Finite Group Databasebeta

← #44562126

Subgroups — #44562126

Order 32128.

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#30632498C211yesmaximal, characteristic, complemented
2#30632497C211yesmaximal, characteristic, complemented
3#30632491C211yesmaximal, characteristic, complemented
4#30632490C211yesmaximal, characteristic, complemented
5#30632074C211yesmaximal, characteristic, complemented
6#30631956C211yesmaximal, characteristic, complemented
7#30631955C211yesmaximal, characteristic, complemented
8#30631949C211yesmaximal, characteristic, complemented
9#30631948C211yesmaximal, characteristic, complemented
10#30631866C211yesmaximal, characteristic, complemented
11#30631861C211yesmaximal, characteristic, complemented
12#30631854C211yesmaximal, characteristic, complemented
13#30631806C211yesmaximal, characteristic, complemented
14#30631806C211yesmaximal, characteristic, complemented
15#453×C251C211yesmaximal, characteristic, complemented
16#22548762C2211yescharacteristic, complemented
17#22548761C2211yescharacteristic
18#22548760C2211yescharacteristic, complemented
19#22548748C2211yescharacteristic, complemented
20#22548745—21no—
21#22548745C2211yescharacteristic, complemented
22#22548744—21no—
23#22548743C2211yescharacteristic, complemented
24#22548742—21no—
25#22548741—21no—
26#22548741—21no—
27#22548741C2211yescharacteristic
28#22548741C2211yescharacteristic, complemented
29#22548740—21no—
30#22548740C2211yescharacteristic, complemented
31#22548739—21no—
32#22548739—21no—
33#22548736—21no—
34#22548735—21no—
35#22548735—21no—
36#22548735C2211yescharacteristic
37#22548735C2211yescharacteristic
38#22548734—21no—
39C2×#16860798—21no—
40C2×#16860798C2211yescharacteristic, complemented
41C2×#16860798C2211yescharacteristic, complemented
42SD16×D251—21no—
43C2×#16860795—21no—
44C2×#16860795C2211yescharacteristic, complemented
45C2×#16860795C2211yescharacteristic, complemented
46D8×D251—21no—
47#22548723C2211yescharacteristic, complemented
48#22548723C2211yescharacteristic, complemented
49#22548722C2211yescharacteristic, complemented
50#22548722C2211yescharacteristic, complemented
51#22548703C2211yescharacteristic, complemented
52#22548650C2211yescharacteristic, complemented
53#22548650C2211yescharacteristic, complemented
54#22548643C2211yescharacteristic, complemented
55#22548643C2211yescharacteristic, complemented
56#22548633C2211yescharacteristic, complemented
57#22548632C2211yescharacteristic, complemented
58#22548631C2211yescharacteristic, complemented
59#22548626C2211yescharacteristic, complemented
60H(5,𝔽2)×C251C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C251C2211yescharacteristic
62AGL(1,ℤ/8)×C251C2211yescharacteristic, complemented
63#127×C251C2211yescharacteristic, complemented
64#104×C251C2211yescharacteristic, complemented
65#104×C251C2211yescharacteristic, complemented
66#100×C251C2211yescharacteristic, complemented
67#16860811—21no—
68#16860810—21no—
69#16860810—21no—
70#16860810C2311yescharacteristic
71#16860810C2311yescharacteristic
72#16860809—21no—
73#16860809—21no—
74#16860809—21no—
75#16860809—41no—
76C2×Dic502—21no—
77C2×Dic502C2311yescharacteristic
78Q8×D251—21no—
79C2×D1004—21no—
80C2×D1004C2311yescharacteristic
81C2×C22⋊D251—41no—
82D4×D251—21no—
83D4×D251—21no—
84D4×D251—21no—
85D4×D251—41no—
86Dic1004—21no—
87#16860800—21no—
88#16860800D411yescharacteristic, complemented
89#16860800D411yescharacteristic, complemented
90#16860799—21no—
91#16860799—21no—
92#16860798—21no—
93#16860798—21no—
94#16860798—21no—
95#16860798—21no—
96#16860798—21no—
97#16860798—21no—
98#16860798—21no—
99#16860798D411yescharacteristic, complemented
100#16860798D411yescharacteristic, complemented
101#16860797—21no—
102#16860797D411yescharacteristic, complemented
103#16860797D411yescharacteristic, complemented
104D2008—21no—
105#16860795—21no—
106#16860795—21no—
107#16860795—21no—
108#16860795—21no—
109#16860795—21no—
110#16860795—21no—
111#16860795—21no—
112#16860795D411yescharacteristic, complemented
113#16860795D411yescharacteristic, complemented
114#16860794—21no—
115#16860794—21no—
116#16860794C2311yescharacteristic
117#16860794C2311yescharacteristic
118#16860793—21no—
119#16860793—21no—
120C2×C251⋊C8—21no—
121C2×C251⋊C8—21no—
122C2×C251⋊C8C2311yescharacteristic
123C2×C251⋊C8C2311yescharacteristic
124C8×D251—21no—
125C8×D251—21no—
126(C2×D251)⋊C4—41no—
127#41×C251—21no—
128#41×C251—21no—
129#41×C251C2311yescharacteristic
130#41×C251C2311yescharacteristic
131C2×D4×C251—21no—
132C2×D4×C251—21no—
133C2×D4×C251—41no—
134C2×D4×C251—41no—
135C2×D4×C251C2311yescharacteristic
136C2×D4×C251C2311yescharacteristic
137SD16×C251—21no—
138SD16×C251—21no—
139D8×C251—21no—
140D8×C251—21no—
141M16×C251C2311yescharacteristic
142M16×C251C2311yescharacteristic
143C42×C251C2311yescharacteristic
144C22×D251—21no—
145C22×D251—41no—
146C22×D251—41no—
147Dic502—21no—
148Dic502—21no—
149Dic502—41no—
150Dic502C2×D411yescharacteristic
151Dic502C2×D411yescharacteristic
152D1004—21no—
153D1004—21no—
154D1004—41no—
155D1004C2×D411yescharacteristic
156D1004C2×D411yescharacteristic
157C22⋊D251—21no—
158C22⋊D251—21no—
159C22⋊D251—41no—
160C22⋊D251—41no—
161C22⋊D251—41no—
162C22⋊D251—41no—
163C2×Dic251—21no—
164C2×Dic251—41no—
165C2×Dic251—41no—
166C4×D251—21no—
167C4×D251—21no—
168C4×D251—41no—
169C4×D251—41no—
170C251⋊C8—21no—
171C251⋊C8—21no—
172C251⋊C8C2×D411yescharacteristic
173C251⋊C8C2×D411yescharacteristic
174C251⋊C8C2×D411yescharacteristic
175C251⋊C8C2×D411yescharacteristic
176C23×C251—21no—
177C23×C251—21no—
178C23×C251—41no—
179Q8×C251C2×D411yescharacteristic
180Q8×C251C2×D411yescharacteristic
181D4×C251—21no—
182D4×C251—21no—
183D4×C251—21no—
184D4×C251—21no—
185D4×C251—21no—
186D4×C251—21no—
187D4×C251—41no—
188D4×C251—41no—
189D4×C251—82no—
190D4×C251C2×D411yescharacteristic
191D4×C251C2×D411yescharacteristic
192C2×C4×C251—21no—
193C2×C4×C251—21no—
194C2×C4×C251—21no—
195C2×C4×C251—41no—
196C2×C4×C251C2411yescharacteristic
197C8×C251—21no—
198C8×C251—21no—
199C22×C251—21no—
200C22×C251—21no—
201C22×C251—41no—
202C22×C251—41no—
203C22×C251—81no—
204C22×C251—81no—
205C22×C251—161no—
206C22×C251H(5,𝔽2)11yescharacteristic
207C2×D251—21no—
208C2×D251—21no—
209C2×D251—41no—
210C2×D251—81no—
211C2×D251—161no—
212C2×D251—161no—
213C4×C251—21no—
214C4×C251—21no—
215C4×C251—41no—
216C4×C251C22×D411yescharacteristic
217C4×C251C22×D411yescharacteristic
218Dic251—21no—
219Dic251—21no—
220Dic251—41no—
221C2×C251—21no—
222C2×C251—41no—
223C2×C251—41no—
224C2×C251—81no—
225C2×C251—81no—
226C2×C251D4211yescharacteristic
227D251—41no—
228D251—41no—
229D251—81no—
230C251#328811yescharacteristic, complemented, Sylow
231#3288—2511nomaximal, Sylow
232#577—2511no—
233#577—2511no—
234#576—2511no—
235#576—2511no—
236#496—2511no—
237#472—2511no—
238#472—2511no—
239#471—2511no—
240#471—2511no—
241#455—2511no—
242#454—2511no—
243#453—2511no—
244#453D25111yescharacteristic, complemented
245#444—2511no—
246#444—2511no—
2472−1+4—2511no—
248H(5,𝔽2)—2511no—
249H(5,𝔽2)—2511no—
250H(5,𝔽2)C2×D25111yescharacteristic, complemented
251#137—5021no—
252#137—5021no—
253#137—2511no—
254#137—2511no—
255AGL(1,ℤ/8)—5021no—
256AGL(1,ℤ/8)—5021no—
257AGL(1,ℤ/8)—5021no—
258AGL(1,ℤ/8)—5021no—
259AGL(1,ℤ/8)—5021no—
260AGL(1,ℤ/8)—5021no—
261AGL(1,ℤ/8)—2511no—
262AGL(1,ℤ/8)—2511no—
263AGL(1,ℤ/8)—2511no—
264AGL(1,ℤ/8)—2511no—
265AGL(1,ℤ/8)C2×D25111yescharacteristic
266AGL(1,ℤ/8)C2×D25111yescharacteristic, complemented
267#135—5021no—
268#135—5021no—
269#135—5021no—
270#135—5021no—
271#135—2511no—
272#135—2511no—
273C2×SD16—5021no—
274C2×SD16—5021no—
275C2×SD16—2511no—
276C2×SD16—2511no—
277C2×D8—5021no—
278C2×D8—5021no—
279C2×D8—2511no—
280C2×D8—2511no—
281#131—2511no—
282#131—2511no—
283#131—2511no—
284#131—2511no—
285#127C2×D25111yescharacteristic, complemented
286#124—2511no—
287#108—2511no—
288#108—2511no—
289#104—2511no—
290#104—2511no—
291#104C2×D25111yescharacteristic, complemented
292#104C2×D25111yescharacteristic, complemented
293#101—2511no—
294#100—2511no—
295#100—2511no—
296#100C2×D25111yescharacteristic, complemented
297#97—2511no—
298#41—21no—
299#41—21no—
300#41—10041no—
301#41—5021no—
302#41—5021no—
303#41—5021no—
304#41—5021no—
305#41—5021no—
306#41—5021no—
307#41—2511no—
308#41—2511no—
309#41C22×D25111yescharacteristic
310#41C22×D25111yescharacteristic
311C2×Q8—5021no—
312C2×Q8—5021no—
313C2×Q8—2511no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—10041no—
319C2×D4—10041no—
320C2×D4—5021no—
321C2×D4—5021no—
322C2×D4—5021no—
323C2×D4—5021no—
324C2×D4—2511no—
325C2×D4C22×D25111yescharacteristic
326C2×D4C22×D25111yescharacteristic
327Q16—5021no—
328Q16—5021no—
329Q16—2511no—
330Q16—2511no—
331SD16—21no—
332SD16—21no—
333SD16—5021no—
334SD16—5021no—
335SD16—5021no—
336SD16—5021no—
337SD16—5021no—
338SD16—5021no—
339SD16—5021no—
340SD16—5021no—
341SD16—5021no—
342SD16—5021no—
343SD16—2511no—
344SD16—2511no—
345SD16—2511no—
346SD16—2511no—
347D8—21no—
348D8—21no—
349D8—5021no—
350D8—5021no—
351D8—5021no—
352D8—5021no—
353D8—5021no—
354D8—5021no—
355D8—5021no—
356D8—5021no—
357D8—2511no—
358D8—2511no—
359M16—5021no—
360M16—5021no—
361M16—5021no—
362M16—5021no—
363M16—2511no—
364M16—2511no—
365M16C22×D25111yescharacteristic
366M16C22×D25111yescharacteristic
367C2×C8—5021no—
368C2×C8—5021no—
369C2×C8—5021no—
370C2×C8—5021no—
371C2×C8—2511no—
372C2×C8—2511no—
373C22⋊C4—10041no—
374C42C22×D25111yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—10041no—
379C23—10041no—
380C23—5021no—
381Q8—10041no—
382Q8—5021no—
383Q8—5021no—
384Q8—2511no—
385Q8—2511no—
386Q8D4×D25111yescharacteristic
387Q8D4×D25111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—10041no—
398D4—10041no—
399D4—10041no—
400D4—10041no—
401D4—10041no—
402D4—5021no—
403D4—5021no—
404D4—5021no—
405D4—5021no—
406D4—2511no—
407D4—2511no—
408D4D4×D25111yescharacteristic
409D4D4×D25111yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—10041no—
415C2×C4—10041no—
416C2×C4—10041no—
417C2×C4—10041no—
418C2×C4—5021no—
419C2×C4—5021no—
420C2×C4—5021no—
421C2×C4C23×D25111yescharacteristic
422C8—21no—
423C8—21no—
424C8—5021no—
425C8—5021no—
426C8—2511no—
427C8—2511no—
428C8—2511no—
429C8—2511no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—40161no—
438C22—40161no—
439C22—20081no—
440C22—10041no—
441C22—5021no—
442C22—5021no—
443C22#2254876011yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—10041no—
448C4—5021no—
449C4—5021no—
450C4C2×D4×D25111yescharacteristic
451C4C2×D4×D25111yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—20081no—
458C2—10041no—
459C2—10041no—
460C2#3063231911yescharacteristic
subgroupG/H
centerC2#30632319
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#453×C251C2
socleC2×C251D42
hypercenter#453D251
hypocenterC251#3288
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #30632498C2semidirect
2. #30632497C2semidirect
3. #30632491C2semidirect
4. #30632490C2semidirect
5. #30632074C2semidirect
6. #30631956C2semidirect
7. #30631955C2semidirect
8. #30631949C2semidirect
9. #30631948C2semidirect
10. #30631866C2semidirect
11. #30631861C2semidirect
12. #30631854C2semidirect
13. #30631806C2semidirect
14. #30631806C2semidirect
15. #453×C251C2semidirect
16. #22548762C22semidirect
18. #22548760C22semidirect
19. #22548748C22semidirect
21. #22548745C22semidirect
23. #22548743C22semidirect
28. #22548741C22semidirect
30. #22548740C22semidirect
40. C2×#16860798C22semidirect
41. C2×#16860798C22semidirect
44. C2×#16860795C22semidirect
45. C2×#16860795C22semidirect
47. #22548723C22semidirect
48. #22548723C22semidirect
49. #22548722C22semidirect
50. #22548722C22semidirect
51. #22548703C22semidirect
52. #22548650C22semidirect
53. #22548650C22semidirect
54. #22548643C22semidirect
55. #22548643C22semidirect
56. #22548633C22semidirect
57. #22548632C22semidirect
58. #22548631C22semidirect
59. #22548626C22semidirect
60. H(5,𝔽2)×C251C22semidirect
62. AGL(1,ℤ/8)×C251C22semidirect
63. #127×C251C22semidirect
64. #104×C251C22semidirect
65. #104×C251C22semidirect
66. #100×C251C22semidirect
88. #16860800D4semidirect
89. #16860800D4semidirect
99. #16860798D4semidirect
100. #16860798D4semidirect
102. #16860797D4semidirect
103. #16860797D4semidirect
112. #16860795D4semidirect
113. #16860795D4semidirect
230. C251#3288semidirect
244. #453D251semidirect
250. H(5,𝔽2)C2×D251semidirect
266. AGL(1,ℤ/8)C2×D251semidirect
285. #127C2×D251semidirect
291. #104C2×D251semidirect
292. #104C2×D251semidirect
296. #100C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324982
#306324972
#306324912
#306324902
#306320742
#306319562
#306319552
#306319492
#306319482
#306318662
#306318612
#306318542
#306318062
#453×C2512
#3288251