411yescharacteristic
128C22×D127C2×D411yescharacteristic
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254C2×D412yes—
133D508—21no—
134D508—21no—
135D508—41no—
136D508—81no—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C22⋊D127—81no—
142C22⋊D127—81no—
143C2×Dic127—21no—
144C2×Dic127—21no—
145C2×Dic127—21no—
146C2×Dic127—42no—
147C2×Dic127—42no—
148C2×Dic127—42no—
149C2×Dic127C2×D411yescharacteristic
150C2×Dic127C2×D411yescharacteristic
151C4×D127—21no—
152C4×D127—21no—
153C4×D127—21no—
154C4×D127—21no—
155C127⋊C8—21no—
156C127⋊C8C2×D412yes—
157C23×C127—21no—
158C23×C127—21no—
159C23×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127—21no—
163D4×C127—21no—
164D4×C127—21no—
165D4×C127—42no—
166D4×C127—81no—
167D4×C127C2×D412yes—
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×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127C22×D411yescharacteristic
185C2×D127—21no—
186C2×D127—21no—
187C2×D127—21no—
188C2×D127—21no—
189C2×D127—21no—
190C2×D127—21no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—84no—
195C4×C127—21no—
196C4×C127—21no—
197C4×C127—41no—
198C4×C127C22×D411yescharacteristic
199C4×C127H(5,𝔽2)11yescharacteristic
200Dic127—21no—
201Dic127—21no—
202Dic127—22no—
203Dic127—41no—
204Dic127—41no—
205C2×C127—42no—
206C2×C127—81no—
207C2×C127—81no—
208C2×C127D4211yescharacteristic
209C2×C127#57511yescharacteristic
210C2×C127#57611yescharacteristic
211D127—41no—
212D127—41no—
213D127—81no—
214#3268—1271nomaximal, Sylow
215C127#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1271no—
217C2×#135—1271no—
218#550—1271no—
219#546—1271no—
220#496—1271no—
221#468—1271no—
222#466—1271no—
223#464—1271no—
224#459D12711yescharacteristic, complemented
225#452—1271no—
226#449—1271no—
227#449—1271no—
228#447—1271no—
229#440—1271no—
230#435—1271no—
231C2×#41—1271no—
232C2×#41—1271no—
233C2×#41—1271no—
234C22×D4—1271no—
235AGL(1,ℤ/8)—2544no—
236#135—2544no—
237C2×Q16—1271no—
238C2×SD16—1271no—
239C2×SD16—1271no—
240C2×SD16—1271no—
241C2×SD16—1271no—
242C2×D8—1271no—
243C2×D8—1271no—
244C2×D8C2×D12711yescharacteristic
245C2×D8C2×D12711yescharacteristic
246C2×M16—1271no—
247C22×C8—1271no—
248#127—2541no—
249#127C2×D12711yescharacteristic, complemented
250#124—1271no—
251#123—2541no—
252C22⋊Q8—1271no—
253#121—1271no—
254#121—1271no—
255#121—1271no—
256#121—2541no—
257#121—2541no—
258#121—2541no—
259#120—2541no—
260C4×Q8—1271no—
261C4×D4—1271no—
262C4×D4—2541no—
263C4×D4C2×D12711yescharacteristic
264C2×C22⋊C4—2541no—
265#107—1271no—
266C4⋊C8C2×D12711yescharacteristic, complemented
267#103—1271no—
268#103—1271no—
269#102—1271no—
270#102—1271no—
271#102C2×D12711yescharacteristic, complemented
272#102C2×D12711yescharacteristic, complemented
273C22⋊C8—1271no—
274C22⋊C8—1271no—
275#97—1271no—
276C24—2541no—
277#41—5081no—
278#41—5081no—
279#41—5081no—
280#41—2542no—
281#41—2542no—
282#41—2542no—
283C2×Q8—1271no—
284C2×Q8—1271no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—1271no—
288C2×D4—1271no—
289C2×D4—5081no—
290C2×D4—2541no—
291C2×D4—2541no—
292C2×D4—2541no—
293C2×D4—2541no—
294C2×D4—2541no—
295C2×D4—2541no—
296C2×D4—2541no—
297C2×D4—2542no—
298C2×D4C22×D12711yescharacteristic
299C2×D4C22×D12711yescharacteristic
300C2×D4C22×D12711yescharacteristic
301C22×C4—21no—
302C22×C4—1271no—
303C22×C4—1271no—
304C22×C4—2541no—
305C22×C4—2541no—
306C22×C4—2541no—
307Q16—2542no—
308SD16—1274no—
309SD16—5081no—
310SD16—2542no—
311SD16—2542no—
312D8—22no—
313D8—22no—
314D8—5081no—
315D8—2542no—
316M16—2542no—
317C2×C8—1271no—
318C2×C8—1271no—
319C2×C8—2542no—
320C2×C8C22×D12711yescharacteristic
321C2×C8C22×D12711yescharacteristic
322C4⋊C4—1271no—
323C4⋊C4—1271no—
324C4⋊C4—2541no—
325C4⋊C4—2541no—
326C4⋊C4C22×D12711yescharacteristic
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—
336C42—2541no—
337C42C22×D12711yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—1271no—
342C23—1271no—
343C23—10161no—
344C23—5082no—
345C23—5082no—
346C23—2541no—
347Q8—1272no—
348Q8—2541no—
349Q8—2541no—
350Q8—2541no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—10161no—
359D4—10161no—
360D4—10161no—
361D4—5081no—
362D4—5082no—
363D4—5082no—
364D4—5082no—
365D4—5082no—
366D4—2541no—
367D4—2541no—
368D4D4×D12712yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—1271no—
372C2×C4—1271no—
373C2×C4—5082no—
374C2×C4—5082no—
375C2×C4—5082no—
376C2×C4—2541no—
377C2×C4—2541no—
378C2×C4—2541no—
379C2×C4—2541no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4D4×D12711yescharacteristic
384C2×C4D4×D12711yescharacteristic
385C2×C4C23×D12711yescharacteristic
386C8—21no—
387C8—21no—
388C8—1272no—
389C8—2541no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—10164no—
400C22—5081no—
401C22—5081no—
402C22—5081no—
403C22—2541no—
404C22—2541no—
405C22—2541no—
406C22—2541no—
407C22—2541no—
408C22—2541no—
409C22C2×D4×D12711yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5081no—
414C4—5081no—
415C4—2541no—
416C4—2541no—
417C4—2542no—
418C4C2×D4×D12711yescharacteristic
419C4#1686125511yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—10161no—
424C2—5081no—
425C2—5081no—
426C2#2255051011yescharacteristic
427C2#2255067211yescharacteristic
428C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3268
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. C2×#16861228C2semidirect
3. #22550546C2semidirect
4. #22550521C2semidirect
5. #22550263C2semidirect
6. #22550121C2semidirect
7. #22550111C2semidirect
8. #22550099C2semidirect
9. #22550039C2semidirect
10. #22550022C2semidirect
11. #22550018C2semidirect
12. #22550008C2semidirect
13. #22549976C2semidirect
14. #22549946C2semidirect
15. #459×C127C2semidirect
18. C2×#14523247C22semidirect
22. C2×Dic508C22semidirect
25. C2×#14523236C22semidirect
26. C2×#14523236C22semidirect
30. C2×C8×D127C22semidirect
40. #16861180C22semidirect
42. C4×Dic254C22semidirect
46. #16861142C22semidirect
47. #16861134C22semidirect
48. #16861132C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861120C22semidirect
56. #127×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #102×C127C22semidirect
60. #102×C127C22semidirect
90. #14523236D4semidirect
215. C127#3268semidirect
224. #459D127semidirect
249. #127C2×D127semidirect
266. C4⋊C8C2×D127semidirect
271. #102C2×D127semidirect
272. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
C2×#168612282
#225505462
#225505212
#225502632
#225501212
#225501112
#225500992
#225500392
#225500222
#225500182
#225500082
#225499762
#225499462
#459×C1272
#3268127