Finite Group Databasebeta

← #44561859

Subgroups — #44561859

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×#22548739C211yesmaximal, characteristic, complemented
2C2×SD16×D251C211yesmaximal, characteristic, complemented
3#30632345C211yesmaximal, characteristic, complemented
4#30632322C211yesmaximal, characteristic, complemented
5#30631930C211yesmaximal, characteristic, complemented
6#30631920C211yesmaximal, characteristic, complemented
7#30631899C211yesmaximal, characteristic, complemented
8#30631884C211yesmaximal, characteristic, complemented
9#30631836C211yesmaximal, characteristic, complemented
10#30631829C211yesmaximal, characteristic, complemented
11#30631824C211yesmaximal, characteristic, complemented
12#30631819C211yesmaximal, characteristic, complemented
13#30631789C211yesmaximal, characteristic, complemented
14#30631760C211yesmaximal, characteristic, complemented
15#496×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic, complemented
17C2×Q8×D251C2211yescharacteristic, complemented
18C2×D4×D251—21no—
19C2×D4×D251C2211yescharacteristic
20C2×D4×D251C2211yescharacteristic
21#22548739—24no—
22C2×#16860799C2211yescharacteristic
23C2×#16860798C2211yescharacteristic, complemented
24C2×#16860798C2211yescharacteristic, complemented
25C2×#16860797C2211yescharacteristic
26SD16×D251—24no—
27C2×#16860795C2211yescharacteristic
28C2×#16860793C2211yescharacteristic, complemented
29C2×C8×D251C2211yescharacteristic, complemented
30#22548701—21no—
31#22548692—21no—
32#22548691C2211yescharacteristic, complemented
33#22548688—21no—
34#22548686—21no—
35#22548686C2211yescharacteristic
36#22548686C2211yescharacteristic
37#22548686C2211yescharacteristic, complemented
38#22548683—21no—
39#22548683—21no—
40C4×Dic502C2211yescharacteristic, complemented
41C4×D1004C2211yescharacteristic
42C4×D1004C2211yescharacteristic
43(C22⋊D251)⋊C4—21no—
44#22548649C2211yescharacteristic, complemented
45C4⋊(C251⋊C8)C2211yescharacteristic, complemented
46#22548640C2211yescharacteristic, complemented
47#22548639C2211yescharacteristic, complemented
48#22548637C2211yescharacteristic
49#22548636C2211yescharacteristic, complemented
50#22548636C2211yescharacteristic, complemented
51#22548636C2211yescharacteristic, complemented
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53(C2×D251)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C251C2211yescharacteristic, complemented
55C2×SD16×C251C2211yescharacteristic, complemented
56C2×D8×C251C2211yescharacteristic, complemented
57C2×D8×C251C2211yescharacteristic, complemented
58#127×C251C2211yescharacteristic
59#124×C251C2211yescharacteristic, complemented
60#97×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C23×D251—21no—
63#16860809—22no—
64#16860809—41no—
65C2×Dic502—21no—
66C2×Dic502C2311yescharacteristic
67Q8×D251—22no—
68Q8×D251—41no—
69C2×D1004D411yescharacteristic
70C2×D1004C2311yescharacteristic
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76C2×C22⋊D251—21no—
77D4×D251—22no—
78D4×D251—22no—
79D4×D251—41no—
80D4×D251—41no—
81D4×D251—44no—
82C22×Dic251—21no—
83C2×C4×D251—21no—
84C2×C4×D251—21no—
85C2×C4×D251C2311yescharacteristic
86C2×C4×D251C2311yescharacteristic
87#16860799D414yescomplemented
88#16860798—22no—
89#16860798—22no—
90#16860797—22no—
91#16860795—22no—
92#16860793—22no—
93C2×C251⋊C8C2311yescharacteristic
94C2×C251⋊C8C2311yescharacteristic
95C8×D251—22no—
96C4⋊Dic251D411yescharacteristic
97C4⋊Dic251C2311yescharacteristic
98C4⋊Dic251C2311yescharacteristic
99Dic251⋊C4—21no—
100Dic251⋊C4—21no—
101C22⋊Dic251—21no—
102C22⋊Dic251—21no—
103C22⋊Dic251—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4D411yescharacteristic
107(C2×D251)⋊C4D411yescharacteristic
108C4×Dic251—21no—
109C2×Q8×C251C2311yescharacteristic
110C2×D4×C251—21no—
111C2×D4×C251—21no—
112C2×D4×C251C2311yescharacteristic
113C2×D4×C251C2311yescharacteristic
114C2×D4×C251C2311yescharacteristic
115SD16×C251—22no—
116SD16×C251—41no—
117D8×C251—22no—
118D8×C251—41no—
119C2×C8×C251C2311yescharacteristic
120C2×C8×C251C2311yescharacteristic
121C22⋊C4×C251—21no—
122C22⋊C4×C251—21no—
123C42×C251C2311yescharacteristic
124C22×D251—21no—
125C22×D251—42no—
126C22×D251—42no—
127C22×D251—42no—
128C22×D251—42no—
129C22×D251—81no—
130C22×D251—81no—
131C22×D251C2×D411yescharacteristic
132C22×D251C2×D411yescharacteristic
133Dic502—21no—
134Dic502—42no—
135Dic502C2×D412yescomplemented
136D1004—22no—
137D1004—21no—
138D1004C2×D412yes—
139C22⋊D251—42no—
140C22⋊D251—42no—
141C22⋊D251—42no—
142C22⋊D251—42no—
143C22⋊D251—42no—
144C22⋊D251—42no—
145C2×Dic251—21no—
146C2×Dic251—21no—
147C2×Dic251—21no—
148C2×Dic251—42no—
149C2×Dic251C2×D411yescharacteristic
150C2×Dic251C2×D411yescharacteristic
151C4×D251—21no—
152C4×D251—21no—
153C4×D251—21no—
154C4×D251—21no—
155C4×D251—42no—
156C4×D251—42no—
157C251⋊C8—21no—
158C251⋊C8—21no—
159C23×C251—21no—
160C23×C251—21no—
161C23×C251—21no—
162Q8×C251—21no—
163Q8×C251—21no—
164D4×C251—21no—
165D4×C251—21no—
166D4×C251—21no—
167D4×C251—21no—
168D4×C251—21no—
169D4×C251—21no—
170D4×C251—42no—
171D4×C251—81no—
172C2×C4×C251—21no—
173C2×C4×C251C2×D411yescharacteristic
174C2×C4×C251C2×D411yescharacteristic
175C2×C4×C251C2411yescharacteristic
176C8×C251—21no—
177C8×C251C2×D412yescomplemented
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251C22×D411yescharacteristic
188C2×D251—21no—
189C2×D251—21no—
190C2×D251—21no—
191C2×D251—22no—
192C2×D251—21no—
193C2×D251—21no—
194C2×D251—41no—
195C2×D251—41no—
196C2×D251—84no—
197C2×D251—84no—
198C2×D251AGL(1,ℤ/8)12yescomplemented
199C4×C251—21no—
200C4×C251—21no—
201C4×C251—41no—
202C4×C251C22×D411yescharacteristic
203C4×C251H(5,𝔽2)11yescharacteristic
204Dic251—21no—
205Dic251—21no—
206Dic251—22no—
207Dic251—41no—
208Dic251—41no—
209C2×C251—81no—
210C2×C251—81no—
211C2×C251—81no—
212C2×C251D4211yescharacteristic
213C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C251#57711yescharacteristic
215D251—22no—
216D251—41no—
217D251—42no—
218C251#326511yescharacteristic, complemented, Sylow
219#3265—2511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2511no—
221C22×SD16—2511no—
222#548—2511no—
223D42—2511no—
224#496D25111yescharacteristic, complemented
225#468—2511no—
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×SD16C2×D25111yescharacteristic, complemented
247C2×SD16C2×D25111yescharacteristic, complemented
248C2×D8—2511no—
249C2×D8C2×D25111yescharacteristic, complemented
250C2×D8C2×D25111yescharacteristic, complemented
251C2×M16—2511no—
252C22×C8—2511no—
253#127C2×D25111yescharacteristic
254#124—5021no—
255#124C2×D25111yescharacteristic, complemented
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×Q8—2511no—
266C4×D4—5021no—
267C4×D4—2511no—
268C4×D4—2511no—
269#107—2511no—
270C4⋊C8—2511no—
271#103—2511no—
272#103—2511no—
273#102—2511no—
274#102—2511no—
275#102—2511no—
276#102—2511no—
277C22⋊C8—2511no—
278C22⋊C8—2511no—
279#97C2×D25111yescharacteristic, complemented
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—5022no—
295C2×D4—5022no—
296C2×D4—5021no—
297C2×D4—5021no—
298C2×D4—5021no—
299C2×D4—5021no—
300C2×D4—5021no—
301C2×D4—5021no—
302C2×D4—2511no—
303C2×D4—2511no—
304C2×D4C22×D25111yescharacteristic
305C2×D4C22×D25111yescharacteristic
306C2×D4C22×D25111yescharacteristic
307C22×C4—5021no—
308C22×C4—5021no—
309C22×C4—5021no—
310C22×C4—2511no—
311C22×C4—2511no—
312SD16—22no—
313SD16—41no—
314SD16—5022no—
315SD16—5022no—
316SD16—5022no—
317SD16—2514no—
318D8—22no—
319D8—41no—
320D8—5022no—
321M16—5022no—
322C2×C8—5022no—
323C2×C8—2511no—
324C2×C8—2511no—
325C2×C8C22×D25111yescharacteristic
326C2×C8C22×D25111yescharacteristic
327C4⋊C4—5021no—
328C4⋊C4—5021no—
329C4⋊C4—2511no—
330C4⋊C4—2511no—
331C4⋊C4—2511no—
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—
342C42C22×D25111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—20081no—
347C23—20081no—
348C23—10042no—
349C23—10042no—
350C23—10042no—
351C23—10042no—
352C23—5021no—
353C23—2511no—
354C23—2511no—
355Q8—21no—
356Q8—21no—
357Q8—10042no—
358Q8—5021no—
359Q8—2512no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—10042no—
369D4—10042no—
370D4—10042no—
371D4—10042no—
372D4—10042no—
373D4—10042no—
374D4—5022no—
375D4—5021no—
376D4—2512no—
377C2×C4—21no—
378C2×C4—10042no—
379C2×C4—10042no—
380C2×C4—10042no—
381C2×C4—5021no—
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—5021no—
395C8—5021no—
396C8D4×D25112yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—20084no—
407C22—20084no—
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—41no—
421C4—10041no—
422C4—10041no—
423C4—5021no—
424C4—5022no—
425C4—5021no—
426C4C2×D4×D25111yescharacteristic
427C4#2254876011yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—10042no—
432C2—10041no—
433C2—5022no—
434C2#3063232211yescharacteristic
435C2AGL(1,ℤ/8)×D25111yescharacteristic
436C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#496×C251C2
socleC22×C251C22×D4
hypercenter#496D251
hypocenterC251#3265
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. C2×SD16×D251C2semidirect
3. #30632345C2semidirect
4. #30632322C2semidirect
5. #30631930C2semidirect
6. #30631920C2semidirect
7. #30631899C2semidirect
8. #30631884C2semidirect
9. #30631836C2semidirect
10. #30631829C2semidirect
11. #30631824C2semidirect
12. #30631819C2semidirect
13. #30631789C2semidirect
14. #30631760C2semidirect
15. #496×C251C2semidirect
16. C2×#16860809C22semidirect
17. C2×Q8×D251C22semidirect
23. C2×#16860798C22semidirect
24. C2×#16860798C22semidirect
28. C2×#16860793C22semidirect
29. C2×C8×D251C22semidirect
32. #22548691C22semidirect
37. #22548686C22semidirect
40. C4×Dic502C22semidirect
44. #22548649C22semidirect
45. C4⋊(C251⋊C8)C22semidirect
46. #22548640C22semidirect
47. #22548639C22semidirect
49. #22548636C22semidirect
50. #22548636C22semidirect
51. #22548636C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. (C2×D251)⋊C8C22semidirect
54. C2×SD16×C251C22semidirect
55. C2×SD16×C251C22semidirect
56. C2×D8×C251C22semidirect
57. C2×D8×C251C22semidirect
59. #124×C251C22semidirect
60. #97×C251C22semidirect
87. #16860799D4semidirect
135. Dic502C2×D4semidirect
177. C8×C251C2×D4semidirect
198. C2×D251AGL(1,ℤ/8)semidirect
218. C251#3265semidirect
224. #496D251semidirect
246. C2×SD16C2×D251semidirect
247. C2×SD16C2×D251semidirect
249. C2×D8C2×D251semidirect
250. C2×D8C2×D251semidirect
255. #124C2×D251semidirect
279. #97C2×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×#225487392
C2×SD16×D2512
#306323452
#306323222
#306319302
#306319202
#306318992
#306318842
#306318362
#306318292
#306318242
#306318192
#306317892
#306317602
#496×C2512
#3265251