Finite Group Databasebeta

← #44561855

Subgroups — #44561855

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×#22548742C211yesmaximal, characteristic, complemented
2C2×SD16×D251C211yesmaximal, characteristic, complemented
3#30632349C211yesmaximal, characteristic, complemented
4#30632322C211yesmaximal, characteristic, complemented
5#30632074C211yesmaximal, characteristic, complemented
6#30631932C211yesmaximal, characteristic, complemented
7#30631922C211yesmaximal, characteristic, complemented
8#30631903C211yesmaximal, characteristic, complemented
9#30631839C211yesmaximal, characteristic, complemented
10#30631831C211yesmaximal, characteristic, complemented
11#30631824C211yesmaximal, characteristic, complemented
12#30631822C211yesmaximal, characteristic, complemented
13#30631785C211yesmaximal, characteristic, complemented
14#30631757C211yesmaximal, characteristic, complemented
15#460×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic, complemented
17C2×Q8×D251C2211yescharacteristic, complemented
18C2×D4×D251—21no—
19C2×D4×D251C2211yescharacteristic
20C2×D4×D251C2211yescharacteristic
21#22548742—24no—
22C2×#16860799C2211yescharacteristic
23C2×#16860798C2211yescharacteristic, complemented
24C2×#16860798C2211yescharacteristic, complemented
25C2×#16860797C2211yescharacteristic
26SD16×D251—24no—
27C2×D2008C2211yescharacteristic
28C2×#16860795C2211yescharacteristic, complemented
29C2×#16860795C2211yescharacteristic, complemented
30C2×#16860793C2211yescharacteristic, complemented
31C2×C8×D251C2211yescharacteristic, complemented
32#22548703—21no—
33#22548703C2211yescharacteristic, complemented
34#22548695—21no—
35#22548695C2211yescharacteristic, complemented
36#22548690—21no—
37#22548690C2211yescharacteristic, complemented
38#22548686—21no—
39#22548686C2211yescharacteristic
40#22548686C2211yescharacteristic
41#22548683—21no—
42#22548683—21no—
43C4×D1004—21no—
44C4×D1004C2211yescharacteristic
45C4×D1004C2211yescharacteristic
46#22548648C2211yescharacteristic, complemented
47#22548640C2211yescharacteristic, complemented
48#22548639C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic, complemented
50#22548635C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53#22548626C2211yescharacteristic, complemented
54C2×SD16×C251C2211yescharacteristic, complemented
55C2×SD16×C251C2211yescharacteristic, complemented
56#127×C251C2211yescharacteristic
57C4×Q8×C251C2211yescharacteristic, complemented
58C4⋊C8×C251C2211yescharacteristic, complemented
59#102×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C23×D251—21no—
63#16860811—22no—
64#16860811—41no—
65C2×Dic502—21no—
66C2×Dic502C2311yescharacteristic
67Q8×D251—22no—
68Q8×D251—41no—
69C2×D1004—21no—
70C2×D1004—21no—
71C2×D1004D411yescharacteristic
72C2×D1004C2311yescharacteristic
73C2×D1004C2311yescharacteristic
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76C2×C22⋊D251—21no—
77C2×C22⋊D251—21no—
78D4×D251—22no—
79D4×D251—22no—
80D4×D251—41no—
81D4×D251—41no—
82D4×D251—44no—
83C2×C4×D251—21no—
84C2×C4×D251—21no—
85C2×C4×D251—21no—
86C2×C4×D251C2311yescharacteristic
87C2×C4×D251C2311yescharacteristic
88#16860799—22no—
89#16860798—22no—
90#16860798—41no—
91#16860797D414yescomplemented
92D2008—22no—
93#16860795—22no—
94#16860795—41no—
95#16860793—22no—
96C2×C251⋊C8C2311yescharacteristic
97C2×C251⋊C8C2311yescharacteristic
98C8×D251—22no—
99C4⋊Dic251—21no—
100C4⋊Dic251D411yescharacteristic
101C4⋊Dic251C2311yescharacteristic
102C22⋊Dic251—21no—
103C22⋊Dic251—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4—21no—
108(C2×D251)⋊C4—21no—
109(C2×D251)⋊C4D411yescharacteristic
110(C2×D251)⋊C4D411yescharacteristic
111C2×Q8×C251C2311yescharacteristic
112C2×D4×C251—21no—
113C2×D4×C251—21no—
114C2×D4×C251C2311yescharacteristic
115C2×D4×C251C2311yescharacteristic
116SD16×C251—22no—
117SD16×C251—22no—
118C2×C8×C251C2311yescharacteristic
119C2×C8×C251C2311yescharacteristic
120C4⋊C4×C251—21no—
121C4⋊C4×C251C2311yescharacteristic
122C42×C251—21no—
123C42×C251C2311yescharacteristic
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—42no—
127C22×D251—42no—
128C22×D251—42no—
129C22×D251—42no—
130C22×D251—81no—
131C22×D251—81no—
132C22×D251C2×D411yescharacteristic
133C22×D251C2×D411yescharacteristic
134Dic502—21no—
135Dic502—21no—
136Dic502—42no—
137D1004—22no—
138D1004—21no—
139D1004—21no—
140D1004—21no—
141D1004—42no—
142D1004—42no—
143D1004C2×D412yes—
144C22⋊D251—42no—
145C22⋊D251—42no—
146C22⋊D251—42no—
147C22⋊D251—42no—
148C2×Dic251—21no—
149C2×Dic251—21no—
150C2×Dic251C2×D411yescharacteristic
151C2×Dic251C2×D411yescharacteristic
152C4×D251—21no—
153C4×D251—21no—
154C4×D251—21no—
155C4×D251—21no—
156C4×D251—42no—
157C4×D251—42no—
158C4×D251—42no—
159C251⋊C8—21no—
160C251⋊C8C2×D412yescomplemented
161C23×C251—21no—
162C23×C251—21no—
163Q8×C251—21no—
164Q8×C251C2×D412yescomplemented
165D4×C251—21no—
166D4×C251—21no—
167D4×C251—21no—
168D4×C251—21no—
169D4×C251—42no—
170D4×C251—81no—
171C2×C4×C251—21no—
172C2×C4×C251—21no—
173C2×C4×C251C2×D411yescharacteristic
174C2×C4×C251C2×D411yescharacteristic
175C2×C4×C251C2411yescharacteristic
176C8×C251—21no—
177C8×C251—21no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251C22×D411yescharacteristic
185C2×D251—21no—
186C2×D251—21no—
187C2×D251—21no—
188C2×D251—22no—
189C2×D251—21no—
190C2×D251—21no—
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—22no—
200C4×C251—21no—
201C4×C251—21no—
202C4×C251—41no—
203C4×C251C22×D411yescharacteristic
204C4×C251H(5,𝔽2)11yescharacteristic
205Dic251—21no—
206Dic251—21no—
207Dic251—41no—
208Dic251—41no—
209C2×C251—81no—
210C2×C251—81no—
211C2×C251D4211yescharacteristic
212C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C251#57711yescharacteristic
214D251—22no—
215D251—41no—
216D251—42no—
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#468—2511no—
226#466—2511no—
227#463—2511no—
228#460D25111yescharacteristic, complemented
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×SD16C2×D25111yescharacteristic, complemented
247C2×SD16C2×D25111yescharacteristic, complemented
248C2×D8—2511no—
249C2×D8—2511no—
250C2×D8—2511no—
251C2×M16—2511no—
252C22×C8—2511no—
253#127C2×D25111yescharacteristic
254#124—5021no—
255#124—2511no—
256C22⋊Q8—5021no—
257C22⋊Q8—2511no—
258#121—5021no—
259#121—5021no—
260#121—2511no—
261#121—2511no—
262#121—2511no—
263#120—5021no—
264#120—5021no—
265C4×Q8C2×D25111yescharacteristic, complemented
266C4×D4—5021no—
267C4×D4—2511no—
268C4×D4—2511no—
269#107—2511no—
270C4⋊C8C2×D25111yescharacteristic, complemented
271#103—2511no—
272#103—2511no—
273#102—2511no—
274#102—2511no—
275#102C2×D25111yescharacteristic, complemented
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—21no—
291C2×D4—10044no—
292C2×D4—10041no—
293C2×D4—10041no—
294C2×D4—5021no—
295C2×D4—5021no—
296C2×D4—5022no—
297C2×D4—5022no—
298C2×D4—5021no—
299C2×D4—5021no—
300C2×D4—5021no—
301C2×D4—5021no—
302C2×D4—2511no—
303C2×D4—2511no—
304C2×D4—2511no—
305C2×D4C22×D25111yescharacteristic
306C2×D4C22×D25111yescharacteristic
307C22×C4—5021no—
308C22×C4—5021no—
309C22×C4—5021no—
310C22×C4—2511no—
311C22×C4—2511no—
312SD16—22no—
313SD16—22no—
314SD16—10041no—
315SD16—5022no—
316SD16—5022no—
317SD16—2514no—
318D8—10041no—
319D8—5022no—
320D8—5022no—
321M16—5022no—
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⋊C4—2511no—
331C4⋊C4C22×D25111yescharacteristic
332C22⋊C4—5021no—
333C22⋊C4—5021no—
334C22⋊C4—5021no—
335C22⋊C4—5021no—
336C22⋊C4—5021no—
337C22⋊C4—5021no—
338C22⋊C4—5021no—
339C22⋊C4—2511no—
340C22⋊C4—2511no—
341C42—21no—
342C42C22×D25111yescharacteristic
343C23—21no—
344C23—21no—
345C23—20081no—
346C23—20081no—
347C23—10042no—
348C23—10042no—
349C23—10042no—
350C23—10042no—
351C23—5021no—
352C23—5021no—
353C23—2511no—
354C23—2511no—
355Q8—21no—
356Q8—10042no—
357Q8—5021no—
358Q8—5021no—
359Q8D4×D25112yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—81no—
366D4—10042no—
367D4—10042no—
368D4—10042no—
369D4—10042no—
370D4—10042no—
371D4—10042no—
372D4—5022no—
373D4—5021no—
374D4—5021no—
375D4—5021no—
376D4—2512no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—10042no—
380C2×C4—10042no—
381C2×C4—10042no—
382C2×C4—5021no—
383C2×C4—5021no—
384C2×C4—5021no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4—2511no—
389C2×C4—2511no—
390C2×C4D4×D25111yescharacteristic
391C2×C4D4×D25111yescharacteristic
392C2×C4C23×D25111yescharacteristic
393C8—21no—
394C8—21no—
395C8—5021no—
396C8—2512no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—20084no—
404C22—20084no—
405C22—10041no—
406C22—10041no—
407C22—10041no—
408C22—10041no—
409C22—10041no—
410C22—5021no—
411C22—5021no—
412C22—5021no—
413C22—5022no—
414C22—5021no—
415C22—5021no—
416C22—2512no—
417C22C2×D4×D25111yescharacteristic
418C4—21no—
419C4—21no—
420C4—22no—
421C4—41no—
422C4—10041no—
423C4—10041no—
424C4—5021no—
425C4—5021no—
426C4C2×D4×D25111yescharacteristic
427C4#2254876111yescharacteristic
428C2—81no—
429C2—81no—
430C2—20081no—
431C2—10042no—
432C2—10041no—
433C2—5022no—
434C2#3063232111yescharacteristic
435C2AGL(1,ℤ/8)×D25111yescharacteristic
436C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#460×C251C2
socleC22×C251C22×D4
hypercenter#460D251
hypocenterC251#3265
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#22548742C2semidirect
2. C2×SD16×D251C2semidirect
3. #30632349C2semidirect
4. #30632322C2semidirect
5. #30632074C2semidirect
6. #30631932C2semidirect
7. #30631922C2semidirect
8. #30631903C2semidirect
9. #30631839C2semidirect
10. #30631831C2semidirect
11. #30631824C2semidirect
12. #30631822C2semidirect
13. #30631785C2semidirect
14. #30631757C2semidirect
15. #460×C251C2semidirect
16. C2×#16860811C22semidirect
17. C2×Q8×D251C22semidirect
23. C2×#16860798C22semidirect
24. C2×#16860798C22semidirect
28. C2×#16860795C22semidirect
29. C2×#16860795C22semidirect
30. C2×#16860793C22semidirect
31. C2×C8×D251C22semidirect
33. #22548703C22semidirect
35. #22548695C22semidirect
37. #22548690C22semidirect
46. #22548648C22semidirect
47. #22548640C22semidirect
48. #22548639C22semidirect
49. #22548637C22semidirect
51. (C2×D251)⋊C8C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. #22548626C22semidirect
54. C2×SD16×C251C22semidirect
55. C2×SD16×C251C22semidirect
57. C4×Q8×C251C22semidirect
58. C4⋊C8×C251C22semidirect
59. #102×C251C22semidirect
60. #102×C251C22semidirect
91. #16860797D4semidirect
160. C251⋊C8C2×D4semidirect
164. Q8×C251C2×D4semidirect
198. C2×D251AGL(1,ℤ/8)semidirect
218. C251#3265semidirect
228. #460D251semidirect
246. C2×SD16C2×D251semidirect
247. C2×SD16C2×D251semidirect
265. C4×Q8C2×D251semidirect
270. C4⋊C8C2×D251semidirect
275. #102C2×D251semidirect
276. #102C2×D251semidirect
359. Q8D4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487422
C2×SD16×D2512
#306323492
#306323222
#306320742
#306319322
#306319222
#306319032
#306318392
#306318312
#306318242
#306318222
#306317852
#306317572
#460×C2512
#3265251