Finite Group Databasebeta

← #44561867

Subgroups — #44561867

Order 32128.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548743C211yesmaximal, characteristic, complemented
2C2×SD16×D251C211yesmaximal, characteristic, complemented
3#30632347C211yesmaximal, characteristic, complemented
4#30632321C211yesmaximal, characteristic, complemented
5#30632073C211yesmaximal, characteristic, complemented
6#30631924C211yesmaximal, characteristic, complemented
7#30631905C211yesmaximal, characteristic, complemented
8#30631886C211yesmaximal, characteristic, complemented
9#30631837C211yesmaximal, characteristic, complemented
10#30631830C211yesmaximal, characteristic, complemented
11#30631827C211yesmaximal, characteristic, complemented
12#30631822C211yesmaximal, characteristic, complemented
13#30631786C211yesmaximal, characteristic, complemented
14#30631756C211yesmaximal, characteristic, complemented
15#468×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic, complemented
17C2×Q8×D251C2211yescharacteristic, complemented
18C22×D1004C2211yescharacteristic
19C2×D4×D251—21no—
20C2×D4×D251C2211yescharacteristic
21#22548743—24no—
22C2×#16860799C2211yescharacteristic
23C2×#16860799C2211yescharacteristic, complemented
24C2×#16860798C2211yescharacteristic, complemented
25C2×#16860797C2211yescharacteristic
26C2×#16860797C2211yescharacteristic, complemented
27SD16×D251—24no—
28C2×D2008C2211yescharacteristic
29C2×D2008C2211yescharacteristic, complemented
30C2×#16860795C2211yescharacteristic, complemented
31C2×C8×D251C2211yescharacteristic, complemented
32#22548711C2211yescharacteristic
33#22548704C2211yescharacteristic, complemented
34#22548701—21no—
35#22548693—21no—
36#22548690C2211yescharacteristic
37#22548690C2211yescharacteristic, complemented
38#22548688—21no—
39#22548688—21no—
40#22548685—21no—
41#22548683—21no—
42#22548680C2211yescharacteristic, complemented
43#22548677—21no—
44#22548677C2211yescharacteristic
45C4×C22⋊D251C2211yescharacteristic
46Dic251⋊C8C2211yescharacteristic, complemented
47#22548638C2211yescharacteristic, complemented
48#22548635C2211yescharacteristic
49#22548635C2211yescharacteristic, complemented
50#22548635C2211yescharacteristic, complemented
51C22⋊(C251⋊C8)C2211yescharacteristic, complemented
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53#22548625C2211yescharacteristic, complemented
54C2×SD16×C251C2211yescharacteristic, complemented
55C2×M16×C251C2211yescharacteristic, complemented
56C22⋊Q8×C251C2211yescharacteristic, complemented
57#121×C251C2211yescharacteristic
58#107×C251C2211yescharacteristic, complemented
59#103×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C23×D251—21no—
63#16860810—22no—
64#16860810—41no—
65C2×Dic502—21no—
66C2×Dic502C2311yescharacteristic
67Q8×D251—22no—
68Q8×D251—41no—
69C2×D1004—22no—
70C2×D1004—21no—
71C2×D1004—21no—
72C2×D1004—21no—
73C2×D1004—41no—
74C2×D1004C2311yescharacteristic
75C2×D1004C2311yescharacteristic
76C2×D1004C2311yescharacteristic
77C2×C22⋊D251—21no—
78C2×C22⋊D251—21no—
79C2×C22⋊D251—21no—
80C2×C22⋊D251—21no—
81C2×C22⋊D251D411yescharacteristic
82D4×D251—22no—
83D4×D251—41no—
84D4×D251—44no—
85C2×C4×D251—21no—
86C2×C4×D251—21no—
87C2×C4×D251—21no—
88C2×C4×D251C2311yescharacteristic
89#16860799—22no—
90#16860799D414yescomplemented
91#16860798—22no—
92#16860797—22no—
93#16860797—41no—
94D2008—22no—
95D2008—22no—
96#16860795—41no—
97C2×C251⋊C8C2311yescharacteristic
98C2×C251⋊C8C2311yescharacteristic
99C8×D251—22no—
100Dic251⋊C4—21no—
101Dic251⋊C4D411yescharacteristic
102C22⋊Dic251D411yescharacteristic
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4—21no—
108(C2×D251)⋊C4D411yescharacteristic
109C4×Dic251—21no—
110C4×Dic251C2311yescharacteristic
111C2×Q8×C251C2311yescharacteristic
112C2×D4×C251—21no—
113C2×D4×C251C2311yescharacteristic
114C22×C4×C251C2311yescharacteristic
115SD16×C251—22no—
116M16×C251—22no—
117C2×C8×C251C2311yescharacteristic
118C2×C8×C251C2311yescharacteristic
119C4⋊C4×C251—21no—
120C4⋊C4×C251C2311yescharacteristic
121C4⋊C4×C251C2311yescharacteristic
122C22⋊C4×C251—21no—
123C22⋊C4×C251—21no—
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—21no—
127C22×D251—42no—
128C22×D251—42no—
129C22×D251—42no—
130C22×D251—42no—
131C22×D251—81no—
132C22×D251—81no—
133C22×D251C2×D411yescharacteristic
134Dic502—21no—
135Dic502—42no—
136Dic502C2×D412yescomplemented
137D1004—21no—
138D1004—21no—
139D1004—21no—
140D1004—21no—
141D1004—21no—
142D1004—42no—
143D1004—42no—
144D1004—42no—
145D1004C2×D412yes—
146C22⋊D251—22no—
147C22⋊D251—42no—
148C22⋊D251—42no—
149C22⋊D251—42no—
150C22⋊D251—81no—
151C2×Dic251—21no—
152C2×Dic251C2×D411yescharacteristic
153C2×Dic251C2×D411yescharacteristic
154C2×Dic251C2×D411yescharacteristic
155C4×D251—21no—
156C4×D251—21no—
157C4×D251—42no—
158C4×D251—42no—
159C4×D251—42no—
160C251⋊C8—21no—
161C251⋊C8—21no—
162C23×C251—21no—
163C23×C251C2×D411yescharacteristic
164Q8×C251—21no—
165Q8×C251—21no—
166D4×C251—21no—
167D4×C251—21no—
168D4×C251—42no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—21no—
173C2×C4×C251—21no—
174C2×C4×C251C2×D411yescharacteristic
175C2×C4×C251C2411yescharacteristic
176C8×C251—21no—
177C8×C251C2×D412yescomplemented
178C22×C251—21no—
179C22×C251—21no—
180C22×C251—21no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251C22×D411yescharacteristic
185C2×D251—21no—
186C2×D251—21no—
187C2×D251—22no—
188C2×D251—41no—
189C2×D251—41no—
190C2×D251—41no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—41no—
195C2×D251—41no—
196C2×D251—84no—
197C2×D251—84no—
198C2×D251AGL(1,ℤ/8)12yescomplemented
199C4×C251—21no—
200C4×C251—41no—
201C4×C251—41no—
202C4×C251—41no—
203C4×C251C22×D411yescharacteristic
204C4×C251H(5,𝔽2)11yescharacteristic
205Dic251—21no—
206Dic251—22no—
207Dic251—21no—
208Dic251—21no—
209C2×C251—41no—
210C2×C251—81no—
211C2×C251D4211yescharacteristic
212C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C251#57711yescharacteristic
214D251—22no—
215D251—42no—
216D251—81no—
217D251—81no—
218C251#326511yescharacteristic, complemented, Sylow
219#3265—2511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2511no—
221C22×SD16—2511no—
222#548—2511no—
223D42—2511no—
224#496—2511no—
225#468D25111yescharacteristic, complemented
226#466—2511no—
227#463—2511no—
228#460—2511no—
229#450—2511no—
230#449—2511no—
231#448—2511no—
232#447—2511no—
233#440—2511no—
234#435—2511no—
235C2×#41—2511no—
236C22×Q8—2511no—
237C22×D4—5021no—
238C22×D4—2511no—
239C22×D4—2511no—
240AGL(1,ℤ/8)—5024no—
241C2×SD16—5024no—
242C2×SD16—2511no—
243C2×SD16—2511no—
244C2×SD16—2511no—
245C2×SD16—2511no—
246C2×SD16—2511no—
247C2×SD16C2×D25111yescharacteristic, complemented
248C2×D8—2511no—
249C2×D8—2511no—
250C2×D8—2511no—
251C2×M16C2×D25111yescharacteristic, complemented
252C22×C8—2511no—
253#127—2511no—
254#124—5021no—
255#124—2511no—
256C22⋊Q8—5021no—
257C22⋊Q8C2×D25111yescharacteristic, complemented
258#121—5021no—
259#121—5021no—
260#121—2511no—
261#121—2511no—
262#121C2×D25111yescharacteristic
263#120—5021no—
264#120—5021no—
265C4×Q8—2511no—
266C4×D4—5021no—
267C4×D4—2511no—
268C4×D4—2511no—
269#107C2×D25111yescharacteristic, complemented
270C4⋊C8—2511no—
271#103—2511no—
272#103C2×D25111yescharacteristic, complemented
273#102—2511no—
274#102—2511no—
275#102—2511no—
276#102C2×D25111yescharacteristic, complemented
277C22⋊C8—2511no—
278C22⋊C8—2511no—
279#97—2511no—
280C24—5021no—
281C24—5021no—
282#41—10041no—
283#41—5022no—
284C2×Q8—10041no—
285C2×Q8—5021no—
286C2×Q8—5022no—
287C2×Q8—2511no—
288C2×Q8C22×D25111yescharacteristic
289C2×D4—21no—
290C2×D4—10041no—
291C2×D4—10044no—
292C2×D4—10041no—
293C2×D4—5021no—
294C2×D4—5021no—
295C2×D4—5022no—
296C2×D4—5022no—
297C2×D4—5021no—
298C2×D4—5021no—
299C2×D4—5021no—
300C2×D4—5021no—
301C2×D4—5021no—
302C2×D4—2511no—
303C2×D4—2511no—
304C2×D4—2511no—
305C2×D4—2511no—
306C2×D4C22×D25111yescharacteristic
307C22×C4—5021no—
308C22×C4—5021no—
309C22×C4—5021no—
310C22×C4—2511no—
311C22×C4C22×D25111yescharacteristic
312SD16—22no—
313SD16—10041no—
314SD16—5022no—
315SD16—5022no—
316SD16—5022no—
317SD16—2514no—
318D8—10041no—
319D8—5022no—
320D8—5022no—
321M16—22no—
322C2×C8—5022no—
323C2×C8—2511no—
324C2×C8—2511no—
325C2×C8C22×D25111yescharacteristic
326C2×C8C22×D25111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—5021no—
329C4⋊C4—2511no—
330C4⋊C4C22×D25111yescharacteristic
331C4⋊C4C22×D25111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—5021no—
335C22⋊C4—5021no—
336C22⋊C4—5021no—
337C22⋊C4—5021no—
338C22⋊C4—5021no—
339C22⋊C4—2511no—
340C22⋊C4—2511no—
341C42—5021no—
342C42—2511no—
343C23—21no—
344C23—20081no—
345C23—20081no—
346C23—10042no—
347C23—10042no—
348C23—10042no—
349C23—10042no—
350C23—5021no—
351C23—5021no—
352C23—5021no—
353C23—2511no—
354C23D4×D25111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—10042no—
358Q8—5021no—
359Q8—2512no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—20081no—
364D4—10042no—
365D4—10042no—
366D4—10042no—
367D4—10042no—
368D4—10042no—
369D4—10042no—
370D4—5022no—
371D4—5021no—
372D4—5021no—
373D4—5021no—
374D4—5021no—
375D4—5021no—
376D4—2512no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—10042no—
383C2×C4—10042no—
384C2×C4—10042no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4—2511no—
389C2×C4—2511no—
390C2×C4—2511no—
391C2×C4D4×D25111yescharacteristic
392C2×C4C23×D25111yescharacteristic
393C8—21no—
394C8—5021no—
395C8—5021no—
396C8D4×D25112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—20084no—
404C22—20084no—
405C22—10041no—
406C22—10041no—
407C22—10041no—
408C22—10041no—
409C22—10041no—
410C22—10041no—
411C22—10041no—
412C22—10041no—
413C22—5022no—
414C22—5021no—
415C22—5021no—
416C22—2512no—
417C22C2×D4×D25111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—5021no—
423C4—5021no—
424C4—5022no—
425C4—5021no—
426C4C2×D4×D25111yescharacteristic
427C4#2254876111yescharacteristic
428C2—41no—
429C2—81no—
430C2—20081no—
431C2—20081no—
432C2—10042no—
433C2—5022no—
434C2#3063232111yescharacteristic
435C2AGL(1,ℤ/8)×D25111yescharacteristic
436C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#468×C251C2
socleC22×C251C22×D4
hypercenter#468D251
hypocenterC251#3265
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C2×SD16×D251C2semidirect
3. #30632347C2semidirect
4. #30632321C2semidirect
5. #30632073C2semidirect
6. #30631924C2semidirect
7. #30631905C2semidirect
8. #30631886C2semidirect
9. #30631837C2semidirect
10. #30631830C2semidirect
11. #30631827C2semidirect
12. #30631822C2semidirect
13. #30631786C2semidirect
14. #30631756C2semidirect
15. #468×C251C2semidirect
16. C2×#16860810C22semidirect
17. C2×Q8×D251C22semidirect
23. C2×#16860799C22semidirect
24. C2×#16860798C22semidirect
26. C2×#16860797C22semidirect
29. C2×D2008C22semidirect
30. C2×#16860795C22semidirect
31. C2×C8×D251C22semidirect
33. #22548704C22semidirect
37. #22548690C22semidirect
42. #22548680C22semidirect
46. Dic251⋊C8C22semidirect
47. #22548638C22semidirect
49. #22548635C22semidirect
50. #22548635C22semidirect
51. C22⋊(C251⋊C8)C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. #22548625C22semidirect
54. C2×SD16×C251C22semidirect
55. C2×M16×C251C22semidirect
56. C22⋊Q8×C251C22semidirect
58. #107×C251C22semidirect
59. #103×C251C22semidirect
60. #102×C251C22semidirect
90. #16860799D4semidirect
136. Dic502C2×D4semidirect
177. C8×C251C2×D4semidirect
198. C2×D251AGL(1,ℤ/8)semidirect
218. C251#3265semidirect
225. #468D251semidirect
247. C2×SD16C2×D251semidirect
251. C2×M16C2×D251semidirect
257. C22⋊Q8C2×D251semidirect
269. #107C2×D251semidirect
272. #103C2×D251semidirect
276. #102C2×D251semidirect
396. C8D4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×SD16×D2512
#306323472
#306323212
#306320732
#306319242
#306319052
#306318862
#306318372
#306318302
#306318272
#306318222
#306317862
#306317562
#468×C2512
#3265251