Finite Group Databasebeta

← #44562156

Subgroups — #44562156

Order 32128.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632498C211yesmaximal, characteristic, complemented
2#30632495C211yesmaximal, characteristic, complemented
3#30632494C211yesmaximal, characteristic, complemented
4#30632058C211yesmaximal, characteristic, complemented
5#30631952C211yesmaximal, characteristic, complemented
6#30631950C211yesmaximal, characteristic, complemented
7#30631948C211yesmaximal, characteristic, complemented
8#30631947C211yesmaximal, characteristic, complemented
9#30631870C211yesmaximal, characteristic, complemented
10#30631868C211yesmaximal, characteristic, complemented
11#30631855C211yesmaximal, characteristic, complemented
12#30631853C211yesmaximal, characteristic, complemented
13#30631804C211yesmaximal, characteristic, complemented
14#30631802C211yesmaximal, characteristic, complemented
15#577×C251C211yesmaximal, characteristic, complemented
16#22548762C2211yescharacteristic
17#22548761C2211yescharacteristic, complemented
18#22548747—21no—
19#22548746—21no—
20#22548746C2211yescharacteristic, complemented
21#22548745—21no—
22#22548745C2211yescharacteristic, complemented
23#22548745C2211yescharacteristic, complemented
24#22548741—21no—
25#22548741C2211yescharacteristic
26#22548740—21no—
27#22548740C2211yescharacteristic
28#22548740C2211yescharacteristic
29#22548739—21no—
30#22548736—21no—
31#22548735—21no—
32#22548735—21no—
33#22548735C2211yescharacteristic
34C2×#16860799C2211yescharacteristic, complemented
35C2×#16860798—21no—
36C2×#16860798C2211yescharacteristic, complemented
37C2×#16860797—21no—
38C2×#16860797C2211yescharacteristic, complemented
39SD16×D251—21no—
40#22548722C2211yescharacteristic, complemented
41#22548722C2211yescharacteristic, complemented
42#22548721C2211yescharacteristic, complemented
43#22548713C2211yescharacteristic, complemented
44#22548711C2211yescharacteristic
45#22548651C2211yescharacteristic, complemented
46#22548650C2211yescharacteristic, complemented
47#22548642C2211yescharacteristic, complemented
48#22548642C2211yescharacteristic, complemented
49#22548642C2211yescharacteristic, complemented
50#22548641C2211yescharacteristic, complemented
51#22548634C2211yescharacteristic, complemented
52#22548633C2211yescharacteristic, complemented
53#22548632C2211yescharacteristic, complemented
54#22548631C2211yescharacteristic, complemented
55C8×Dic251C2211yescharacteristic, complemented
562−1+4×C251C2211yescharacteristic
57H(5,𝔽2)×C251C2211yescharacteristic, complemented
58#137×C251—21no—
59#137×C251C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C251—21no—
61AGL(1,ℤ/8)×C251C2211yescharacteristic
62#135×C251—21no—
63#135×C251C2211yescharacteristic
64C2×SD16×C251—21no—
65C2×SD16×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#16860800—21no—
84#16860800D411yescharacteristic, complemented
85#16860799—21no—
86#16860799—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#16860797—21no—
96#16860797—21no—
97#16860797—21no—
98#16860797D411yescharacteristic, complemented
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—
113Dic251⋊C4—41no—
114C4×Dic251C2311yescharacteristic
115#41×C251—21no—
116#41×C251—21no—
117#41×C251—21no—
118#41×C251—21no—
119#41×C251—41no—
120#41×C251C2311yescharacteristic
121#41×C251C2311yescharacteristic
122#41×C251C2311yescharacteristic
123C2×Q8×C251—21no—
124C2×Q8×C251—21no—
125C2×Q8×C251C2311yescharacteristic
126C2×D4×C251—21no—
127C2×D4×C251—21no—
128C2×D4×C251—41no—
129C2×D4×C251C2311yescharacteristic
130Q16×C251—21no—
131Q16×C251D411yescharacteristic, complemented
132SD16×C251—21no—
133SD16×C251—21no—
134SD16×C251—21no—
135SD16×C251—21no—
136SD16×C251D411yescharacteristic, complemented
137SD16×C251D411yescharacteristic, complemented
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—41no—
171Q8×C251—21no—
172Q8×C251—21no—
173Q8×C251—41no—
174Q8×C251C2×D411yescharacteristic
175Q8×C251C2×D411yescharacteristic
176Q8×C251C2×D411yescharacteristic
177D4×C251—21no—
178D4×C251—21no—
179D4×C251—21no—
180D4×C251—21no—
181D4×C251—21no—
182D4×C251—41no—
183D4×C251—41no—
184D4×C251—41no—
185D4×C251C2×D411yescharacteristic
186D4×C251C2×D411yescharacteristic
187D4×C251C2×D411yescharacteristic
188C2×C4×C251—21no—
189C2×C4×C251—21no—
190C2×C4×C251—21no—
191C2×C4×C251—21no—
192C2×C4×C251—41no—
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—81no—
204C22×C251—161no—
205C22×C251H(5,𝔽2)11yescharacteristic
206C2×D251—21no—
207C2×D251—41no—
208C2×D251—81no—
209C2×D251—161no—
210C4×C251—21no—
211C4×C251—21no—
212C4×C251—21no—
213C4×C251—41no—
214C4×C251C22×D411yescharacteristic
215C4×C251C22×D411yescharacteristic
216Dic251—22no—
217Dic251—21no—
218Dic251—41no—
219C2×C251—21no—
220C2×C251—41no—
221C2×C251—41no—
222C2×C251—41no—
223C2×C251—81no—
224C2×C251D4211yescharacteristic
225D251—41no—
226D251—81no—
227C251#329111yescharacteristic, complemented, Sylow
228#3291—2511nomaximal, Sylow
229#577—2511no—
230#577—2511no—
231#577—2511no—
232#577D25111yescharacteristic, complemented
233#492—2511no—
234#471—2511no—
235#471—2511no—
236#471—2511no—
237#471—2511no—
238#456—2511no—
239#456—2511no—
240#453—2511no—
241#453—2511no—
242#443—2511no—
243#443—2511no—
2442−1+4—2511no—
2452−1+4C2×D25111yescharacteristic
246H(5,𝔽2)—2511no—
247H(5,𝔽2)C2×D25111yescharacteristic, complemented
248#137—21no—
249#137—5021no—
250#137—5021no—
251#137—5021no—
252#137—2511no—
253#137—2511no—
254#137—2511no—
255#137C2×D25111yescharacteristic, complemented
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—5021no—
258AGL(1,ℤ/8)—5021no—
259AGL(1,ℤ/8)—5021no—
260AGL(1,ℤ/8)—2511no—
261AGL(1,ℤ/8)—2511no—
262AGL(1,ℤ/8)—2511no—
263AGL(1,ℤ/8)C2×D25111yescharacteristic
264#135—21no—
265#135—5021no—
266#135—5021no—
267#135—5021no—
268#135—2511no—
269#135C2×D25111yescharacteristic
270C2×SD16—21no—
271C2×SD16—5021no—
272C2×SD16—5021no—
273C2×SD16—5021no—
274C2×SD16—2511no—
275C2×SD16—2511no—
276C2×SD16—2511no—
277C2×SD16C2×D25111yescharacteristic, complemented
278#131—2511no—
279#131—2511no—
280#131—2511no—
281#131C2×D25111yescharacteristic, complemented
282C4⋊Q8—2511no—
283#127—2511no—
284#108—2511no—
285#108—2511no—
286#104—2511no—
287#104—2511no—
288#104—2511no—
289#104—2511no—
290#101—2511no—
291#101—2511no—
292#100—2511no—
293#100—2511no—
294C4×C8—2511no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—10041no—
301#41—5021no—
302#41—5021no—
303#41—5021no—
304#41—5021no—
305#41—2511no—
306#41C22×D25111yescharacteristic
307#41C22×D25111yescharacteristic
308#41C22×D25111yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—5021no—
312C2×Q8—5021no—
313C2×Q8—2511no—
314C2×Q8C22×D25111yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—10041no—
319C2×D4—10041no—
320C2×D4—5021no—
321C2×D4—5021no—
322C2×D4—2511no—
323C2×D4C22×D25111yescharacteristic
324Q16—21no—
325Q16—5021no—
326Q16—5021no—
327Q16—5021no—
328Q16—2511no—
329Q16C22⋊D25111yescharacteristic, complemented
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—5021no—
335SD16—5021no—
336SD16—5021no—
337SD16—5021no—
338SD16—5021no—
339SD16—5021no—
340SD16—5021no—
341SD16—5021no—
342SD16—5021no—
343SD16—5021no—
344SD16—5021no—
345SD16—5021no—
346SD16—2511no—
347SD16—2511no—
348SD16C22⋊D25111yescharacteristic, complemented
349SD16C22⋊D25111yescharacteristic, complemented
350D8—21no—
351D8—5021no—
352D8—5021no—
353D8—5021no—
354D8—2511no—
355D8C22⋊D25111yescharacteristic, complemented
356M16—21no—
357M16—5021no—
358M16—5021no—
359M16—5021no—
360M16—2511no—
361M16—2511no—
362M16—2511no—
363M16C22×D25111yescharacteristic
364C2×C8—21no—
365C2×C8—5021no—
366C2×C8—5021no—
367C2×C8—5021no—
368C2×C8—2511no—
369C2×C8C22×D25111yescharacteristic
370C4⋊C4—10041no—
371C42—2511no—
372C23—21no—
373C23—41no—
374C23—10041no—
375C23—5021no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—10041no—
380Q8—5021no—
381Q8—5021no—
382Q8—2511no—
383Q8D4×D25111yescharacteristic
384Q8C2×C22⋊D25111yescharacteristic
385Q8C2×C22⋊D25111yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—20082no—
395D4—10041no—
396D4—10041no—
397D4—10041no—
398D4—5021no—
399D4—5021no—
400D4—5021no—
401D4—2511no—
402D4D4×D25111yescharacteristic
403D4C2×C22⋊D25111yescharacteristic
404D4C2×C22⋊D25111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—10041no—
413C2×C4—10041no—
414C2×C4—10041no—
415C2×C4—5021no—
416C2×C4—5021no—
417C2×C4—5021no—
418C2×C4C23×D25111yescharacteristic
419C8—21no—
420C8—5021no—
421C8—5021no—
422C8—5021no—
423C8—2511no—
424C8—2511no—
425C8C2×C22⋊D25111yescharacteristic
426C8C2×C22⋊D25111yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—40161no—
434C22—20081no—
435C22—10041no—
436C22—5021no—
437C22#2254876011yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—10041no—
443C4—5022no—
444C4—5021no—
445C4C2×D4×D25111yescharacteristic
446C4C22×C22⋊D25111yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—20081no—
453C2—10041no—
454C2D4×C22⋊D25111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#577×C251C2
socleC2×C251D42
hypercenter#577D251
hypocenterC251#3291
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #30632498C2semidirect
2. #30632495C2semidirect
3. #30632494C2semidirect
4. #30632058C2semidirect
5. #30631952C2semidirect
6. #30631950C2semidirect
7. #30631948C2semidirect
8. #30631947C2semidirect
9. #30631870C2semidirect
10. #30631868C2semidirect
11. #30631855C2semidirect
12. #30631853C2semidirect
13. #30631804C2semidirect
14. #30631802C2semidirect
15. #577×C251C2semidirect
17. #22548761C22semidirect
20. #22548746C22semidirect
22. #22548745C22semidirect
23. #22548745C22semidirect
34. C2×#16860799C22semidirect
36. C2×#16860798C22semidirect
38. C2×#16860797C22semidirect
40. #22548722C22semidirect
41. #22548722C22semidirect
42. #22548721C22semidirect
43. #22548713C22semidirect
45. #22548651C22semidirect
46. #22548650C22semidirect
47. #22548642C22semidirect
48. #22548642C22semidirect
49. #22548642C22semidirect
50. #22548641C22semidirect
51. #22548634C22semidirect
52. #22548633C22semidirect
53. #22548632C22semidirect
54. #22548631C22semidirect
55. C8×Dic251C22semidirect
57. H(5,𝔽2)×C251C22semidirect
59. #137×C251C22semidirect
65. C2×SD16×C251C22semidirect
66. #131×C251C22semidirect
84. #16860800D4semidirect
92. #16860798D4semidirect
98. #16860797D4semidirect
102. #16860795D4semidirect
131. Q16×C251D4semidirect
136. SD16×C251D4semidirect
137. SD16×C251D4semidirect
139. D8×C251D4semidirect
227. C251#3291semidirect
232. #577D251semidirect
247. H(5,𝔽2)C2×D251semidirect
255. #137C2×D251semidirect
277. C2×SD16C2×D251semidirect
281. #131C2×D251semidirect
329. Q16C22⋊D251semidirect
348. SD16C22⋊D251semidirect
349. SD16C22⋊D251semidirect
355. D8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306324982
#306324952
#306324942
#306320582
#306319522
#306319502
#306319482
#306319472
#306318702
#306318682
#306318552
#306318532
#306318042
#306318022
#577×C2512
#3291251