Finite Group Databasebeta

← #44559628

Subgroups — #44559628

Order 32128.

784 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548760C211yesmaximal, characteristic, complemented
2C2×#22548739C211yesmaximal, characteristic, complemented
3C2×D8×D251C211yesmaximal, characteristic, complemented
4#30632253C211yesmaximal, characteristic, complemented
5#30631834C211yesmaximal, characteristic, complemented
6#30631833C211yesmaximal, characteristic, complemented
7#30631830C211yesmaximal, characteristic, complemented
8#30631829C211yesmaximal, characteristic, complemented
9#30631822C211yesmaximal, characteristic, complemented
10#30631821C211yesmaximal, characteristic, complemented
11#30631819C211yesmaximal, characteristic, complemented
12#30631690C211yesmaximal, characteristic, complemented
13#30631680C211yesmaximal, characteristic, complemented
14#30631652C211yesmaximal, characteristic, complemented
15#447×C251C211yesmaximal, characteristic, complemented
16#22548760—24no—
17#22548760—24no—
18C2×#16860810C2211yescharacteristic
19C2×#16860810C2211yescharacteristic, complemented
20C2×#16860809—21no—
21C2×#16860809C2211yescharacteristic
22C2×#16860809C2211yescharacteristic, complemented
23C22×C22⋊D251—21no—
24C22×C22⋊D251—21no—
25C2×D4×D251—21no—
26C2×D4×D251C2211yescharacteristic
27C2×D4×D251C2211yescharacteristic, complemented
28C2×D4×D251C2211yescharacteristic, complemented
29#22548739—24no—
30C2×#16860799C2211yescharacteristic, complemented
31C2×#16860798C2211yescharacteristic, complemented
32C2×D2008C2211yescharacteristic, complemented
33C2×#16860795C2211yescharacteristic, complemented
34C2×#16860795C2211yescharacteristic, complemented
35C2×#16860795C2211yescharacteristic, complemented
36D8×D251—24no—
37C2×#16860793C2211yescharacteristic, complemented
38C2×C8×D251C2211yescharacteristic, complemented
39#22548711—21no—
40#22548701—21no—
41#22548690C2211yescharacteristic, complemented
42#22548689C2211yescharacteristic, complemented
43#22548686C2211yescharacteristic, complemented
44#22548685—21no—
45#22548683—21no—
46#22548673C2211yescharacteristic, complemented
47#22548640C2211yescharacteristic, complemented
48#22548638C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic, complemented
50#22548636C2211yescharacteristic, complemented
51#22548636C2211yescharacteristic, complemented
52#22548635C2211yescharacteristic, complemented
53C22⋊(C251⋊C8)C2211yescharacteristic, complemented
54(C2×D251)⋊C8C2211yescharacteristic, complemented
55(C2×D251)⋊C8C2211yescharacteristic, complemented
56C22×D4×C251C2211yescharacteristic, complemented
57C2×D8×C251C2211yescharacteristic, complemented
58C2×D8×C251C2211yescharacteristic, complemented
59#121×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic, complemented
61#102×C251C2211yescharacteristic, complemented
62C22⋊C8×C251C2211yescharacteristic, complemented
63C23×D251—21no—
64C23×D251—21no—
65C23×D251—21no—
66#16860810—22no—
67#16860810—22no—
68#16860810—22no—
69#16860810—22no—
70#16860809—28no—
71#16860809—22no—
72#16860809—22no—
73#16860809—22no—
74#16860809D414yescomplemented
75C2×Dic502D411yescharacteristic
76C2×Dic502C2311yescharacteristic
77C2×D1004—21no—
78C2×D1004D411yescharacteristic
79C2×D1004C2311yescharacteristic
80C2×D1004C2311yescharacteristic
81C2×C22⋊D251—28no—
82C2×C22⋊D251—28no—
83C2×C22⋊D251—21no—
84C2×C22⋊D251—21no—
85C2×C22⋊D251—21no—
86C2×C22⋊D251—21no—
87C2×C22⋊D251—21no—
88C2×C22⋊D251—21no—
89C2×C22⋊D251—21no—
90C2×C22⋊D251—21no—
91C2×C22⋊D251D411yescharacteristic
92C2×C22⋊D251D411yescharacteristic
93D4×D251—28no—
94D4×D251—22no—
95D4×D251—22no—
96D4×D251—22no—
97D4×D251—22no—
98D4×D251—22no—
99D4×D251D414yescomplemented
100C22×Dic251—21no—
101C22×Dic251—21no—
102C2×C4×D251—21no—
103C2×C4×D251C2311yescharacteristic
104C2×C4×D251C2311yescharacteristic
105#16860799—22no—
106#16860798—22no—
107D2008—22no—
108#16860795—22no—
109#16860795—22no—
110#16860795—22no—
111#16860793—22no—
112C2×C251⋊C8C2311yescharacteristic
113C2×C251⋊C8C2311yescharacteristic
114C8×D251—22no—
115C4⋊Dic251C2311yescharacteristic
116C22⋊Dic251—21no—
117(C2×D251)⋊C4—21no—
118(C2×D251)⋊C4—21no—
119(C2×D251)⋊C4—21no—
120C4×Dic251—21no—
121C24×C251—21no—
122C2×D4×C251—22no—
123C2×D4×C251—22no—
124C2×D4×C251—21no—
125C2×D4×C251—21no—
126C2×D4×C251C2311yescharacteristic
127C2×D4×C251C2311yescharacteristic
128C2×D4×C251C2311yescharacteristic
129C22×C4×C251C2311yescharacteristic
130D8×C251—22no—
131D8×C251—22no—
132C2×C8×C251C2311yescharacteristic
133C2×C8×C251C2311yescharacteristic
134C4⋊C4×C251C2311yescharacteristic
135C22⋊C4×C251—21no—
136C22×D251—21no—
137C22×D251—24no—
138C22×D251—24no—
139C22×D251—21no—
140C22×D251—42no—
141C22×D251—81no—
142C22×D251—81no—
143C22×D251—81no—
144C22×D251—81no—
145C22×D251—81no—
146C22×D251—81no—
147C22×D251C2×D411yescharacteristic
148C22×D251C2×D411yescharacteristic
149Dic502—21no—
150Dic502—22no—
151Dic502C2×D412yes—
152D1004—22no—
153D1004—21no—
154D1004—21no—
155D1004—21no—
156D1004—81no—
157D1004C2×D412yes—
158C22⋊D251—24no—
159C22⋊D251—24no—
160C22⋊D251—24no—
161C22⋊D251—24no—
162C22⋊D251—22no—
163C22⋊D251—24no—
164C22⋊D251—22no—
165C22⋊D251—42no—
166C22⋊D251—81no—
167C22⋊D251—81no—
168C2×Dic251—24no—
169C2×Dic251—21no—
170C2×Dic251—24no—
171C2×Dic251—21no—
172C2×Dic251C2×D411yescharacteristic
173C2×Dic251C2×D411yescharacteristic
174C4×D251—24no—
175C4×D251—21no—
176C4×D251—21no—
177C4×D251C2×D412yes—
178C4×D251C2×D412yes—
179C251⋊C8—21no—
180C251⋊C8—21no—
181C23×C251—24no—
182C23×C251—21no—
183C23×C251—21no—
184C23×C251—21no—
185C23×C251—81no—
186C23×C251—81no—
187C23×C251C2×D411yescharacteristic
188D4×C251—24no—
189D4×C251—21no—
190D4×C251—21no—
191D4×C251—21no—
192D4×C251—21no—
193D4×C251—81no—
194D4×C251C2×D412yes—
195D4×C251C2×D412yes—
196C2×C4×C251—21no—
197C2×C4×C251—21no—
198C2×C4×C251—21no—
199C2×C4×C251C2×D411yescharacteristic
200C2×C4×C251C2411yescharacteristic
201C8×C251—21no—
202C8×C251—21no—
203C22×C251—21no—
204C22×C251—22no—
205C22×C251—22no—
206C22×C251—21no—
207C22×C251—21no—
208C22×C251—41no—
209C22×C251—41no—
210C22×C251—41no—
211C22×C251—41no—
212C22×C251—41no—
213C22×C251—41no—
214C22×C251—41no—
215C22×C251—84no—
216C22×C251C22×D411yescharacteristic
217C2×D251—21no—
218C2×D251—21no—
219C2×D251—21no—
220C2×D251—22no—
221C2×D251—21no—
222C2×D251—21no—
223C2×D251—41no—
224C2×D251—41no—
225C2×D251—41no—
226C2×D251—41no—
227C2×D251—41no—
228C2×D251—84no—
229C2×D251—84no—
230C2×D251—84no—
231C2×D251#12012yes—
232C4×C251—21no—
233C4×C251—41no—
234C4×C251C22×D411yescharacteristic
235C4×C251C22×D411yescharacteristic
236Dic251—22no—
237Dic251—21no—
238Dic251—41no—
239Dic251#12012yes—
240C2×C251—42no—
241C2×C251—42no—
242C2×C251—41no—
243C2×C251—81no—
244C2×C251C2×#12011yescharacteristic
245C2×C251#57611yescharacteristic
246C2×C251#57611yescharacteristic
247D251—22no—
248D251—41no—
249D251—42no—
250D251—81no—
251C251#300811yescharacteristic, complemented, Sylow
252#3008—2511nomaximal, Sylow
253C2×H(5,𝔽2)—2511no—
254C2×AGL(1,ℤ/8)—2511no—
255C22×D8—2511no—
256#535—2511no—
257#449—2511no—
258#449—2511no—
259#449—2511no—
260#449—2511no—
261#447—2511no—
262#447—2511no—
263#447—2511no—
264#447D25111yescharacteristic, complemented
265#418—2511no—
266#416—2511no—
267#408—2511no—
268H(5,𝔽2)—5024no—
269H(5,𝔽2)—5024no—
270C2×#41—5021no—
271C2×#41—2511no—
272C2×#41—2511no—
273C2×#41—2511no—
274C2×#41—2511no—
275C22×D4—5021no—
276C22×D4—5021no—
277C22×D4—5021no—
278C22×D4—2511no—
279C22×D4—2511no—
280C22×D4—2511no—
281C22×D4C2×D25111yescharacteristic, complemented
282AGL(1,ℤ/8)—5024no—
283C2×SD16—2511no—
284C2×SD16—2511no—
285C2×D8—5024no—
286C2×D8—2511no—
287C2×D8—2511no—
288C2×D8—2511no—
289C2×D8—2511no—
290C2×D8C2×D25111yescharacteristic, complemented
291C2×D8C2×D25111yescharacteristic, complemented
292C2×M16—2511no—
293C22×C8—2511no—
294#127—5021no—
295#124—5021no—
296#121—2511no—
297#121—2511no—
298#121—2511no—
299#121C2×D25111yescharacteristic, complemented
300#120—5021no—
301#120—5021no—
302#117—2511no—
303#103—2511no—
304#103—2511no—
305#102—2511no—
306#102—2511no—
307#102—2511no—
308#102—2511no—
309#102C2×D25111yescharacteristic, complemented
310#102C2×D25111yescharacteristic, complemented
311C22⋊C8—2511no—
312C22⋊C8—2511no—
313C22⋊C8—2511no—
314C22⋊C8C2×D25111yescharacteristic, complemented
315C24—21no—
316C24—5021no—
317C24—5021no—
318C24—5021no—
319#41—5022no—
320#41—5022no—
321#41—5022no—
322#41—5022no—
323#41—5028no—
324#41—5022no—
325#41—5022no—
326#41—5022no—
327#41—2514no—
328C2×Q8—2511no—
329C2×Q8—2511no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—5021no—
335C2×D4—5028no—
336C2×D4—5022no—
337C2×D4—5021no—
338C2×D4—5021no—
339C2×D4—5028no—
340C2×D4—5028no—
341C2×D4—5021no—
342C2×D4—5021no—
343C2×D4—5021no—
344C2×D4—5022no—
345C2×D4—5022no—
346C2×D4—5022no—
347C2×D4—5022no—
348C2×D4—5021no—
349C2×D4—5021no—
350C2×D4—5021no—
351C2×D4—2511no—
352C2×D4—2511no—
353C2×D4—2514no—
354C2×D4—2511no—
355C2×D4—2511no—
356C2×D4—2511no—
357C2×D4C22×D25111yescharacteristic
358C2×D4C22×D25111yescharacteristic
359C2×D4C22×D25111yescharacteristic
360C22×C4—5021no—
361C22×C4—5021no—
362C22×C4—5021no—
363C22×C4—2511no—
364C22×C4—2511no—
365C22×C4C22×D25111yescharacteristic
366SD16—5022no—
367SD16—5022no—
368D8—22no—
369D8—22no—
370D8—5022no—
371D8—5022no—
372D8—5022no—
373D8—5022no—
374M16—5022no—
375C2×C8—5022no—
376C2×C8—2511no—
377C2×C8—2511no—
378C2×C8C22×D25111yescharacteristic
379C2×C8C22×D25111yescharacteristic
380C4⋊C4—2511no—
381C4⋊C4C22×D25111yescharacteristic
382C22⋊C4—21no—
383C22⋊C4—5021no—
384C22⋊C4—5021no—
385C22⋊C4—5021no—
386C22⋊C4—5021no—
387C42—5021no—
388C23—24no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—81no—
393C23—81no—
394C23—20081no—
395C23—20081no—
396C23—20081no—
397C23—20081no—
398C23—20081no—
399C23—20081no—
400C23—10042no—
401C23—5024no—
402C23—5024no—
403C23—5021no—
404C23—5021no—
405C23—2511no—
406C23—2511no—
407C23D4×D25111yescharacteristic
408Q8—5022no—
409Q8—5021no—
410Q8—2512no—
411D4—24no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—21no—
416D4—81no—
417D4—20081no—
418D4—20081no—
419D4—20081no—
420D4—10042no—
421D4—5022no—
422D4—5024no—
423D4—5024no—
424D4—5024no—
425D4—5024no—
426D4—5022no—
427D4—5022no—
428D4—5024no—
429D4—5021no—
430D4—5021no—
431D4—5021no—
432D4—2512no—
433D4D4×D25112yes—
434D4D4×D25112yes—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—5024no—
439C2×C4—5021no—
440C2×C4—5024no—
441C2×C4—5021no—
442C2×C4—5024no—
443C2×C4—5021no—
444C2×C4—5021no—
445C2×C4—2511no—
446C2×C4—2511no—
447C2×C4—2512no—
448C2×C4—2512no—
449C2×C4D4×D25111yescharacteristic
450C2×C4C23×D25111yescharacteristic
451C8—21no—
452C8—21no—
453C8—5021no—
454C8—5021no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—84no—
468C22—20084no—
469C22—20084no—
470C22—20084no—
471C22—10041no—
472C22—10041no—
473C22—10041no—
474C22—10041no—
475C22—10041no—
476C22—5021no—
477C22—5021no—
478C22—5021no—
479C22—5022no—
480C22—5021no—
481C22—5021no—
482C22—2512no—
483C22C2×D4×D25111yescharacteristic
484C4—21no—
485C4—41no—
486C4—10041no—
487C4—5022no—
488C4—5021no—
489C4—2512no—
490C4C2×D4×D25111yescharacteristic
491C4C2×D4×D25111yescharacteristic
492C2—41no—
493C2—42no—
494C2—42no—
495C2—81no—
496C2—20081no—
497C2—10042no—
498C2—10041no—
499C2—5022no—
500C2#120×D25111yescharacteristic
501C2#3063248811yescharacteristic
502C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251#3008
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#22548760C2semidirect
2. C2×#22548739C2semidirect
3. C2×D8×D251C2semidirect
4. #30632253C2semidirect
5. #30631834C2semidirect
6. #30631833C2semidirect
7. #30631830C2semidirect
8. #30631829C2semidirect
9. #30631822C2semidirect
10. #30631821C2semidirect
11. #30631819C2semidirect
12. #30631690C2semidirect
13. #30631680C2semidirect
14. #30631652C2semidirect
15. #447×C251C2semidirect
19. C2×#16860810C22semidirect
22. C2×#16860809C22semidirect
27. C2×D4×D251C22semidirect
28. C2×D4×D251C22semidirect
30. C2×#16860799C22semidirect
31. C2×#16860798C22semidirect
32. C2×D2008C22semidirect
33. C2×#16860795C22semidirect
34. C2×#16860795C22semidirect
35. C2×#16860795C22semidirect
37. C2×#16860793C22semidirect
38. C2×C8×D251C22semidirect
41. #22548690C22semidirect
42. #22548689C22semidirect
43. #22548686C22semidirect
46. #22548673C22semidirect
47. #22548640C22semidirect
48. #22548638C22semidirect
49. #22548637C22semidirect
50. #22548636C22semidirect
51. #22548636C22semidirect
52. #22548635C22semidirect
53. C22⋊(C251⋊C8)C22semidirect
54. (C2×D251)⋊C8C22semidirect
55. (C2×D251)⋊C8C22semidirect
56. C22×D4×C251C22semidirect
57. C2×D8×C251C22semidirect
58. C2×D8×C251C22semidirect
59. #121×C251C22semidirect
60. #102×C251C22semidirect
61. #102×C251C22semidirect
62. C22⋊C8×C251C22semidirect
74. #16860809D4semidirect
99. D4×D251D4semidirect
251. C251#3008semidirect
264. #447D251semidirect
281. C22×D4C2×D251semidirect
290. C2×D8C2×D251semidirect
291. C2×D8C2×D251semidirect
299. #121C2×D251semidirect
309. #102C2×D251semidirect
310. #102C2×D251semidirect
314. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
C2×#225487392
C2×D8×D2512
#306322532
#306318342
#306318332
#306318302
#306318292
#306318222
#306318212
#306318192
#306316902
#306316802
#306316522
#447×C2512
#3008251