Finite Group Databasebeta

← #30650368

Subgroups — #30650368

Order 16256.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861242C211yesmaximal, characteristic, complemented
2C2×#16861233C211yesmaximal, characteristic, complemented
3#22550532C211yesmaximal, characteristic, complemented
4#22550523C211yesmaximal, characteristic, complemented
5#22550259C211yesmaximal, characteristic, complemented
6#22550135C211yesmaximal, characteristic, complemented
7#22550097C211yesmaximal, characteristic, complemented
8#22550090C211yesmaximal, characteristic, complemented
9#22550039C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22550019C211yesmaximal, characteristic, complemented
12#22550014C211yesmaximal, characteristic, complemented
13#22549963C211yesmaximal, characteristic, complemented
14#22549952C211yesmaximal, characteristic, complemented
15#469×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C22×Dic254C2211yescharacteristic, complemented
19C2×D4×D127C2211yescharacteristic
20#16861242—24no—
21#16861233—24no—
22C2×Dic508C2211yescharacteristic, complemented
23C2×Dic508C2211yescharacteristic, complemented
24C2×#14523237C2211yescharacteristic
25C2×#14523237C2211yescharacteristic
26C2×#14523236C2211yescharacteristic
27C2×#14523236C2211yescharacteristic
28C2×#14523236C2211yescharacteristic, complemented
29C2×#14523233C2211yescharacteristic
30C2×#14523231C2211yescharacteristic
31#16861196C2211yescharacteristic
32#16861196C2211yescharacteristic, complemented
33#16861195—21no—
34#16861191—21no—
35#16861189C2211yescharacteristic
36#16861189C2211yescharacteristic
37#16861186—21no—
38#16861181—21no—
39#16861180—21no—
40#16861177—21no—
41#16861174C2211yescharacteristic, complemented
42D4×Dic127—21no—
43#16861171C2211yescharacteristic
44C4×C22⋊D127C2211yescharacteristic
45C22⋊C4×D127—21no—
46Dic127⋊C8C2211yescharacteristic, complemented
47#16861132C2211yescharacteristic, complemented
48#16861132C2211yescharacteristic, complemented
49#16861132C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic
51C22⋊(C127⋊C8)C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic
53C8×Dic127C2211yescharacteristic, complemented
54C2×D8×C127C2211yescharacteristic
55C2×M16×C127C2211yescharacteristic, complemented
56#121×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic, complemented
58#106×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523248—22no—
63#14523248—41no—
64#14523247—22no—
65#14523247—41no—
66C2×Dic254—22no—
67C2×Dic254—21no—
68C2×Dic254—41no—
69C2×Dic254C2311yescharacteristic
70C2×Dic254C2311yescharacteristic
71C2×Dic254C2311yescharacteristic
72C2×D508C2311yescharacteristic
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127D411yescharacteristic
77D4×D127—22no—
78D4×D127—41no—
79C22×Dic127—21no—
80C22×Dic127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127C2311yescharacteristic
84Dic508—22no—
85Dic508—22no—
86#14523237—22no—
87#14523237D414yescomplemented
88#14523236—22no—
89#14523236—41no—
90#14523236—41no—
91#14523233—22no—
92#14523231—22no—
93C2×C127⋊C8C2311yescharacteristic
94C2×C127⋊C8C2311yescharacteristic
95C4⋊Dic127—21no—
96C4⋊Dic127—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4D411yescharacteristic
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127D411yescharacteristic
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108C4×Dic127—21no—
109C4×Dic127C2311yescharacteristic
110C2×D4×C127—21no—
111C2×D4×C127—21no—
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127C2311yescharacteristic
115D8×C127—22no—
116M16×C127—22no—
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C4⋊C4×C127C2311yescharacteristic
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127—42no—
125C22×D127—42no—
126C22×D127—81no—
127C22×D127C2×D411yescharacteristic
128Dic254—21no—
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254—21no—
133Dic254—42no—
134Dic254C2×D412yes—
135D508—21no—
136D508C2×D412yes—
137C22⋊D127—41no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—42no—
145C2×Dic127—42no—
146C2×Dic127C2×D411yescharacteristic
147C2×Dic127C2×D411yescharacteristic
148C2×Dic127C2×D411yescharacteristic
149C4×D127—21no—
150C4×D127—21no—
151C4×D127—42no—
152C4×D127—81no—
153C127⋊C8—21no—
154C127⋊C8—21no—
155C23×C127—21no—
156C23×C127—21no—
157C23×C127C2×D411yescharacteristic
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127—42no—
163D4×C127—81no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127C2×D412yes—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
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—84no—
189C4×C127—21no—
190C4×C127—41no—
191C4×C127—41no—
192C4×C127C22×D411yescharacteristic
193C4×C127H(5,𝔽2)11yescharacteristic
194Dic127—21no—
195Dic127—22no—
196Dic127—21no—
197Dic127—21no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—41no—
201C2×C127—81no—
202C2×C127—81no—
203C2×C127D4211yescharacteristic
204C2×C127#57511yescharacteristic
205C2×C127#57711yescharacteristic
206D127—41no—
207D127—42no—
208#3275—1271nomaximal, Sylow
209C127#327511yescharacteristic, complemented, Sylow
210C2×#137—1271no—
211C2×AGL(1,ℤ/8)—1271no—
212#548—1271no—
213#546—1271no—
214#495—1271no—
215#470—1271no—
216#469D12711yescharacteristic, complemented
217#464—1271no—
218#463—1271no—
219#452—1271no—
220#450—1271no—
221#449—1271no—
222#448—1271no—
223C4×SD16—1271no—
224#436—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C22×Q8—1271no—
228C22×D4—1271no—
229#137—2544no—
230AGL(1,ℤ/8)—2544no—
231C2×Q16—1271no—
232C2×Q16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16—1271no—
237C2×SD16—1271no—
238C2×D8—1271no—
239C2×D8C2×D12711yescharacteristic
240C2×M16—1271no—
241C2×M16C2×D12711yescharacteristic, complemented
242#124—1271no—
243#124—1271no—
244#124—2541no—
245#123—2541no—
246C22⋊Q8—1271no—
247C22⋊Q8—1271no—
248C22⋊Q8—2541no—
249#121—2541no—
250#121—2541no—
251#121C2×D12711yescharacteristic
252#121C2×D12711yescharacteristic, complemented
253#120—2541no—
254C4×Q8—1271no—
255C4×D4—1271no—
256C4×D4—1271no—
257C4×D4—2541no—
258C2×C22⋊C4—2541no—
259#106C2×D12711yescharacteristic, complemented
260C4⋊C8—1271no—
261#103—1271no—
262#103—1271no—
263#103—1271no—
264#102—1271no—
265#102C2×D12711yescharacteristic
266#102C2×D12711yescharacteristic, complemented
267C22⋊C8—1271no—
268C22⋊C8—1271no—
269C4×C8—1271no—
270C24—2541no—
271#41—5081no—
272#41—5081no—
273#41—2542no—
274#41—2542no—
275C2×Q8—1271no—
276C2×Q8—1271no—
277C2×Q8—1271no—
278C2×Q8—5081no—
279C2×Q8—2542no—
280C2×Q8—2541no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—1271no—
284C2×D4—1271no—
285C2×D4—5081no—
286C2×D4—2541no—
287C2×D4—2542no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4C22×D12711yescharacteristic
291C2×D4C22×D12711yescharacteristic
292C22×C4—1271no—
293C22×C4—2541no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4C22×D12711yescharacteristic
298Q16—2542no—
299Q16—2542no—
300SD16—1274no—
301SD16—5081no—
302SD16—5081no—
303SD16—2542no—
304SD16—2542no—
305D8—22no—
306D8—2542no—
307M16—22no—
308M16—2542no—
309C2×C8—1271no—
310C2×C8—1271no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—1271no—
314C4⋊C4—2541no—
315C4⋊C4—2541no—
316C4⋊C4—2541no—
317C4⋊C4C22×D12711yescharacteristic
318C4⋊C4C22×D12711yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—1271no—
322C22⋊C4—1271no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C42—1271no—
331C42—2541no—
332C23—21no—
333C23—21no—
334C23—1271no—
335C23—10161no—
336C23—5082no—
337C23—5082no—
338C23—2541no—
339C23D4×D12711yescharacteristic
340Q8—1272no—
341Q8—5082no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345Q8—2541no—
346Q8—2541no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—81no—
353D4—1272no—
354D4—5082no—
355D4—5081no—
356D4—5082no—
357D4—5082no—
358D4—2541no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—1271no—
364C2×C4—1271no—
365C2×C4—1271no—
366C2×C4—10161no—
367C2×C4—5082no—
368C2×C4—5082no—
369C2×C4—5082no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4—2541no—
375C2×C4D4×D12711yescharacteristic
376C2×C4C23×D12711yescharacteristic
377C8—21no—
378C8—2541no—
379C8—2541no—
380C8D4×D12712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—10164no—
391C22—5081no—
392C22—5081no—
393C22—2541no—
394C22—2541no—
395C22—2542no—
396C22—2541no—
397C22C2×D4×D12711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—5081no—
402C4—5081no—
403C4—2541no—
404C4—2541no—
405C4—2541no—
406C4—2542no—
407C4C2×D4×D12711yescharacteristic
408C4#1686125511yescharacteristic
409C2—41no—
410C2—81no—
411C2—81no—
412C2—5081no—
413C2—5082no—
414C2#2255051011yescharacteristic
415C2#2255067211yescharacteristic
416C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#469×C127C2
socleC22×C127C22×D4
hypercenter#469D127
hypocenterC127#3275
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×#16861242C2semidirect
2. C2×#16861233C2semidirect
3. #22550532C2semidirect
4. #22550523C2semidirect
5. #22550259C2semidirect
6. #22550135C2semidirect
7. #22550097C2semidirect
8. #22550090C2semidirect
9. #22550039C2semidirect
10. #22550028C2semidirect
11. #22550019C2semidirect
12. #22550014C2semidirect
13. #22549963C2semidirect
14. #22549952C2semidirect
15. #469×C127C2semidirect
18. C22×Dic254C22semidirect
22. C2×Dic508C22semidirect
23. C2×Dic508C22semidirect
28. C2×#14523236C22semidirect
32. #16861196C22semidirect
41. #16861174C22semidirect
46. Dic127⋊C8C22semidirect
47. #16861132C22semidirect
48. #16861132C22semidirect
49. #16861132C22semidirect
51. C22⋊(C127⋊C8)C22semidirect
53. C8×Dic127C22semidirect
55. C2×M16×C127C22semidirect
57. #121×C127C22semidirect
58. #106×C127C22semidirect
60. #102×C127C22semidirect
87. #14523237D4semidirect
209. C127#3275semidirect
216. #469D127semidirect
241. C2×M16C2×D127semidirect
252. #121C2×D127semidirect
259. #106C2×D127semidirect
266. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×#168612332
#225505322
#225505232
#225502592
#225501352
#225500972
#225500902
#225500392
#225500282
#225500192
#225500142
#225499632
#225499522
#469×C1272
#3275127