Finite Group Databasebeta

← #44562129

Subgroups — #44562129

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#30632494C211yesmaximal, characteristic, complemented
3#30632490C211yesmaximal, characteristic, complemented
4#30632075C211yesmaximal, characteristic, complemented
5#30631955C211yesmaximal, characteristic, complemented
6#30631954C211yesmaximal, characteristic, complemented
7#30631950C211yesmaximal, characteristic, complemented
8#30631948C211yesmaximal, characteristic, complemented
9#30631865C211yesmaximal, characteristic, complemented
10#30631858C211yesmaximal, characteristic, complemented
11#30631855C211yesmaximal, characteristic, complemented
12#30631853C211yesmaximal, characteristic, complemented
13#30631809C211yesmaximal, characteristic, complemented
14#30631807C211yesmaximal, characteristic, complemented
15#576×C251C211yesmaximal, characteristic, complemented
16#22548762C2211yescharacteristic, complemented
17#22548761C2211yescharacteristic, complemented
18#22548747—21no—
19#22548745—21no—
20#22548745C2211yescharacteristic, complemented
21#22548745C2211yescharacteristic, complemented
22#22548741—21no—
23#22548741—21no—
24#22548741C2211yescharacteristic, complemented
25#22548741C2211yescharacteristic, complemented
26#22548740—21no—
27#22548740C2211yescharacteristic
28#22548740C2211yescharacteristic, complemented
29#22548739—21no—
30#22548736—21no—
31#22548735—21no—
32#22548735—21no—
33#22548735C2211yescharacteristic
34C2×#16860799C2211yescharacteristic, complemented
35C2×#16860798—21no—
36C2×#16860798C2211yescharacteristic, complemented
37SD16×D251—21no—
38C2×#16860795—21no—
39C2×#16860795C2211yescharacteristic, complemented
40#22548722C2211yescharacteristic, complemented
41#22548722C2211yescharacteristic, complemented
42#22548721C2211yescharacteristic, complemented
43#22548711C2211yescharacteristic, complemented
44#22548702C2211yescharacteristic, complemented
45#22548651C2211yescharacteristic, complemented
46#22548650C2211yescharacteristic, complemented
47#22548642C2211yescharacteristic, complemented
48#22548642C2211yescharacteristic, complemented
49#22548642C2211yescharacteristic, complemented
50#22548641C2211yescharacteristic, complemented
51#22548633C2211yescharacteristic, complemented
52#22548632C2211yescharacteristic, complemented
53#22548632C2211yescharacteristic, complemented
54#22548631C2211yescharacteristic, complemented
55#22548625C2211yescharacteristic, complemented
56H(5,𝔽2)×C251C2211yescharacteristic
57H(5,𝔽2)×C251C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C251—21no—
59AGL(1,ℤ/8)×C251—21no—
60AGL(1,ℤ/8)×C251C2211yescharacteristic
61AGL(1,ℤ/8)×C251C2211yescharacteristic, complemented
62#135×C251—21no—
63#135×C251C2211yescharacteristic
64C2×D8×C251—21no—
65C2×D8×C251C2211yescharacteristic, complemented
66#131×C251C2211yescharacteristic, complemented
67#16860811—21no—
68#16860810—21no—
69#16860810—21no—
70#16860810C2311yescharacteristic
71#16860809—21no—
72#16860809—41no—
73C2×Dic502—21no—
74C2×Dic502C2311yescharacteristic
75Q8×D251—21no—
76C2×D1004—21no—
77C2×D1004C2311yescharacteristic
78C2×C22⋊D251—41no—
79D4×D251—21no—
80D4×D251—41no—
81#16860800—21no—
82#16860800—21no—
83#16860800D411yescharacteristic, complemented
84#16860799—21no—
85#16860799—21no—
86#16860798—21no—
87#16860798—21no—
88#16860798—21no—
89#16860798—21no—
90#16860798—21no—
91#16860798—21no—
92#16860798D411yescharacteristic, complemented
93#16860797—21no—
94#16860797—21no—
95#16860797D411yescharacteristic, complemented
96#16860795—21no—
97#16860795—21no—
98#16860795—21no—
99#16860795—21no—
100#16860795—21no—
101#16860795—21no—
102#16860795D411yescharacteristic, complemented
103#16860794—21no—
104#16860794—21no—
105#16860794C2311yescharacteristic
106#16860794C2311yescharacteristic
107#16860794C2311yescharacteristic
108#16860793—21no—
109C2×C251⋊C8—21no—
110C2×C251⋊C8—21no—
111C2×C251⋊C8C2311yescharacteristic
112C8×D251—21no—
113C22⋊Dic251—41no—
114C4×Dic251C2311yescharacteristic
115#41×C251—21no—
116#41×C251—21no—
117#41×C251—21no—
118#41×C251—21no—
119#41×C251C2311yescharacteristic
120#41×C251C2311yescharacteristic
121#41×C251C2311yescharacteristic
122C2×D4×C251—21no—
123C2×D4×C251—21no—
124C2×D4×C251—21no—
125C2×D4×C251—21no—
126C2×D4×C251—41no—
127C2×D4×C251—41no—
128C2×D4×C251C2311yescharacteristic
129C2×D4×C251C2311yescharacteristic
130Q16×C251D411yescharacteristic, complemented
131SD16×C251—21no—
132SD16×C251—21no—
133SD16×C251D411yescharacteristic, complemented
134SD16×C251D411yescharacteristic, complemented
135D8×C251—21no—
136D8×C251—21no—
137D8×C251—21no—
138D8×C251—21no—
139D8×C251D411yescharacteristic, complemented
140M16×C251—21no—
141M16×C251C2311yescharacteristic
142C2×C8×C251—21no—
143C2×C8×C251C2311yescharacteristic
144C22×D251—21no—
145C22×D251—41no—
146Dic502—21no—
147Dic502—21no—
148Dic502—41no—
149Dic502C2×D411yescharacteristic
150D1004—21no—
151D1004—21no—
152D1004—41no—
153D1004C2×D411yescharacteristic
154C22⋊D251—21no—
155C22⋊D251—41no—
156C22⋊D251—41no—
157C22⋊D251—82no—
158C2×Dic251—21no—
159C2×Dic251—21no—
160C2×Dic251—41no—
161C4×D251—21no—
162C4×D251—41no—
163C4×D251—41no—
164C251⋊C8—21no—
165C251⋊C8—21no—
166C251⋊C8—21no—
167C251⋊C8C2×D411yescharacteristic
168C251⋊C8C2×D411yescharacteristic
169C23×C251—21no—
170C23×C251—21no—
171C23×C251—41no—
172C23×C251—41no—
173Q8×C251C2×D411yescharacteristic
174Q8×C251C2×D411yescharacteristic
175Q8×C251C2×D411yescharacteristic
176D4×C251—21no—
177D4×C251—21no—
178D4×C251—21no—
179D4×C251—21no—
180D4×C251—21no—
181D4×C251—21no—
182D4×C251—21no—
183D4×C251—41no—
184D4×C251—41no—
185D4×C251—41no—
186D4×C251—41no—
187D4×C251C2×D411yescharacteristic
188D4×C251C2×D411yescharacteristic
189D4×C251C2×D411yescharacteristic
190C2×C4×C251—21no—
191C2×C4×C251—21no—
192C2×C4×C251—21no—
193C2×C4×C251—41no—
194C2×C4×C251—41no—
195C2×C4×C251C2411yescharacteristic
196C8×C251—21no—
197C8×C251C2×D411yescharacteristic
198C8×C251C2×D411yescharacteristic
199C22×C251—21no—
200C22×C251—21no—
201C22×C251—21no—
202C22×C251—41no—
203C22×C251—41no—
204C22×C251—81no—
205C22×C251—81no—
206C22×C251—161no—
207C22×C251—161no—
208C22×C251H(5,𝔽2)11yescharacteristic
209C2×D251—21no—
210C2×D251—41no—
211C2×D251—81no—
212C2×D251—161no—
213C4×C251—21no—
214C4×C251—21no—
215C4×C251—21no—
216C4×C251C22×D411yescharacteristic
217C4×C251C22×D411yescharacteristic
218Dic251—21no—
219Dic251—41no—
220Dic251—41no—
221C2×C251—21no—
222C2×C251—41no—
223C2×C251—41no—
224C2×C251—41no—
225C2×C251—81no—
226C2×C251—81no—
227C2×C251D4211yescharacteristic
228D251—41no—
229D251—81no—
230C251#328811yescharacteristic, complemented, Sylow
231#3288—2511nomaximal, Sylow
232#577—2511no—
233#577—2511no—
234#576—2511no—
235#576D25111yescharacteristic, complemented
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#453—2511no—
245#444—2511no—
246#444—2511no—
2472−1+4—2511no—
248H(5,𝔽2)—2511no—
249H(5,𝔽2)C2×D25111yescharacteristic
250H(5,𝔽2)C2×D25111yescharacteristic, complemented
251#137—5021no—
252#137—5021no—
253#137—2511no—
254#137—2511no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
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—21no—
268#135—5021no—
269#135—5021no—
270#135—5021no—
271#135—2511no—
272#135C2×D25111yescharacteristic
273C2×SD16—5021no—
274C2×SD16—5021no—
275C2×SD16—2511no—
276C2×SD16—2511no—
277C2×D8—21no—
278C2×D8—5021no—
279C2×D8—2511no—
280C2×D8C2×D25111yescharacteristic, complemented
281#131—2511no—
282#131—2511no—
283#131—2511no—
284#131C2×D25111yescharacteristic, complemented
285#127—2511no—
286#124—2511no—
287#108—2511no—
288#108—2511no—
289#104—2511no—
290#104—2511no—
291#104—2511no—
292#104—2511no—
293#101—2511no—
294#100—2511no—
295#100—2511no—
296#100—2511no—
297#97—2511no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—10041no—
303#41—5021no—
304#41—5021no—
305#41—5021no—
306#41—5021no—
307#41—2511no—
308#41C22×D25111yescharacteristic
309#41C22×D25111yescharacteristic
310#41C22×D25111yescharacteristic
311C2×Q8—5021no—
312C2×Q8—5021no—
313C2×Q8—2511no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—10041no—
321C2×D4—10041no—
322C2×D4—5021no—
323C2×D4—5021no—
324C2×D4—2511no—
325C2×D4C22×D25111yescharacteristic
326C2×D4C22×D25111yescharacteristic
327Q16—5021no—
328Q16—5021no—
329Q16—2511no—
330Q16C22⋊D25111yescharacteristic, complemented
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—
345SD16C22⋊D25111yescharacteristic, complemented
346SD16C22⋊D25111yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—5021no—
352D8—5021no—
353D8—5021no—
354D8—5021no—
355D8—5021no—
356D8—5021no—
357D8—2511no—
358D8C22⋊D25111yescharacteristic, complemented
359M16—21no—
360M16—5021no—
361M16—5021no—
362M16—5021no—
363M16—2511no—
364M16—2511no—
365M16—2511no—
366M16C22×D25111yescharacteristic
367C2×C8—21no—
368C2×C8—5021no—
369C2×C8—5021no—
370C2×C8—5021no—
371C2×C8—2511no—
372C2×C8C22×D25111yescharacteristic
373C22⋊C4—10041no—
374C42—2511no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—10041no—
380C23—5021no—
381Q8—10041no—
382Q8—5021no—
383Q8—5021no—
384Q8—2511no—
385Q8D4×D25111yescharacteristic
386Q8C2×C22⋊D25111yescharacteristic
387Q8C2×C22⋊D25111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—20082no—
400D4—10041no—
401D4—10041no—
402D4—10041no—
403D4—5021no—
404D4—5021no—
405D4—5021no—
406D4—2511no—
407D4D4×D25111yescharacteristic
408D4C2×C22⋊D25111yescharacteristic
409D4C2×C22⋊D25111yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—10041no—
416C2×C4—10041no—
417C2×C4—10041no—
418C2×C4—5021no—
419C2×C4—5021no—
420C2×C4—5021no—
421C2×C4C23×D25111yescharacteristic
422C8—21no—
423C8—5021no—
424C8—5021no—
425C8—5021no—
426C8—2511no—
427C8—2511no—
428C8C2×C22⋊D25111yescharacteristic
429C8C2×C22⋊D25111yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—40161no—
440C22—20081no—
441C22—10041no—
442C22—5021no—
443C22#2254876011yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—10041no—
448C4—10041no—
449C4—5021no—
450C4C2×D4×D25111yescharacteristic
451C4C22×C22⋊D25111yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—20081no—
459C2—10041no—
460C2D4×C22⋊D25111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#576×C251C2
socleC2×C251D42
hypercenter#576D251
hypocenterC251#3288
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #30632498C2semidirect
2. #30632494C2semidirect
3. #30632490C2semidirect
4. #30632075C2semidirect
5. #30631955C2semidirect
6. #30631954C2semidirect
7. #30631950C2semidirect
8. #30631948C2semidirect
9. #30631865C2semidirect
10. #30631858C2semidirect
11. #30631855C2semidirect
12. #30631853C2semidirect
13. #30631809C2semidirect
14. #30631807C2semidirect
15. #576×C251C2semidirect
16. #22548762C22semidirect
17. #22548761C22semidirect
20. #22548745C22semidirect
21. #22548745C22semidirect
24. #22548741C22semidirect
25. #22548741C22semidirect
28. #22548740C22semidirect
34. C2×#16860799C22semidirect
36. C2×#16860798C22semidirect
39. C2×#16860795C22semidirect
40. #22548722C22semidirect
41. #22548722C22semidirect
42. #22548721C22semidirect
43. #22548711C22semidirect
44. #22548702C22semidirect
45. #22548651C22semidirect
46. #22548650C22semidirect
47. #22548642C22semidirect
48. #22548642C22semidirect
49. #22548642C22semidirect
50. #22548641C22semidirect
51. #22548633C22semidirect
52. #22548632C22semidirect
53. #22548632C22semidirect
54. #22548631C22semidirect
55. #22548625C22semidirect
57. H(5,𝔽2)×C251C22semidirect
61. AGL(1,ℤ/8)×C251C22semidirect
65. C2×D8×C251C22semidirect
66. #131×C251C22semidirect
83. #16860800D4semidirect
92. #16860798D4semidirect
95. #16860797D4semidirect
102. #16860795D4semidirect
130. Q16×C251D4semidirect
133. SD16×C251D4semidirect
134. SD16×C251D4semidirect
139. D8×C251D4semidirect
230. C251#3288semidirect
235. #576D251semidirect
250. H(5,𝔽2)C2×D251semidirect
266. AGL(1,ℤ/8)C2×D251semidirect
280. C2×D8C2×D251semidirect
284. #131C2×D251semidirect
330. Q16C22⋊D251semidirect
345. SD16C22⋊D251semidirect
346. SD16C22⋊D251semidirect
358. D8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306324982
#306324942
#306324902
#306320752
#306319552
#306319542
#306319502
#306319482
#306318652
#306318582
#306318552
#306318532
#306318092
#306318072
#576×C2512
#3288251