Finite Group Databasebeta

← #44559664

Subgroups — #44559664

Order 32128.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 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×#22548734C211yesmaximal, characteristic, complemented
4#30632266C211yesmaximal, characteristic, complemented
5#30631851C211yesmaximal, characteristic, complemented
6#30631850C211yesmaximal, characteristic, complemented
7#30631839C211yesmaximal, characteristic, complemented
8#30631838C211yesmaximal, characteristic, complemented
9#30631830C211yesmaximal, characteristic, complemented
10#30631829C211yesmaximal, characteristic, complemented
11#30631819C211yesmaximal, characteristic, complemented
12#30631685C211yesmaximal, characteristic, complemented
13#102×D251C211yesmaximal, characteristic, complemented
14#30631652C211yesmaximal, characteristic, complemented
15#447×C251C211yesmaximal, characteristic, complemented
16#22548760—24no—
17#22548760—24no—
18C2×#16860810C2211yescharacteristic
19C2×#16860810C2211yescharacteristic
20C2×#16860809—21no—
21C2×#16860809C2211yescharacteristic
22C2×#16860809C2211yescharacteristic
23C2×#16860809C2211yescharacteristic, complemented
24C22×C22⋊D251—21no—
25C22×C22⋊D251—21no—
26C2×D4×D251—21no—
27C2×D4×D251C2211yescharacteristic
28C2×D4×D251C2211yescharacteristic
29#22548739—24no—
30#22548734—24no—
31C2×Dic1004C2211yescharacteristic, complemented
32C2×#16860799C2211yescharacteristic, complemented
33C2×#16860798C2211yescharacteristic, complemented
34C2×#16860798C2211yescharacteristic, complemented
35C2×#16860798C2211yescharacteristic, complemented
36C2×#16860795C2211yescharacteristic, complemented
37C2×#16860793C2211yescharacteristic, complemented
38C2×C8×D251C2211yescharacteristic, complemented
39#22548695C2211yescharacteristic, complemented
40#22548694C2211yescharacteristic, complemented
41#22548692—21no—
42#22548688—21no—
43#22548687—21no—
44#22548686—21no—
45#22548686C2211yescharacteristic, complemented
46C4⋊C4×D251C2211yescharacteristic, 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
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—
65#16860810—22no—
66#16860810—22no—
67#16860810—22no—
68#16860810—22no—
69#16860809—28no—
70#16860809—22no—
71#16860809—22no—
72#16860809—22no—
73#16860809—22no—
74#16860809—22no—
75#16860809D414yescomplemented
76C2×Dic502D411yescharacteristic
77C2×Dic502C2311yescharacteristic
78C2×Dic502C2311yescharacteristic
79C2×D1004D411yescharacteristic
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⋊D251D411yescharacteristic
91C2×C22⋊D251D411yescharacteristic
92D4×D251—28no—
93D4×D251—22no—
94D4×D251—22no—
95D4×D251—22no—
96D4×D251D414yescomplemented
97C22×Dic251—21no—
98C22×Dic251—21no—
99C22×Dic251—21no—
100C2×C4×D251—21no—
101C2×C4×D251—21no—
102C2×C4×D251C2311yescharacteristic
103C2×C4×D251C2311yescharacteristic
104Dic1004—22no—
105#16860799—22no—
106#16860798—22no—
107#16860798—22no—
108#16860798—22no—
109#16860795—22no—
110#16860793—22no—
111C2×C251⋊C8C2311yescharacteristic
112C2×C251⋊C8C2311yescharacteristic
113C8×D251—22no—
114C4⋊Dic251—21no—
115C4⋊Dic251C2311yescharacteristic
116Dic251⋊C4—21no—
117Dic251⋊C4—21no—
118C22⋊Dic251—21no—
119C22⋊Dic251—21no—
120(C2×D251)⋊C4—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—81no—
140C22×D251—81no—
141C22×D251—81no—
142C22×D251—81no—
143C22×D251C2×D411yescharacteristic
144C22×D251C2×D411yescharacteristic
145Dic502—21no—
146Dic502—22no—
147Dic502—21no—
148Dic502—21no—
149Dic502C2×D412yes—
150D1004—22no—
151D1004—21no—
152D1004C2×D412yes—
153C22⋊D251—24no—
154C22⋊D251—24no—
155C22⋊D251—24no—
156C22⋊D251—24no—
157C22⋊D251—22no—
158C22⋊D251—24no—
159C22⋊D251—22no—
160C22⋊D251—42no—
161C22⋊D251—81no—
162C2×Dic251—24no—
163C2×Dic251—21no—
164C2×Dic251—24no—
165C2×Dic251—21no—
166C2×Dic251—21no—
167C2×Dic251—42no—
168C2×Dic251C2×D411yescharacteristic
169C2×Dic251C2×D411yescharacteristic
170C4×D251—24no—
171C4×D251—21no—
172C4×D251—21no—
173C4×D251—81no—
174C4×D251C2×D412yes—
175C4×D251C2×D412yes—
176C251⋊C8—21no—
177C251⋊C8—21no—
178C23×C251—24no—
179C23×C251—21no—
180C23×C251—21no—
181C23×C251—21no—
182C23×C251—81no—
183C23×C251—81no—
184C23×C251C2×D411yescharacteristic
185D4×C251—24no—
186D4×C251—21no—
187D4×C251—21no—
188D4×C251—21no—
189D4×C251—21no—
190D4×C251—81no—
191D4×C251C2×D412yes—
192D4×C251C2×D412yes—
193C2×C4×C251—21no—
194C2×C4×C251—21no—
195C2×C4×C251—21no—
196C2×C4×C251C2×D411yescharacteristic
197C2×C4×C251C2411yescharacteristic
198C8×C251—21no—
199C8×C251—21no—
200C22×C251—21no—
201C22×C251—22no—
202C22×C251—22no—
203C22×C251—21no—
204C22×C251—21no—
205C22×C251—41no—
206C22×C251—41no—
207C22×C251—41no—
208C22×C251—41no—
209C22×C251—41no—
210C22×C251—41no—
211C22×C251—41no—
212C22×C251—84no—
213C22×C251C22×D411yescharacteristic
214C2×D251—21no—
215C2×D251—21no—
216C2×D251—21no—
217C2×D251—22no—
218C2×D251—21no—
219C2×D251—21no—
220C2×D251—41no—
221C2×D251—41no—
222C2×D251—84no—
223C2×D251—84no—
224C2×D251#12012yes—
225C4×C251—21no—
226C4×C251—41no—
227C4×C251C22×D411yescharacteristic
228C4×C251C22×D411yescharacteristic
229Dic251—22no—
230Dic251—21no—
231Dic251—41no—
232Dic251—41no—
233Dic251#12012yes—
234C2×C251—42no—
235C2×C251—42no—
236C2×C251—41no—
237C2×C251—81no—
238C2×C251C2×#12011yescharacteristic
239C2×C251#57611yescharacteristic
240C2×C251#57711yescharacteristic
241D251—22no—
242D251—41no—
243D251—42no—
244C251#301211yescharacteristic, complemented, Sylow
245#3012—2511nomaximal, Sylow
246C2×H(5,𝔽2)—2511no—
247C2×AGL(1,ℤ/8)—2511no—
248C2×#135—2511no—
249#537—2511no—
250#452—2511no—
251#452—2511no—
252#450—2511no—
253#450—2511no—
254#449—2511no—
255#449—2511no—
256#447—2511no—
257#447D25111yescharacteristic, complemented
258#417—2511no—
259C2×#102—2511no—
260#408—2511no—
261H(5,𝔽2)—5024no—
262H(5,𝔽2)—5024no—
263C2×#41—5021no—
264C2×#41—2511no—
265C2×#41—2511no—
266C2×#41—2511no—
267C2×#41—2511no—
268C2×#41—2511no—
269C22×D4—5021no—
270C22×D4—5021no—
271C22×D4—5021no—
272C22×D4—2511no—
273C22×D4—2511no—
274C22×D4C2×D25111yescharacteristic
275AGL(1,ℤ/8)—5024no—
276#135—5024no—
277C2×Q16—2511no—
278C2×SD16—2511no—
279C2×SD16—2511no—
280C2×SD16—2511no—
281C2×SD16—2511no—
282C2×D8—2511no—
283C2×D8C2×D25111yescharacteristic, complemented
284C2×D8C2×D25111yescharacteristic, complemented
285C2×M16—2511no—
286C22×C8—2511no—
287C22⋊Q8—5021no—
288C22⋊Q8—2511no—
289C22⋊Q8—2511no—
290#121—5021no—
291#121—5021no—
292#121—5021no—
293#121—2511no—
294#121C2×D25111yescharacteristic, complemented
295C2×C4⋊C4—2511no—
296#103—2511no—
297#103—2511no—
298#102—2511no—
299#102—2511no—
300#102—2511no—
301#102—2511no—
302#102C2×D25111yescharacteristic, complemented
303#102C2×D25111yescharacteristic, complemented
304C22⋊C8—2511no—
305C22⋊C8—2511no—
306C22⋊C8—2511no—
307C22⋊C8C2×D25111yescharacteristic, complemented
308C24—21no—
309C24—5021no—
310C24—5021no—
311#41—5022no—
312#41—5022no—
313#41—5022no—
314#41—5022no—
315#41—5022no—
316#41—5022no—
317#41—5028no—
318#41—5022no—
319#41—5022no—
320#41—5022no—
321#41—2514no—
322C2×Q8—2511no—
323C2×Q8—2511no—
324C2×Q8—2511no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—5028no—
330C2×D4—5022no—
331C2×D4—5021no—
332C2×D4—5021no—
333C2×D4—5028no—
334C2×D4—5028no—
335C2×D4—5021no—
336C2×D4—5021no—
337C2×D4—5021no—
338C2×D4—5022no—
339C2×D4—5022no—
340C2×D4—5021no—
341C2×D4—5021no—
342C2×D4—2511no—
343C2×D4—2511no—
344C2×D4—2514no—
345C2×D4—2511no—
346C2×D4—2511no—
347C2×D4C22×D25111yescharacteristic
348C2×D4C22×D25111yescharacteristic
349C2×D4C22×D25111yescharacteristic
350C22×C4—5021no—
351C22×C4—5021no—
352C22×C4—5021no—
353C22×C4—5021no—
354C22×C4—5021no—
355C22×C4—2511no—
356C22×C4—2511no—
357C22×C4C22×D25111yescharacteristic
358Q16—5022no—
359SD16—5022no—
360SD16—5022no—
361SD16—5022no—
362SD16—5022no—
363D8—22no—
364D8—22no—
365D8—5022no—
366M16—5022no—
367C2×C8—5022no—
368C2×C8—2511no—
369C2×C8—2511no—
370C2×C8C22×D25111yescharacteristic
371C2×C8C22×D25111yescharacteristic
372C4⋊C4—5021no—
373C4⋊C4—5021no—
374C4⋊C4—5021no—
375C4⋊C4—2511no—
376C4⋊C4C22×D25111yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—5021no—
379C22⋊C4—5021no—
380C22⋊C4—5021no—
381C23—24no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—81no—
386C23—81no—
387C23—20081no—
388C23—20081no—
389C23—20081no—
390C23—20081no—
391C23—5024no—
392C23—5024no—
393C23—5021no—
394C23—2511no—
395C23—2511no—
396C23D4×D25111yescharacteristic
397Q8—5022no—
398Q8—5021no—
399Q8—5021no—
400Q8—5021no—
401Q8—2512no—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—21no—
406D4—21no—
407D4—81no—
408D4—20081no—
409D4—10042no—
410D4—5022no—
411D4—5024no—
412D4—5024no—
413D4—5024no—
414D4—5024no—
415D4—5022no—
416D4—5022no—
417D4—5024no—
418D4—5021no—
419D4—2512no—
420D4D4×D25112yes—
421D4D4×D25112yes—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—20081no—
426C2×C4—10042no—
427C2×C4—5024no—
428C2×C4—5021no—
429C2×C4—5021no—
430C2×C4—5024no—
431C2×C4—5021no—
432C2×C4—5024no—
433C2×C4—5021no—
434C2×C4—5021no—
435C2×C4—2511no—
436C2×C4—2511no—
437C2×C4—2512no—
438C2×C4—2512no—
439C2×C4D4×D25111yescharacteristic
440C2×C4C23×D25111yescharacteristic
441C8—21no—
442C8—21no—
443C8—5021no—
444C8—5021no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—84no—
458C22—20084no—
459C22—20084no—
460C22—10041no—
461C22—10041no—
462C22—5021no—
463C22—5021no—
464C22—5021no—
465C22—5022no—
466C22—5021no—
467C22—5021no—
468C22—2512no—
469C22C2×D4×D25111yescharacteristic
470C4—21no—
471C4—41no—
472C4—10041no—
473C4—10041no—
474C4—5022no—
475C4—5021no—
476C4—2512no—
477C4C2×D4×D25111yescharacteristic
478C4C2×D4×D25111yescharacteristic
479C2—41no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2—10042no—
484C2—10041no—
485C2—5022no—
486C2#120×D25111yescharacteristic
487C2#3063248811yescharacteristic
488C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251#3012
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×#22548734C2semidirect
4. #30632266C2semidirect
5. #30631851C2semidirect
6. #30631850C2semidirect
7. #30631839C2semidirect
8. #30631838C2semidirect
9. #30631830C2semidirect
10. #30631829C2semidirect
11. #30631819C2semidirect
12. #30631685C2semidirect
13. #102×D251C2semidirect
14. #30631652C2semidirect
15. #447×C251C2semidirect
23. C2×#16860809C22semidirect
31. C2×Dic1004C22semidirect
32. C2×#16860799C22semidirect
33. C2×#16860798C22semidirect
34. C2×#16860798C22semidirect
35. C2×#16860798C22semidirect
36. C2×#16860795C22semidirect
37. C2×#16860793C22semidirect
38. C2×C8×D251C22semidirect
39. #22548695C22semidirect
40. #22548694C22semidirect
45. #22548686C22semidirect
46. C4⋊C4×D251C22semidirect
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
57. C2×D8×C251C22semidirect
58. C2×D8×C251C22semidirect
59. #121×C251C22semidirect
60. #102×C251C22semidirect
61. #102×C251C22semidirect
62. C22⋊C8×C251C22semidirect
75. #16860809D4semidirect
96. D4×D251D4semidirect
244. C251#3012semidirect
257. #447D251semidirect
283. C2×D8C2×D251semidirect
284. C2×D8C2×D251semidirect
294. #121C2×D251semidirect
302. #102C2×D251semidirect
303. #102C2×D251semidirect
307. 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×#225487342
#306322662
#306318512
#306318502
#306318392
#306318382
#306318302
#306318292
#306318192
#306316852
#102×D2512
#306316522
#447×C2512
#3012251