Finite Group Databasebeta

← #44559630

Subgroups — #44559630

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×#22548761C211yesmaximal, characteristic, complemented
2C2×#22548742C211yesmaximal, characteristic, complemented
3C2×D8×D251C211yesmaximal, characteristic, complemented
4#30632253C211yesmaximal, characteristic, complemented
5#30631835C211yesmaximal, characteristic, complemented
6#30631832C211yesmaximal, characteristic, complemented
7#30631831C211yesmaximal, characteristic, complemented
8#30631823C211yesmaximal, characteristic, complemented
9#30631822C211yesmaximal, characteristic, complemented
10#30631820C211yesmaximal, characteristic, complemented
11#30631819C211yesmaximal, characteristic, complemented
12#30631690C211yesmaximal, characteristic, complemented
13#30631682C211yesmaximal, characteristic, complemented
14#30631652C211yesmaximal, characteristic, complemented
15#449×C251C211yesmaximal, characteristic, complemented
16#22548761—24no—
17#22548761—24no—
18C2×#16860811C2211yescharacteristic
19C2×#16860811C2211yescharacteristic, complemented
20C2×#16860810—21no—
21C2×#16860810C2211yescharacteristic
22C22×D1004—21no—
23C22×D1004C2211yescharacteristic, complemented
24C2×D4×D251—21no—
25C2×D4×D251—21no—
26C2×D4×D251C2211yescharacteristic
27C2×D4×D251C2211yescharacteristic, complemented
28C2×D4×D251C2211yescharacteristic, complemented
29#22548742—24no—
30C2×#16860797C2211yescharacteristic, complemented
31C2×D2008C2211yescharacteristic, 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#22548639C2211yescharacteristic, complemented
48#22548637C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic, complemented
50#22548636C2211yescharacteristic, complemented
51#22548635C2211yescharacteristic, complemented
52#22548635C2211yescharacteristic, complemented
53C22⋊(C251⋊C8)C2211yescharacteristic, complemented
54(C2×D251)⋊C8C2211yescharacteristic, complemented
55(C2×D251)⋊C8C2211yescharacteristic, complemented
56C2×#41×C251C2211yescharacteristic, complemented
57C2×SD16×C251C2211yescharacteristic, complemented
58C2×D8×C251C2211yescharacteristic, complemented
59#121×C251C2211yescharacteristic, complemented
60#103×C251C2211yescharacteristic, complemented
61#102×C251C2211yescharacteristic, complemented
62C22⋊C8×C251C2211yescharacteristic, complemented
63C23×D251—21no—
64C23×D251—21no—
65C23×D251—21no—
66C23×D251—21no—
67#16860811—22no—
68#16860811—22no—
69#16860811—22no—
70#16860811D414yescomplemented
71#16860810—28no—
72#16860810—22no—
73#16860810—22no—
74C2×Dic502D411yescharacteristic
75C2×D1004—28no—
76C2×D1004—21no—
77C2×D1004—21no—
78C2×D1004—22no—
79C2×D1004—22no—
80C2×D1004—21no—
81C2×D1004—21no—
82C2×D1004D411yescharacteristic
83C2×D1004C2311yescharacteristic
84C2×D1004C2311yescharacteristic
85C2×D1004C2311yescharacteristic
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—28no—
95D4×D251—22no—
96D4×D251—22no—
97D4×D251—22no—
98D4×D251—22no—
99D4×D251—22no—
100D4×D251D414yescomplemented
101C2×C4×D251—21no—
102C2×C4×D251—21no—
103C2×C4×D251C2311yescharacteristic
104C2×C4×D251C2311yescharacteristic
105#16860797—22no—
106D2008—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—
121#41×C251—22no—
122#41×C251—22no—
123C2×Q8×C251C2311yescharacteristic
124C2×D4×C251—21no—
125C2×D4×C251—21no—
126C2×D4×C251C2311yescharacteristic
127C2×D4×C251C2311yescharacteristic
128C22×C4×C251—21no—
129C22×C4×C251C2311yescharacteristic
130SD16×C251—22no—
131D8×C251—22no—
132C2×C8×C251C2311yescharacteristic
133C2×C8×C251C2311yescharacteristic
134C4⋊C4×C251C2311yescharacteristic
135C22⋊C4×C251—21no—
136C22×D251—21no—
137C22×D251—21no—
138C22×D251—24no—
139C22×D251—24no—
140C22×D251—24no—
141C22×D251—21no—
142C22×D251—42no—
143C22×D251—81no—
144C22×D251—81no—
145C22×D251—81no—
146C22×D251—81no—
147C22×D251—81no—
148C22×D251—81no—
149C22×D251—81no—
150C22×D251—81no—
151C22×D251C2×D411yescharacteristic
152C22×D251C2×D411yescharacteristic
153Dic502—22no—
154D1004—24no—
155D1004—24no—
156D1004—24no—
157D1004—22no—
158D1004—21no—
159D1004—21no—
160D1004—21no—
161D1004—21no—
162D1004—81no—
163D1004C2×D412yes—
164D1004C2×D412yes—
165C22⋊D251—24no—
166C22⋊D251—22no—
167C22⋊D251—22no—
168C22⋊D251—24no—
169C22⋊D251—42no—
170C22⋊D251—81no—
171C22⋊D251—81no—
172C2×Dic251—21no—
173C2×Dic251C2×D411yescharacteristic
174C2×Dic251C2×D411yescharacteristic
175C4×D251—24no—
176C4×D251—21no—
177C4×D251—21no—
178C4×D251—24no—
179C4×D251C2×D412yes—
180C4×D251C2×D412yes—
181C251⋊C8—21no—
182C251⋊C8—21no—
183C23×C251—21no—
184C23×C251—21no—
185C23×C251C2×D411yescharacteristic
186Q8×C251—21no—
187Q8×C251C2×D412yes—
188D4×C251—24no—
189D4×C251—21no—
190D4×C251—21no—
191D4×C251—21no—
192D4×C251—81no—
193D4×C251C2×D412yes—
194C2×C4×C251—24no—
195C2×C4×C251—21no—
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—21no—
206C22×C251—21no—
207C22×C251—41no—
208C22×C251—41no—
209C22×C251—41no—
210C22×C251—41no—
211C22×C251—41no—
212C22×C251C22×D411yescharacteristic
213C2×D251—21no—
214C2×D251—21no—
215C2×D251—21no—
216C2×D251—22no—
217C2×D251—22no—
218C2×D251—21no—
219C2×D251—21no—
220C2×D251—41no—
221C2×D251—41no—
222C2×D251—41no—
223C2×D251—41no—
224C2×D251—41no—
225C2×D251—41no—
226C2×D251—41no—
227C2×D251—84no—
228C2×D251—84no—
229C2×D251—84no—
230C2×D251—84no—
231C2×D251#12012yes—
232C4×C251—21no—
233C4×C251—22no—
234C4×C251—41no—
235C4×C251C22×D411yescharacteristic
236C4×C251C22×D411yescharacteristic
237Dic251—21no—
238Dic251—41no—
239Dic251#12012yes—
240C2×C251—41no—
241C2×C251—42no—
242C2×C251—81no—
243C2×C251C2×#12011yescharacteristic
244C2×C251#57611yescharacteristic
245C2×C251#57611yescharacteristic
246D251—22no—
247D251—41no—
248D251—42no—
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#449D25111yescharacteristic, complemented
261#447—2511no—
262#447—2511no—
263#447—2511no—
264#447—2511no—
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×#41C2×D25111yescharacteristic, complemented
275C22×D4—5021no—
276C22×D4—5021no—
277C22×D4—5021no—
278C22×D4—2511no—
279C22×D4—2511no—
280C22×D4—2511no—
281C22×D4—2511no—
282AGL(1,ℤ/8)—5024no—
283C2×SD16—2511no—
284C2×SD16C2×D25111yescharacteristic, complemented
285C2×D8—5024no—
286C2×D8—2511no—
287C2×D8—2511no—
288C2×D8—2511no—
289C2×D8—2511no—
290C2×D8—2511no—
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#103C2×D25111yescharacteristic, complemented
305#102—2511no—
306#102—2511no—
307#102—2511no—
308#102—2511no—
309#102—2511no—
310#102C2×D25111yescharacteristic, complemented
311C22⋊C8—2511no—
312C22⋊C8—2511no—
313C22⋊C8—2511no—
314C22⋊C8C2×D25111yescharacteristic, complemented
315C24—5021no—
316C24—5021no—
317C24—5021no—
318C24—5021no—
319#41—22no—
320#41—22no—
321#41—5028no—
322#41—5022no—
323#41—5022no—
324#41—5022no—
325#41—5022no—
326#41—5022no—
327#41—2514no—
328C2×Q8—2511no—
329C2×Q8C22×D25111yescharacteristic
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—5021no—
333C2×D4—5028no—
334C2×D4—5021no—
335C2×D4—5021no—
336C2×D4—5022no—
337C2×D4—5022no—
338C2×D4—5028no—
339C2×D4—5028no—
340C2×D4—5022no—
341C2×D4—5021no—
342C2×D4—5021no—
343C2×D4—5022no—
344C2×D4—5022no—
345C2×D4—5022no—
346C2×D4—5022no—
347C2×D4—5021no—
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×D4—2511no—
358C2×D4C22×D25111yescharacteristic
359C2×D4C22×D25111yescharacteristic
360C22×C4—21no—
361C22×C4—5021no—
362C22×C4—5021no—
363C22×C4—2511no—
364C22×C4—2511no—
365C22×C4C22×D25111yescharacteristic
366SD16—22no—
367SD16—5022no—
368D8—22no—
369D8—5022no—
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—21no—
389C23—21no—
390C23—20081no—
391C23—20081no—
392C23—20081no—
393C23—20081no—
394C23—20081no—
395C23—20081no—
396C23—20081no—
397C23—20081no—
398C23—10042no—
399C23—5024no—
400C23—5024no—
401C23—5024no—
402C23—5021no—
403C23—5021no—
404C23—5021no—
405C23—2511no—
406C23—2511no—
407C23D4×D25111yescharacteristic
408Q8—21no—
409Q8—5022no—
410Q8D4×D25112yes—
411D4—24no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—81no—
416D4—20081no—
417D4—20081no—
418D4—20081no—
419D4—10042no—
420D4—5024no—
421D4—5024no—
422D4—5024no—
423D4—5022no—
424D4—5024no—
425D4—5022no—
426D4—5022no—
427D4—5024no—
428D4—5021no—
429D4—5021no—
430D4—5021no—
431D4—5021no—
432D4—2512no—
433D4—2512no—
434D4D4×D25112yes—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—5024no—
441C2×C4—5024no—
442C2×C4—5021no—
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—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—20084no—
465C22—20084no—
466C22—20084no—
467C22—20084no—
468C22—10041no—
469C22—10041no—
470C22—10041no—
471C22—10041no—
472C22—10041no—
473C22—10041no—
474C22—10041no—
475C22—5021no—
476C22—5021no—
477C22—5021no—
478C22—5022no—
479C22—5022no—
480C22—5021no—
481C22—5021no—
482C22—2512no—
483C22C2×D4×D25111yescharacteristic
484C4—21no—
485C4—22no—
486C4—41no—
487C4—10041no—
488C4—5021no—
489C4—2512no—
490C4C2×D4×D25111yescharacteristic
491C4C2×D4×D25111yescharacteristic
492C2—41no—
493C2—42no—
494C2—81no—
495C2—20081no—
496C2—10042no—
497C2—10042no—
498C2—10041no—
499C2—5022no—
500C2#120×D25111yescharacteristic
501C2#3063248911yescharacteristic
502C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#449×C251C2
socleC22×C251C22×D4
hypercenter#449D251
hypocenterC251#3008
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2semidirect
2. C2×#22548742C2semidirect
3. C2×D8×D251C2semidirect
4. #30632253C2semidirect
5. #30631835C2semidirect
6. #30631832C2semidirect
7. #30631831C2semidirect
8. #30631823C2semidirect
9. #30631822C2semidirect
10. #30631820C2semidirect
11. #30631819C2semidirect
12. #30631690C2semidirect
13. #30631682C2semidirect
14. #30631652C2semidirect
15. #449×C251C2semidirect
19. C2×#16860811C22semidirect
23. C22×D1004C22semidirect
27. C2×D4×D251C22semidirect
28. C2×D4×D251C22semidirect
30. C2×#16860797C22semidirect
31. C2×D2008C22semidirect
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. #22548639C22semidirect
48. #22548637C22semidirect
49. #22548637C22semidirect
50. #22548636C22semidirect
51. #22548635C22semidirect
52. #22548635C22semidirect
53. C22⋊(C251⋊C8)C22semidirect
54. (C2×D251)⋊C8C22semidirect
55. (C2×D251)⋊C8C22semidirect
56. C2×#41×C251C22semidirect
57. C2×SD16×C251C22semidirect
58. C2×D8×C251C22semidirect
59. #121×C251C22semidirect
60. #103×C251C22semidirect
61. #102×C251C22semidirect
62. C22⋊C8×C251C22semidirect
70. #16860811D4semidirect
100. D4×D251D4semidirect
251. C251#3008semidirect
260. #449D251semidirect
274. C2×#41C2×D251semidirect
284. C2×SD16C2×D251semidirect
291. C2×D8C2×D251semidirect
299. #121C2×D251semidirect
304. #103C2×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×#225487612
C2×#225487422
C2×D8×D2512
#306322532
#306318352
#306318322
#306318312
#306318232
#306318222
#306318202
#306318192
#306316902
#306316822
#306316522
#449×C2512
#3008251