Finite Group Databasebeta

← #30648231

Subgroups — #30648231

Order 16256.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2C2×D8×D127C211yesmaximal, characteristic, complemented
3#22550444C211yesmaximal, characteristic, complemented
4#22550348C211yesmaximal, characteristic, complemented
5#22550094C211yesmaximal, characteristic, complemented
6#22550093C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550072C211yesmaximal, characteristic, complemented
9#22550071C211yesmaximal, characteristic, complemented
10#22550070C211yesmaximal, characteristic, complemented
11#22550069C211yesmaximal, characteristic, complemented
12#22549901C211yesmaximal, characteristic, complemented
13#22549879C211yesmaximal, characteristic, complemented
14#22549871C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861236—24no—
20C2×#14523235C2211yescharacteristic, complemented
21C2×D1016C2211yescharacteristic
22C2×D1016C2211yescharacteristic, complemented
23C2×#14523233C2211yescharacteristic
24C2×#14523233C2211yescharacteristic
25C2×#14523233C2211yescharacteristic, complemented
26D8×D127—24no—
27C2×#14523231C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861206C2211yescharacteristic, complemented
30#16861205C2211yescharacteristic, complemented
31#16861198C2211yescharacteristic, complemented
32#16861182—21no—
33#16861182—21no—
34#16861179—21no—
35#16861179—21no—
36C4×D508C2211yescharacteristic, complemented
37D4×Dic127C2211yescharacteristic, complemented
38#16861171C2211yescharacteristic, complemented
39C4×C22⋊D127—21no—
40(C22⋊D127)⋊C4—21no—
41#16861167C2211yescharacteristic
42#16861166C2211yescharacteristic
43C2×(C2×D127)⋊C4—21no—
44C22⋊C4×D127—21no—
45C4⋊(C127⋊C8)C2211yescharacteristic, complemented
46Dic127⋊C8C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861133C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic, complemented
52#16861129C2211yescharacteristic, complemented
53#16861129C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×D8×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic, complemented
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63#14523249—22no—
64#14523249—22no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508—21no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75D4×D127—22no—
76D4×D127—22no—
77D4×D127—22no—
78D4×D127D414yescomplemented
79C22×Dic127—21no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127C2311yescharacteristic
84#14523235—22no—
85D1016—22no—
86D1016—22no—
87#14523233—22no—
88#14523233—22no—
89#14523233—22no—
90#14523231—22no—
91C2×C127⋊C8C2311yescharacteristic
92C2×C127⋊C8C2311yescharacteristic
93C8×D127—22no—
94C4⋊Dic127C2311yescharacteristic
95Dic127⋊C4D411yescharacteristic
96Dic127⋊C4D411yescharacteristic
97C22⋊Dic127—21no—
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)⋊C4D411yescharacteristic
104(C2×D127)⋊C4D411yescharacteristic
105C4×Dic127—21no—
106C4×Dic127C2311yescharacteristic
107C2×Q8×C127C2311yescharacteristic
108C2×D4×C127C2311yescharacteristic
109C2×D4×C127C2311yescharacteristic
110C22×C4×C127—21no—
111SD16×C127—22no—
112D8×C127—22no—
113C2×C8×C127C2311yescharacteristic
114C2×C8×C127C2311yescharacteristic
115C4⋊C4×C127C2311yescharacteristic
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C42×C127C2311yescharacteristic
120C22×D127—21no—
121C22×D127—24no—
122C22×D127—21no—
123C22×D127—21no—
124C22×D127—42no—
125C22×D127—81no—
126C22×D127—81no—
127C22×D127—81no—
128C22×D127—81no—
129C22×D127C2×D411yescharacteristic
130D508—21no—
131D508—21no—
132D508—21no—
133D508—21no—
134D508—21no—
135D508—42no—
136D508—81no—
137D508—81no—
138D508C2×D412yes—
139C22⋊D127—24no—
140C22⋊D127—42no—
141C22⋊D127—81no—
142C22⋊D127—81no—
143C2×Dic127—21no—
144C2×Dic127—81no—
145C2×Dic127C2×D411yescharacteristic
146C2×Dic127C2×D411yescharacteristic
147C2×Dic127C2×D411yescharacteristic
148C4×D127—42no—
149C4×D127—81no—
150C4×D127—81no—
151C4×D127C2×D412yes—
152C4×D127#4112yes—
153C127⋊C8—21no—
154C127⋊C8—21no—
155C23×C127—21no—
156C23×C127—21no—
157Q8×C127—21no—
158Q8×C127—21no—
159D4×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127C2×D412yes—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—81no—
166C2×C4×C127C2×D411yescharacteristic
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2411yescharacteristic
169C8×C127—21no—
170C8×C127—21no—
171C22×C127—22no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127C22×D411yescharacteristic
178C2×D127—21no—
179C2×D127—21no—
180C2×D127—22no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C2×D127—41no—
189C2×D127—84no—
190C2×D127—84no—
191C2×D127#12112yes—
192C4×C127—21no—
193C4×C127—21no—
194C4×C127—41no—
195C4×C127—41no—
196C4×C127C22×D411yescharacteristic
197C4×C127C2×#4111yescharacteristic
198Dic127—21no—
199Dic127—21no—
200Dic127—41no—
201Dic127#12112yes—
202C2×C127—42no—
203C2×C127—81no—
204C2×C127C2×#12111yescharacteristic
205C2×C127#57611yescharacteristic
206C2×C127#57611yescharacteristic
207D127—22no—
208D127—42no—
209D127—81no—
210D127—81no—
211#3041—1271nomaximal, Sylow
212C127#304111yescharacteristic, complemented, Sylow
213C2×AGL(1,ℤ/8)—1271no—
214C22×D8—1271no—
215#535—1271no—
216#518—1271no—
217#463—1271no—
218#463—1271no—
219#463—1271no—
220#463D12711yescharacteristic, complemented
221#459—1271no—
222#459—1271no—
223#459—1271no—
224#459—1271no—
225#424—1271no—
226#418—1271no—
227#416—1271no—
228C2×#41—1271no—
229C22×D4—1271no—
230C22×D4—1271no—
231AGL(1,ℤ/8)—2544no—
232C2×SD16—1271no—
233C2×SD16C2×D12711yescharacteristic, complemented
234C2×D8—1271no—
235C2×D8—1271no—
236C2×D8—1271no—
237C2×D8—1271no—
238C2×D8—1271no—
239C2×D8—2544no—
240C2×D8C2×D12711yescharacteristic
241C2×M16—1271no—
242C22×C8—1271no—
243#127—1271no—
244#127—1271no—
245#124—1271no—
246#124C2×D12711yescharacteristic, complemented
247#121—2541no—
248#121—2541no—
249#120—2541no—
250#120—2541no—
251C4×D4—1271no—
252C4×D4—1271no—
253C4×D4—1271no—
254C4×D4—2541no—
255C4×D4—2541no—
256C4×D4C2×D12711yescharacteristic, complemented
257#117—1271no—
258#117—1271no—
259C2×C22⋊C4—2541no—
260C2×C22⋊C4—2541no—
261C4⋊C8—1271no—
262C4⋊C8—1271no—
263C4⋊C8—1271no—
264C4⋊C8C2×D12711yescharacteristic, complemented
265#103—1271no—
266#103C2×D12711yescharacteristic, complemented
267#102—1271no—
268#102—1271no—
269#102—1271no—
270#102—1271no—
271#102—1271no—
272#102C2×D12711yescharacteristic, complemented
273C24—2541no—
274C24—2541no—
275#41—2542no—
276#41—2542no—
277C2×Q8C22×D12711yescharacteristic
278C2×D4—1274no—
279C2×D4—1271no—
280C2×D4—1271no—
281C2×D4—1271no—
282C2×D4—2541no—
283C2×D4—2542no—
284C2×D4—2542no—
285C2×D4—2542no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4—2541no—
291C2×D4—2541no—
292C2×D4C22×D12711yescharacteristic
293C2×D4C22×D12711yescharacteristic
294C22×C4—21no—
295C22×C4—1271no—
296C22×C4—2541no—
297C22×C4—2541no—
298C22×C4—2541no—
299C22×C4—2541no—
300SD16—22no—
301SD16—2542no—
302D8—22no—
303D8—2542no—
304D8—2542no—
305D8—2542no—
306D8—2542no—
307D8—2542no—
308M16—2542no—
309C2×C8—1271no—
310C2×C8—1271no—
311C2×C8—2542no—
312C2×C8C22×D12711yescharacteristic
313C2×C8C22×D12711yescharacteristic
314C4⋊C4—1271no—
315C4⋊C4—1271no—
316C4⋊C4—1271no—
317C4⋊C4C22×D12711yescharacteristic
318C22⋊C4—21no—
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—
329C42—1271no—
330C42—2541no—
331C42C22×D12711yescharacteristic
332C23—21no—
333C23—21no—
334C23—1271no—
335C23—10161no—
336C23—10161no—
337C23—10161no—
338C23—10161no—
339C23—5082no—
340C23—2544no—
341C23—2541no—
342C23—2541no—
343C23—2541no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—1272no—
350D4—10161no—
351D4—10161no—
352D4—10161no—
353D4—10161no—
354D4—5082no—
355D4—5082no—
356D4—2544no—
357D4—2541no—
358D4—2541no—
359D4—2541no—
360D4—2541no—
361D4—2541no—
362D4D4×D12712yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—81no—
366C2×C4—1271no—
367C2×C4—1272no—
368C2×C4—1272no—
369C2×C4—1271no—
370C2×C4—1271no—
371C2×C4—10161no—
372C2×C4—10161no—
373C2×C4—10161no—
374C2×C4—5082no—
375C2×C4—2541no—
376C2×C4D4×D12711yescharacteristic
377C2×C4D4×D12711yescharacteristic
378C2×C4C23×D12711yescharacteristic
379C8—21no—
380C8—21no—
381C8—2541no—
382C8—2541no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—1272no—
390C22—10164no—
391C22—10164no—
392C22—5081no—
393C22—5081no—
394C22—5081no—
395C22—5081no—
396C22—5081no—
397C22—5081no—
398C22—5081no—
399C22—5081no—
400C22—2542no—
401C22—2541no—
402C22—2541no—
403C22C2×D4×D12711yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—1272no—
409C4—5081no—
410C4—2541no—
411C4—2541no—
412C4C2×D4×D12711yescharacteristic
413C4#41×D12711yescharacteristic
414C2—42no—
415C2—81no—
416C2—10161no—
417C2—10161no—
418C2—5082no—
419C2—2542no—
420C2#121×D12711yescharacteristic
421C2#2255067811yescharacteristic
422C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
hypercenter#463D127
hypocenterC127#3041
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×D8×D127C2semidirect
3. #22550444C2semidirect
4. #22550348C2semidirect
5. #22550094C2semidirect
6. #22550093C2semidirect
7. #22550088C2semidirect
8. #22550072C2semidirect
9. #22550071C2semidirect
10. #22550070C2semidirect
11. #22550069C2semidirect
12. #22549901C2semidirect
13. #22549879C2semidirect
14. #22549871C2semidirect
15. #463×C127C2semidirect
20. C2×#14523235C22semidirect
22. C2×D1016C22semidirect
25. C2×#14523233C22semidirect
29. #16861206C22semidirect
30. #16861205C22semidirect
31. #16861198C22semidirect
36. C4×D508C22semidirect
37. D4×Dic127C22semidirect
38. #16861171C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. Dic127⋊C8C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861133C22semidirect
49. #16861131C22semidirect
50. #16861131C22semidirect
51. #16861130C22semidirect
52. #16861129C22semidirect
53. #16861129C22semidirect
54. C2×SD16×C127C22semidirect
56. #124×C127C22semidirect
57. C4×D4×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
78. D4×D127D4semidirect
212. C127#3041semidirect
220. #463D127semidirect
233. C2×SD16C2×D127semidirect
246. #124C2×D127semidirect
256. C4×D4C2×D127semidirect
264. C4⋊C8C2×D127semidirect
266. #103C2×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×#168612362
C2×D8×D1272
#225504442
#225503482
#225500942
#225500932
#225500882
#225500722
#225500712
#225500702
#225500692
#225499012
#225498792
#225498712
#463×C1272
#3041127