Finite Group Databasebeta

← #30650263

Subgroups — #30650263

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×#16861236C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550545C211yesmaximal, characteristic, complemented
4#22550523C211yesmaximal, characteristic, complemented
5#22550264C211yesmaximal, characteristic, complemented
6#22550121C211yesmaximal, characteristic, complemented
7#22550110C211yesmaximal, characteristic, complemented
8#22550096C211yesmaximal, characteristic, complemented
9#22550070C211yesmaximal, characteristic, complemented
10#22550041C211yesmaximal, characteristic, complemented
11#22550023C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549979C211yesmaximal, characteristic, complemented
14#22549944C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861236—24no—
20#16861229—24no—
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523236C2211yescharacteristic, complemented
24C2×#14523235C2211yescharacteristic
25C2×D1016C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523231C2211yescharacteristic
29C22×C127⋊C8C2211yescharacteristic, complemented
30#16861205—21no—
31#16861205C2211yescharacteristic, complemented
32#16861196C2211yescharacteristic
33#16861191—21no—
34#16861189C2211yescharacteristic
35#16861183—21no—
36#16861183C2211yescharacteristic, complemented
37#16861180—21no—
38#16861180—21no—
39#16861180C2211yescharacteristic
40#16861177—21no—
41Q8×Dic127C2211yescharacteristic, complemented
42#16861171C2211yescharacteristic
43C4×C22⋊D127—21no—
44C4×C22⋊D127C2211yescharacteristic
45C22⋊C4×D127—21no—
46#16861142C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic
50#16861130C2211yescharacteristic
51#16861129C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic
53#16861119C2211yescharacteristic, complemented
54C2×#41×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic
56C2×D8×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic, complemented
60C22⋊C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—41no—
64#14523248—22no—
65#14523248—41no—
66C2×Dic254C2311yescharacteristic
67C2×D508—21no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127D411yescharacteristic
77D4×D127—22no—
78D4×D127—41no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127C2311yescharacteristic
83#14523238—22no—
84#14523237—41no—
85#14523236—22no—
86#14523235D414yescomplemented
87D1016—22no—
88#14523233—22no—
89#14523233—22no—
90#14523231—22no—
91C2×C127⋊C8—22no—
92C2×C127⋊C8C2311yescharacteristic
93C2×C127⋊C8C2311yescharacteristic
94C4⋊Dic127—21no—
95C4⋊Dic127—21no—
96C4⋊Dic127C2311yescharacteristic
97Dic127⋊C4D411yescharacteristic
98C22⋊Dic127—21no—
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127D411yescharacteristic
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4D411yescharacteristic
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108#41×C127—22no—
109#41×C127—41no—
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127—21no—
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127—21no—
116C22×C4×C127C2311yescharacteristic
117SD16×C127—22no—
118D8×C127—41no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—81no—
128C22×D127C2×D411yescharacteristic
129Dic254—21no—
130Dic254—21no—
131D508—21no—
132D508—21no—
133D508—21no—
134D508—42no—
135D508C2×D412yes—
136C22⋊D127—41no—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—81no—
141C22⋊D127—81no—
142C22⋊D127—81no—
143C2×Dic127—21no—
144C2×Dic127—21no—
145C2×Dic127C2×D411yescharacteristic
146C2×Dic127C2×D411yescharacteristic
147C2×Dic127C2×D411yescharacteristic
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—42no—
151C4×D127—42no—
152C4×D127—81no—
153C127⋊C8—21no—
154C127⋊C8C2×D412yes—
155C23×C127—21no—
156C23×C127—21no—
157C23×C127C2×D411yescharacteristic
158Q8×C127—21no—
159Q8×C127C2×D412yes—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127—21no—
163D4×C127—21no—
164D4×C127—42no—
165D4×C127—81no—
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—21no—
177C22×C127—21no—
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—22no—
187C2×D127—21no—
188C2×D127—21no—
189C2×D127—41no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—84no—
195C4×C127—21no—
196C4×C127—22no—
197C4×C127—41no—
198C4×C127C22×D411yescharacteristic
199C4×C127H(5,𝔽2)11yescharacteristic
200Dic127—21no—
201Dic127—21no—
202Dic127—21no—
203Dic127—41no—
204Dic127—41no—
205C2×C127—41no—
206C2×C127—81no—
207C2×C127—81no—
208C2×C127D4211yescharacteristic
209C2×C127#57511yescharacteristic
210C2×C127#57611yescharacteristic
211D127—41no—
212D127—42no—
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#459—1271no—
225#452—1271no—
226#449—1271no—
227#449D12711yescharacteristic, complemented
228#447—1271no—
229#440—1271no—
230#435—1271no—
231C2×#41—1271no—
232C2×#41—1271no—
233C2×#41C2×D12711yescharacteristic, complemented
234C22×D4—1271no—
235AGL(1,ℤ/8)—2544no—
236#135—2544no—
237C2×Q16—1271no—
238C2×SD16—1271no—
239C2×SD16—1271no—
240C2×SD16—1271no—
241C2×SD16C2×D12711yescharacteristic
242C2×D8—1271no—
243C2×D8—1271no—
244C2×D8—1271no—
245C2×D8C2×D12711yescharacteristic
246C2×M16—1271no—
247C22×C8—1271no—
248#127—1271no—
249#127—2541no—
250#124—1271no—
251#123—2541no—
252C22⋊Q8—1271no—
253#121—1271no—
254#121—1271no—
255#121—2541no—
256#121—2541no—
257#121—2541no—
258#121C2×D12711yescharacteristic
259#120—2541no—
260C4×Q8—1271no—
261C4×D4—1271no—
262C4×D4—1271no—
263C4×D4—2541no—
264C2×C22⋊C4—2541no—
265#107—1271no—
266C4⋊C8—1271no—
267#103—1271no—
268#103C2×D12711yescharacteristic, complemented
269#102—1271no—
270#102—1271no—
271#102—1271no—
272#102C2×D12711yescharacteristic, complemented
273C22⋊C8—1271no—
274C22⋊C8C2×D12711yescharacteristic, complemented
275#97—1271no—
276C24—2541no—
277#41—22no—
278#41—41no—
279#41—5081no—
280#41—5081no—
281#41—2542no—
282#41—2542no—
283C2×Q8—1271no—
284C2×Q8C22×D12711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—1271no—
288C2×D4—1271no—
289C2×D4—1271no—
290C2×D4—5081no—
291C2×D4—2541no—
292C2×D4—2541no—
293C2×D4—2541no—
294C2×D4—2541no—
295C2×D4—2542no—
296C2×D4—2541no—
297C2×D4—2541no—
298C2×D4—2541no—
299C2×D4C22×D12711yescharacteristic
300C2×D4C22×D12711yescharacteristic
301C22×C4—21no—
302C22×C4—1271no—
303C22×C4—2541no—
304C22×C4—2541no—
305C22×C4—2541no—
306C22×C4C22×D12711yescharacteristic
307Q16—2542no—
308SD16—22no—
309SD16—1274no—
310SD16—5081no—
311SD16—2542no—
312D8—41no—
313D8—2542no—
314D8—2542no—
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—1271no—
337C42—2541no—
338C23—21no—
339C23—21no—
340C23—1271no—
341C23—10161no—
342C23—5082no—
343C23—5082no—
344C23—2541no—
345C23—2541no—
346C23D4×D12711yescharacteristic
347Q8—21no—
348Q8—2541no—
349Q8—2541no—
350Q8D4×D12712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—81no—
357D4—1272no—
358D4—10161no—
359D4—10161no—
360D4—10161no—
361D4—5082no—
362D4—5082no—
363D4—5081no—
364D4—5082no—
365D4—5082no—
366D4—2541no—
367D4—2541no—
368D4—2541no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—1271no—
375C2×C4—1271no—
376C2×C4—1271no—
377C2×C4—10161no—
378C2×C4—5082no—
379C2×C4—5082no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4D4×D12711yescharacteristic
385C2×C4C23×D12711yescharacteristic
386C8—21no—
387C8—21no—
388C8—1272no—
389C8—2541no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—10164no—
400C22—5081no—
401C22—5081no—
402C22—5081no—
403C22—5081no—
404C22—5081no—
405C22—2541no—
406C22—2541no—
407C22—2542no—
408C22—2541no—
409C22C2×D4×D12711yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—5081no—
414C4—5081no—
415C4—2541no—
416C4—2541no—
417C4—2541no—
418C4C2×D4×D12711yescharacteristic
419C4#1686125411yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—10161no—
424C2—5081no—
425C2—5082no—
426C2#2255050811yescharacteristic
427C2#2255067311yescharacteristic
428C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3268
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×#16861229C2semidirect
3. #22550545C2semidirect
4. #22550523C2semidirect
5. #22550264C2semidirect
6. #22550121C2semidirect
7. #22550110C2semidirect
8. #22550096C2semidirect
9. #22550070C2semidirect
10. #22550041C2semidirect
11. #22550023C2semidirect
12. #22550008C2semidirect
13. #22549979C2semidirect
14. #22549944C2semidirect
15. #449×C127C2semidirect
21. C2×#14523238C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523236C22semidirect
29. C22×C127⋊C8C22semidirect
31. #16861205C22semidirect
36. #16861183C22semidirect
41. Q8×Dic127C22semidirect
46. #16861142C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
51. #16861129C22semidirect
53. #16861119C22semidirect
54. C2×#41×C127C22semidirect
58. #103×C127C22semidirect
59. #102×C127C22semidirect
60. C22⋊C8×C127C22semidirect
86. #14523235D4semidirect
215. C127#3268semidirect
227. #449D127semidirect
233. C2×#41C2×D127semidirect
268. #103C2×D127semidirect
272. #102C2×D127semidirect
274. 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×#168612292
#225505452
#225505232
#225502642
#225501212
#225501102
#225500962
#225500702
#225500412
#225500232
#225500082
#225499792
#225499442
#449×C1272
#3268127