Finite Group Databasebeta

← #30650334

Subgroups — #30650334

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×#16861234C211yesmaximal, characteristic, complemented
3#22550517C211yesmaximal, characteristic, complemented
4#22550514C211yesmaximal, characteristic, complemented
5#22550261C211yesmaximal, characteristic, complemented
6#22550125C211yesmaximal, characteristic, complemented
7#22550124C211yesmaximal, characteristic, complemented
8#22550094C211yesmaximal, characteristic, complemented
9#22550093C211yesmaximal, characteristic, complemented
10#22550037C211yesmaximal, characteristic, complemented
11#22550012C211yesmaximal, characteristic, complemented
12#22550011C211yesmaximal, characteristic, complemented
13#22549956C211yesmaximal, characteristic, complemented
14#22549951C211yesmaximal, characteristic, complemented
15#452×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C22×D508C2211yescharacteristic, complemented
18C2×D4×D127C2211yescharacteristic, complemented
19#16861236—24no—
20#16861234—24no—
21C2×#14523235C2211yescharacteristic, complemented
22C2×#14523235C2211yescharacteristic, complemented
23C2×#14523235C2211yescharacteristic, complemented
24C2×D1016C2211yescharacteristic
25C2×D1016C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523232C2211yescharacteristic, complemented
29C2×#14523231C2211yescharacteristic, complemented
30#16861198C2211yescharacteristic, complemented
31#16861198C2211yescharacteristic, complemented
32#16861194—21no—
33#16861192—21no—
34#16861185C2211yescharacteristic, complemented
35#16861184—21no—
36#16861184C2211yescharacteristic
37#16861183C2211yescharacteristic
38#16861181—21no—
39#16861179—21no—
40#16861177—21no—
41D4×Dic127C2211yescharacteristic, complemented
42#16861171C2211yescharacteristic, complemented
43C4×C22⋊D127C2211yescharacteristic
44C2×(C2×D127)⋊C4—21no—
45C22⋊C4×D127—21no—
46#16861142C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861133C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic, complemented
51#16861129C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53C8×Dic127C2211yescharacteristic, complemented
54C2×#41×C127C2211yescharacteristic
55C2×Q16×C127C2211yescharacteristic, complemented
56C2×SD16×C127C2211yescharacteristic, complemented
57C22⋊Q8×C127C2211yescharacteristic, complemented
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⋊D127D411yescharacteristic
76D4×D127—22no—
77D4×D127—41no—
78C22×Dic127—21no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127C2311yescharacteristic
82#14523235—22no—
83#14523235—22no—
84#14523235—22no—
85D1016—22no—
86D1016—41no—
87#14523233—22no—
88#14523233D414yescomplemented
89#14523232—22no—
90#14523231—22no—
91C2×C127⋊C8C2311yescharacteristic
92C2×C127⋊C8C2311yescharacteristic
93C4⋊Dic127C2311yescharacteristic
94Dic127⋊C4—21no—
95Dic127⋊C4D411yescharacteristic
96C22⋊Dic127—21no—
97C22⋊Dic127D411yescharacteristic
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4D411yescharacteristic
107C4×Dic127C2311yescharacteristic
108#41×C127—22no—
109#41×C127—41no—
110C2×Q8×C127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116Q16×C127—41no—
117SD16×C127—22no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127—21no—
121C4⋊C4×C127C2311yescharacteristic
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
133D508—21no—
134D508—21no—
135D508—21no—
136D508—21no—
137D508—21no—
138D508—42no—
139D508—42no—
140D508—81no—
141D508C2×D412yes—
142C22⋊D127—41no—
143C22⋊D127—42no—
144C22⋊D127—81no—
145C2×Dic127—21no—
146C2×Dic127—81no—
147C2×Dic127C2×D411yescharacteristic
148C2×Dic127C2×D411yescharacteristic
149C2×Dic127C2×D411yescharacteristic
150C4×D127—21no—
151C4×D127—21no—
152C4×D127—42no—
153C4×D127—81no—
154C127⋊C8—21no—
155C127⋊C8C2×D412yes—
156C23×C127—21no—
157C23×C127C2×D411yescharacteristic
158Q8×C127—21no—
159Q8×C127—21no—
160Q8×C127—21no—
161Q8×C127—21no—
162D4×C127—21no—
163D4×C127—42no—
164D4×C127C2×D412yes—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—21no—
170C2×C4×C127—42no—
171C2×C4×C127C2×D411yescharacteristic
172C2×C4×C127C2411yescharacteristic
173C8×C127—21no—
174C8×C127—21no—
175C22×C127—21no—
176C22×C127—22no—
177C22×C127—21no—
178C22×C127—21no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127C22×D411yescharacteristic
182C2×D127—21no—
183C2×D127—22no—
184C2×D127—21no—
185C2×D127—21no—
186C2×D127—41no—
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—42no—
208C2×C127D4211yescharacteristic
209C2×C127#57511yescharacteristic
210C2×C127#57611yescharacteristic
211D127—41no—
212D127—42no—
213D127—81no—
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#463—1271no—
226#452—1271no—
227#452D12711yescharacteristic, complemented
228#447—1271no—
229#447—1271no—
230C4×D8—1271no—
231#436—1271no—
232C2×#41—1271no—
233C2×#41C2×D12711yescharacteristic
234C22×D4—1271no—
235C22×D4—1271no—
236AGL(1,ℤ/8)—2544no—
237AGL(1,ℤ/8)—2544no—
238C2×Q16C2×D12711yescharacteristic, complemented
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×M16—1271no—
249#124—1271no—
250#124—1271no—
251#123—2541no—
252#123—2541no—
253C22⋊Q8—1271no—
254C22⋊Q8C2×D12711yescharacteristic, complemented
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×D4—1271no—
264C2×C22⋊C4—2541no—
265C2×C22⋊C4—2541no—
266#107—1271no—
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⋊C8C2×D12711yescharacteristic, complemented
276C4×C8—1271no—
277C24—2541no—
278C24—2541no—
279#41—22no—
280#41—41no—
281#41—5081no—
282#41—2542no—
283C2×Q8C22×D12711yescharacteristic
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—2541no—
297C2×D4—2542no—
298C2×D4—2541no—
299C2×D4C22×D12711yescharacteristic
300C22×C4—21no—
301C22×C4—1271no—
302C22×C4—2541no—
303C22×C4—2541no—
304C22×C4—2541no—
305C22×C4C22×D12711yescharacteristic
306Q16—41no—
307SD16—22no—
308SD16—2542no—
309SD16—2542no—
310SD16—2542no—
311D8—1274no—
312D8—5081no—
313D8—2542no—
314D8—2542no—
315M16—2542no—
316M16—2542no—
317C2×C8—1271no—
318C2×C8—1271no—
319C2×C8C22×D12711yescharacteristic
320C2×C8C22×D12711yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—1271no—
323C4⋊C4—1271no—
324C4⋊C4—2541no—
325C4⋊C4C22×D12711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—1271no—
328C22⋊C4—1271no—
329C22⋊C4—2541no—
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—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—1272no—
358D4—10161no—
359D4—10161no—
360D4—5082no—
361D4—5082no—
362D4—5082no—
363D4—5081no—
364D4—2541no—
365D4—2541no—
366D4—2541no—
367D4—2541no—
368D4—2541no—
369D4D4×D12712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—1271no—
377C2×C4—1271no—
378C2×C4—1271no—
379C2×C4—10161no—
380C2×C4—10161no—
381C2×C4—5082no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4—2541no—
385C2×C4D4×D12711yescharacteristic
386C2×C4C23×D12711yescharacteristic
387C8—21no—
388C8—21no—
389C8—1272no—
390C8—2541no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—10164no—
398C22—10164no—
399C22—5081no—
400C22—5081no—
401C22—5081no—
402C22—5081no—
403C22—5081no—
404C22—5081no—
405C22—5081no—
406C22—5081no—
407C22—2541no—
408C22—2541no—
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#1686125411yescharacteristic
422C2—41no—
423C2—42no—
424C2—10161no—
425C2—10161no—
426C2—5081no—
427C2—5082no—
428C2#2255050811yescharacteristic
429C2#2255067511yescharacteristic
430C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#452×C127C2
socleC22×C127C22×D4
hypercenter#452D127
hypocenterC127#3273
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×#16861234C2semidirect
3. #22550517C2semidirect
4. #22550514C2semidirect
5. #22550261C2semidirect
6. #22550125C2semidirect
7. #22550124C2semidirect
8. #22550094C2semidirect
9. #22550093C2semidirect
10. #22550037C2semidirect
11. #22550012C2semidirect
12. #22550011C2semidirect
13. #22549956C2semidirect
14. #22549951C2semidirect
15. #452×C127C2semidirect
17. C22×D508C22semidirect
18. C2×D4×D127C22semidirect
21. C2×#14523235C22semidirect
22. C2×#14523235C22semidirect
23. C2×#14523235C22semidirect
28. C2×#14523232C22semidirect
29. C2×#14523231C22semidirect
30. #16861198C22semidirect
31. #16861198C22semidirect
34. #16861185C22semidirect
41. D4×Dic127C22semidirect
42. #16861171C22semidirect
46. #16861142C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861133C22semidirect
49. #16861131C22semidirect
50. #16861131C22semidirect
51. #16861129C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. C8×Dic127C22semidirect
55. C2×Q16×C127C22semidirect
56. C2×SD16×C127C22semidirect
57. C22⋊Q8×C127C22semidirect
58. #103×C127C22semidirect
59. #102×C127C22semidirect
60. C22⋊C8×C127C22semidirect
88. #14523233D4semidirect
216. C127#3273semidirect
227. #452D127semidirect
238. C2×Q16C2×D127semidirect
242. C2×SD16C2×D127semidirect
254. C22⋊Q8C2×D127semidirect
269. #103C2×D127semidirect
273. #102C2×D127semidirect
275. C22⋊C8C2×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
#225505172
#225505142
#225502612
#225501252
#225501242
#225500942
#225500932
#225500372
#225500122
#225500112
#225499562
#225499512
#452×C1272
#3273127