Finite Group Databasebeta

← #46848125

Subgroups — #46848125

Order 34688.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774708C211yesmaximal, characteristic, complemented
2#31774707C211yesmaximal, characteristic, complemented
3#31774701C211yesmaximal, characteristic, complemented
4#31774700C211yesmaximal, characteristic, complemented
5#31774284C211yesmaximal, characteristic, complemented
6#31774166C211yesmaximal, characteristic, complemented
7#31774165C211yesmaximal, characteristic, complemented
8#31774159C211yesmaximal, characteristic, complemented
9#31774158C211yesmaximal, characteristic, complemented
10#31774076C211yesmaximal, characteristic, complemented
11#31774071C211yesmaximal, characteristic, complemented
12#31774064C211yesmaximal, characteristic, complemented
13#31774016C211yesmaximal, characteristic, complemented
14#31774016C211yesmaximal, characteristic, complemented
15#453×C271C211yesmaximal, characteristic, complemented
16#22576207C2211yescharacteristic, complemented
17#22576206C2211yescharacteristic
18#22576205C2211yescharacteristic, complemented
19#22576193C2211yescharacteristic, complemented
20#22576190—21no—
21#22576190C2211yescharacteristic, complemented
22#22576189—21no—
23#22576188C2211yescharacteristic, complemented
24#22576187—21no—
25#22576186—21no—
26#22576186—21no—
27#22576186C2211yescharacteristic
28#22576186C2211yescharacteristic, complemented
29#22576185—21no—
30#22576185C2211yescharacteristic, complemented
31#22576184—21no—
32#22576184—21no—
33#22576181—21no—
34#22576180—21no—
35#22576180—21no—
36#22576180C2211yescharacteristic
37#22576180C2211yescharacteristic
38#22576179—21no—
39C2×#16864642—21no—
40C2×#16864642C2211yescharacteristic, complemented
41C2×#16864642C2211yescharacteristic, complemented
42SD16×D271—21no—
43C2×#16864639—21no—
44C2×#16864639C2211yescharacteristic, complemented
45C2×#16864639C2211yescharacteristic, complemented
46D8×D271—21no—
47#22576168C2211yescharacteristic, complemented
48#22576168C2211yescharacteristic, complemented
49#22576167C2211yescharacteristic, complemented
50#22576167C2211yescharacteristic, complemented
51#22576148C2211yescharacteristic, complemented
52#22576095C2211yescharacteristic, complemented
53#22576095C2211yescharacteristic, complemented
54#22576088C2211yescharacteristic, complemented
55#22576088C2211yescharacteristic, complemented
56#22576078C2211yescharacteristic, complemented
57#22576077C2211yescharacteristic, complemented
58#22576076C2211yescharacteristic, complemented
59#22576071C2211yescharacteristic, complemented
60H(5,𝔽2)×C271C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C271C2211yescharacteristic
62AGL(1,ℤ/8)×C271C2211yescharacteristic, complemented
63#127×C271C2211yescharacteristic, complemented
64#104×C271C2211yescharacteristic, complemented
65#104×C271C2211yescharacteristic, complemented
66#100×C271C2211yescharacteristic, complemented
67#16864655—21no—
68#16864654—21no—
69#16864654—21no—
70#16864654C2311yescharacteristic
71#16864654C2311yescharacteristic
72#16864653—21no—
73#16864653—21no—
74#16864653—21no—
75#16864653—41no—
76C2×Dic542—21no—
77C2×Dic542C2311yescharacteristic
78Q8×D271—21no—
79C2×D1084—21no—
80C2×D1084C2311yescharacteristic
81C2×C22⋊D271—41no—
82D4×D271—21no—
83D4×D271—21no—
84D4×D271—21no—
85D4×D271—41no—
86Dic1084—21no—
87#16864644—21no—
88#16864644D411yescharacteristic, complemented
89#16864644D411yescharacteristic, complemented
90#16864643—21no—
91#16864643—21no—
92#16864642—21no—
93#16864642—21no—
94#16864642—21no—
95#16864642—21no—
96#16864642—21no—
97#16864642—21no—
98#16864642—21no—
99#16864642D411yescharacteristic, complemented
100#16864642D411yescharacteristic, complemented
101#16864641—21no—
102#16864641D411yescharacteristic, complemented
103#16864641D411yescharacteristic, complemented
104D2168—21no—
105#16864639—21no—
106#16864639—21no—
107#16864639—21no—
108#16864639—21no—
109#16864639—21no—
110#16864639—21no—
111#16864639—21no—
112#16864639D411yescharacteristic, complemented
113#16864639D411yescharacteristic, complemented
114#16864638—21no—
115#16864638—21no—
116#16864638C2311yescharacteristic
117#16864638C2311yescharacteristic
118#16864637—21no—
119#16864637—21no—
120C2×C271⋊C8—21no—
121C2×C271⋊C8—21no—
122C2×C271⋊C8C2311yescharacteristic
123C2×C271⋊C8C2311yescharacteristic
124C8×D271—21no—
125C8×D271—21no—
126(C2×D271)⋊C4—41no—
127#41×C271—21no—
128#41×C271—21no—
129#41×C271C2311yescharacteristic
130#41×C271C2311yescharacteristic
131C2×D4×C271—21no—
132C2×D4×C271—21no—
133C2×D4×C271—41no—
134C2×D4×C271—41no—
135C2×D4×C271C2311yescharacteristic
136C2×D4×C271C2311yescharacteristic
137SD16×C271—21no—
138SD16×C271—21no—
139D8×C271—21no—
140D8×C271—21no—
141M16×C271C2311yescharacteristic
142M16×C271C2311yescharacteristic
143C42×C271C2311yescharacteristic
144C22×D271—21no—
145C22×D271—41no—
146C22×D271—41no—
147Dic542—21no—
148Dic542—21no—
149Dic542—41no—
150Dic542C2×D411yescharacteristic
151Dic542C2×D411yescharacteristic
152D1084—21no—
153D1084—21no—
154D1084—41no—
155D1084C2×D411yescharacteristic
156D1084C2×D411yescharacteristic
157C22⋊D271—21no—
158C22⋊D271—21no—
159C22⋊D271—41no—
160C22⋊D271—41no—
161C22⋊D271—41no—
162C22⋊D271—41no—
163C2×Dic271—21no—
164C2×Dic271—41no—
165C2×Dic271—41no—
166C4×D271—21no—
167C4×D271—21no—
168C4×D271—41no—
169C4×D271—41no—
170C271⋊C8—21no—
171C271⋊C8—21no—
172C271⋊C8C2×D411yescharacteristic
173C271⋊C8C2×D411yescharacteristic
174C271⋊C8C2×D411yescharacteristic
175C271⋊C8C2×D411yescharacteristic
176C23×C271—21no—
177C23×C271—21no—
178C23×C271—41no—
179Q8×C271C2×D411yescharacteristic
180Q8×C271C2×D411yescharacteristic
181D4×C271—21no—
182D4×C271—21no—
183D4×C271—21no—
184D4×C271—21no—
185D4×C271—21no—
186D4×C271—21no—
187D4×C271—41no—
188D4×C271—41no—
189D4×C271—82no—
190D4×C271C2×D411yescharacteristic
191D4×C271C2×D411yescharacteristic
192C2×C4×C271—21no—
193C2×C4×C271—21no—
194C2×C4×C271—21no—
195C2×C4×C271—41no—
196C2×C4×C271C2411yescharacteristic
197C8×C271—21no—
198C8×C271—21no—
199C22×C271—21no—
200C22×C271—21no—
201C22×C271—41no—
202C22×C271—41no—
203C22×C271—81no—
204C22×C271—81no—
205C22×C271—161no—
206C22×C271H(5,𝔽2)11yescharacteristic
207C2×D271—21no—
208C2×D271—21no—
209C2×D271—41no—
210C2×D271—81no—
211C2×D271—161no—
212C2×D271—161no—
213C4×C271—21no—
214C4×C271—21no—
215C4×C271—41no—
216C4×C271C22×D411yescharacteristic
217C4×C271C22×D411yescharacteristic
218Dic271—21no—
219Dic271—21no—
220Dic271—41no—
221C2×C271—21no—
222C2×C271—41no—
223C2×C271—41no—
224C2×C271—81no—
225C2×C271—81no—
226C2×C271D4211yescharacteristic
227D271—41no—
228D271—41no—
229D271—81no—
230C271#328811yescharacteristic, complemented, Sylow
231#3288—2711nomaximal, Sylow
232#577—2711no—
233#577—2711no—
234#576—2711no—
235#576—2711no—
236#496—2711no—
237#472—2711no—
238#472—2711no—
239#471—2711no—
240#471—2711no—
241#455—2711no—
242#454—2711no—
243#453—2711no—
244#453D27111yescharacteristic, complemented
245#444—2711no—
246#444—2711no—
2472−1+4—2711no—
248H(5,𝔽2)—2711no—
249H(5,𝔽2)—2711no—
250H(5,𝔽2)C2×D27111yescharacteristic, complemented
251#137—2711no—
252#137—2711no—
253#137—5421no—
254#137—5421no—
255AGL(1,ℤ/8)—2711no—
256AGL(1,ℤ/8)—2711no—
257AGL(1,ℤ/8)—2711no—
258AGL(1,ℤ/8)—2711no—
259AGL(1,ℤ/8)—5421no—
260AGL(1,ℤ/8)—5421no—
261AGL(1,ℤ/8)—5421no—
262AGL(1,ℤ/8)—5421no—
263AGL(1,ℤ/8)—5421no—
264AGL(1,ℤ/8)—5421no—
265AGL(1,ℤ/8)C2×D27111yescharacteristic
266AGL(1,ℤ/8)C2×D27111yescharacteristic, complemented
267#135—2711no—
268#135—2711no—
269#135—5421no—
270#135—5421no—
271#135—5421no—
272#135—5421no—
273C2×SD16—2711no—
274C2×SD16—2711no—
275C2×SD16—5421no—
276C2×SD16—5421no—
277C2×D8—2711no—
278C2×D8—2711no—
279C2×D8—5421no—
280C2×D8—5421no—
281#131—2711no—
282#131—2711no—
283#131—2711no—
284#131—2711no—
285#127C2×D27111yescharacteristic, complemented
286#124—2711no—
287#108—2711no—
288#108—2711no—
289#104—2711no—
290#104—2711no—
291#104C2×D27111yescharacteristic, complemented
292#104C2×D27111yescharacteristic, complemented
293#101—2711no—
294#100—2711no—
295#100—2711no—
296#100C2×D27111yescharacteristic, complemented
297#97—2711no—
298#41—21no—
299#41—21no—
300#41—2711no—
301#41—2711no—
302#41—5421no—
303#41—5421no—
304#41—5421no—
305#41—5421no—
306#41—5421no—
307#41—5421no—
308#41—10841no—
309#41C22×D27111yescharacteristic
310#41C22×D27111yescharacteristic
311C2×Q8—2711no—
312C2×Q8—5421no—
313C2×Q8—5421no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—2711no—
319C2×D4—5421no—
320C2×D4—5421no—
321C2×D4—5421no—
322C2×D4—5421no—
323C2×D4—10841no—
324C2×D4—10841no—
325C2×D4C22×D27111yescharacteristic
326C2×D4C22×D27111yescharacteristic
327Q16—2711no—
328Q16—2711no—
329Q16—5421no—
330Q16—5421no—
331SD16—21no—
332SD16—21no—
333SD16—2711no—
334SD16—2711no—
335SD16—2711no—
336SD16—2711no—
337SD16—5421no—
338SD16—5421no—
339SD16—5421no—
340SD16—5421no—
341SD16—5421no—
342SD16—5421no—
343SD16—5421no—
344SD16—5421no—
345SD16—5421no—
346SD16—5421no—
347D8—21no—
348D8—21no—
349D8—2711no—
350D8—2711no—
351D8—5421no—
352D8—5421no—
353D8—5421no—
354D8—5421no—
355D8—5421no—
356D8—5421no—
357D8—5421no—
358D8—5421no—
359M16—2711no—
360M16—2711no—
361M16—5421no—
362M16—5421no—
363M16—5421no—
364M16—5421no—
365M16C22×D27111yescharacteristic
366M16C22×D27111yescharacteristic
367C2×C8—2711no—
368C2×C8—2711no—
369C2×C8—5421no—
370C2×C8—5421no—
371C2×C8—5421no—
372C2×C8—5421no—
373C22⋊C4—10841no—
374C42C22×D27111yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—5421no—
379C23—10841no—
380C23—10841no—
381Q8—2711no—
382Q8—2711no—
383Q8—5421no—
384Q8—5421no—
385Q8—10841no—
386Q8D4×D27111yescharacteristic
387Q8D4×D27111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—2711no—
398D4—2711no—
399D4—5421no—
400D4—5421no—
401D4—5421no—
402D4—5421no—
403D4—10841no—
404D4—10841no—
405D4—10841no—
406D4—10841no—
407D4—10841no—
408D4D4×D27111yescharacteristic
409D4D4×D27111yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—5421no—
415C2×C4—5421no—
416C2×C4—5421no—
417C2×C4—10841no—
418C2×C4—10841no—
419C2×C4—10841no—
420C2×C4—10841no—
421C2×C4C23×D27111yescharacteristic
422C8—21no—
423C8—21no—
424C8—2711no—
425C8—2711no—
426C8—2711no—
427C8—2711no—
428C8—5421no—
429C8—5421no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—5421no—
438C22—5421no—
439C22—10841no—
440C22—21681no—
441C22—43361no—
442C22—43361no—
443C22#2257620511yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—5421no—
448C4—5421no—
449C4—10841no—
450C4C2×D4×D27111yescharacteristic
451C4C2×D4×D27111yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—10841no—
458C2—10841no—
459C2—21681no—
460C2#3177452911yescharacteristic
subgroupG/H
centerC2#31774529
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#453×C271C2
socleC2×C271D42
hypercenter#453D271
hypocenterC271#3288
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #31774708C2semidirect
2. #31774707C2semidirect
3. #31774701C2semidirect
4. #31774700C2semidirect
5. #31774284C2semidirect
6. #31774166C2semidirect
7. #31774165C2semidirect
8. #31774159C2semidirect
9. #31774158C2semidirect
10. #31774076C2semidirect
11. #31774071C2semidirect
12. #31774064C2semidirect
13. #31774016C2semidirect
14. #31774016C2semidirect
15. #453×C271C2semidirect
16. #22576207C22semidirect
18. #22576205C22semidirect
19. #22576193C22semidirect
21. #22576190C22semidirect
23. #22576188C22semidirect
28. #22576186C22semidirect
30. #22576185C22semidirect
40. C2×#16864642C22semidirect
41. C2×#16864642C22semidirect
44. C2×#16864639C22semidirect
45. C2×#16864639C22semidirect
47. #22576168C22semidirect
48. #22576168C22semidirect
49. #22576167C22semidirect
50. #22576167C22semidirect
51. #22576148C22semidirect
52. #22576095C22semidirect
53. #22576095C22semidirect
54. #22576088C22semidirect
55. #22576088C22semidirect
56. #22576078C22semidirect
57. #22576077C22semidirect
58. #22576076C22semidirect
59. #22576071C22semidirect
60. H(5,𝔽2)×C271C22semidirect
62. AGL(1,ℤ/8)×C271C22semidirect
63. #127×C271C22semidirect
64. #104×C271C22semidirect
65. #104×C271C22semidirect
66. #100×C271C22semidirect
88. #16864644D4semidirect
89. #16864644D4semidirect
99. #16864642D4semidirect
100. #16864642D4semidirect
102. #16864641D4semidirect
103. #16864641D4semidirect
112. #16864639D4semidirect
113. #16864639D4semidirect
230. C271#3288semidirect
244. #453D271semidirect
250. H(5,𝔽2)C2×D271semidirect
266. AGL(1,ℤ/8)C2×D271semidirect
285. #127C2×D271semidirect
291. #104C2×D271semidirect
292. #104C2×D271semidirect
296. #100C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747082
#317747072
#317747012
#317747002
#317742842
#317741662
#317741652
#317741592
#317741582
#317740762
#317740712
#317740642
#317740162
#453×C2712
#3288271