Finite Group Databasebeta

← #44561851

Subgroups — #44561851

Order 32128.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548742C211yesmaximal, characteristic, complemented
2C2×D8×D251C211yesmaximal, characteristic, complemented
3#30632330C211yesmaximal, characteristic, complemented
4D4×D1004C211yesmaximal, characteristic, complemented
5#30632077C211yesmaximal, characteristic, complemented
6#30631939C211yesmaximal, characteristic, complemented
7#30631919C211yesmaximal, characteristic, complemented
8#30631883C211yesmaximal, characteristic, complemented
9#30631836C211yesmaximal, characteristic, complemented
10#30631835C211yesmaximal, characteristic, complemented
11#30631822C211yesmaximal, characteristic, complemented
12#30631819C211yesmaximal, characteristic, complemented
13#30631797C211yesmaximal, characteristic, complemented
14#30631757C211yesmaximal, characteristic, complemented
15#463×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic, complemented
17C22×D1004—21no—
18C2×D4×D251C2211yescharacteristic
19C2×D4×D251C2211yescharacteristic, complemented
20C2×D4×D251C2211yescharacteristic, complemented
21#22548742—24no—
22C2×#16860797C2211yescharacteristic, complemented
23C2×#16860797C2211yescharacteristic, complemented
24C2×D2008C2211yescharacteristic, complemented
25C2×D2008C2211yescharacteristic, complemented
26C2×#16860795C2211yescharacteristic
27C2×#16860795C2211yescharacteristic, complemented
28C2×#16860795C2211yescharacteristic, complemented
29D8×D251—24no—
30C2×#16860793C2211yescharacteristic, complemented
31C2×C8×D251C2211yescharacteristic, complemented
32#22548712C2211yescharacteristic, complemented
33#22548697—21no—
34#22548691C2211yescharacteristic, complemented
35#22548690C2211yescharacteristic
36#22548690C2211yescharacteristic, complemented
37#22548689—21no—
38#22548688—21no—
39#22548686C2211yescharacteristic, complemented
40#22548685—21no—
41#22548685—21no—
42#22548685—21no—
43C4×D1004C2211yescharacteristic
44C4×D1004C2211yescharacteristic, complemented
45C22⋊C4×D251—21no—
46#22548646C2211yescharacteristic, complemented
47#22548637C2211yescharacteristic, complemented
48#22548637C2211yescharacteristic, complemented
49#22548636C2211yescharacteristic, complemented
50#22548635C2211yescharacteristic, complemented
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53#22548626C2211yescharacteristic, complemented
54C2×SD16×C251C2211yescharacteristic, complemented
55C2×D8×C251C2211yescharacteristic
56#124×C251C2211yescharacteristic, complemented
57C4×D4×C251C2211yescharacteristic, complemented
58C4⋊C8×C251C2211yescharacteristic, complemented
59#103×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C23×D251—21no—
63C23×D251—21no—
64#16860811—22no—
65#16860811—41no—
66C2×D1004—21no—
67C2×D1004—21no—
68C2×D1004—21no—
69C2×D1004—21no—
70C2×D1004—21no—
71C2×D1004—44no—
72C2×D1004D411yescharacteristic
73C2×D1004C2311yescharacteristic
74C2×D1004C2311yescharacteristic
75C2×D1004C2311yescharacteristic
76C2×C22⋊D251—21no—
77C2×C22⋊D251—21no—
78C2×C22⋊D251—21no—
79C2×C22⋊D251—21no—
80D4×D251—22no—
81D4×D251—22no—
82D4×D251—22no—
83D4×D251—41no—
84D4×D251—41no—
85D4×D251—41no—
86C2×C4×D251—21no—
87C2×C4×D251—21no—
88C2×C4×D251C2311yescharacteristic
89C2×C4×D251C2311yescharacteristic
90#16860797—22no—
91#16860797—41no—
92D2008—22no—
93D2008—22no—
94#16860795—22no—
95#16860795—41no—
96#16860795D414yescomplemented
97#16860793—22no—
98C2×C251⋊C8C2311yescharacteristic
99C2×C251⋊C8C2311yescharacteristic
100C8×D251—22no—
101C4⋊Dic251D411yescharacteristic
102C4⋊Dic251C2311yescharacteristic
103Dic251⋊C4—21no—
104C22⋊Dic251—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×C251C2311yescharacteristic
113C2×D4×C251C2311yescharacteristic
114C22×C4×C251—21no—
115SD16×C251—22no—
116D8×C251—22no—
117C2×C8×C251C2311yescharacteristic
118C2×C8×C251C2311yescharacteristic
119C4⋊C4×C251C2311yescharacteristic
120C22⋊C4×C251—21no—
121C22⋊C4×C251—21no—
122C22⋊C4×C251—21no—
123C42×C251C2311yescharacteristic
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—21no—
127C22×D251—42no—
128C22×D251—42no—
129C22×D251—42no—
130C22×D251—42no—
131C22×D251—42no—
132C22×D251—42no—
133C22×D251—81no—
134C22×D251—81no—
135C22×D251—81no—
136C22×D251C2×D411yescharacteristic
137C22×D251C2×D411yescharacteristic
138D1004—22no—
139D1004—21no—
140D1004—21no—
141D1004—21no—
142D1004—21no—
143D1004—21no—
144D1004—42no—
145D1004—42no—
146D1004—42no—
147D1004—42no—
148D1004—81no—
149D1004C2×D412yescomplemented
150C22⋊D251—42no—
151C22⋊D251—42no—
152C22⋊D251—42no—
153C22⋊D251—81no—
154C2×Dic251—21no—
155C2×Dic251C2×D411yescharacteristic
156C2×Dic251C2×D411yescharacteristic
157C4×D251—21no—
158C4×D251—21no—
159C4×D251—21no—
160C4×D251—21no—
161C4×D251—42no—
162C4×D251—81no—
163C251⋊C8—21no—
164C251⋊C8C2×D412yescomplemented
165C23×C251—21no—
166C23×C251—21no—
167Q8×C251—21no—
168Q8×C251—21no—
169D4×C251—21no—
170D4×C251—21no—
171D4×C251—21no—
172D4×C251C2×D412yes—
173C2×C4×C251—21no—
174C2×C4×C251—21no—
175C2×C4×C251—42no—
176C2×C4×C251C2×D411yescharacteristic
177C2×C4×C251C2×D411yescharacteristic
178C2×C4×C251C2411yescharacteristic
179C8×C251—21no—
180C8×C251—21no—
181C22×C251—22no—
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—41no—
197C2×D251—41no—
198C2×D251—41no—
199C2×D251—41no—
200C2×D251—41no—
201C2×D251—41no—
202C2×D251—84no—
203C2×D251—84no—
204C2×D251—84no—
205C2×D251AGL(1,ℤ/8)12yescomplemented
206C4×C251—21no—
207C4×C251—21no—
208C4×C251—41no—
209C4×C251—41no—
210C4×C251C22×D411yescharacteristic
211C4×C251H(5,𝔽2)11yescharacteristic
212Dic251—21no—
213Dic251—21no—
214Dic251—41no—
215C2×C251—42no—
216C2×C251—81no—
217C2×C251D4211yescharacteristic
218C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C251#57611yescharacteristic
220D251—22no—
221D251—41no—
222D251—42no—
223D251—81no—
224D251—81no—
225C251#326411yescharacteristic, complemented, Sylow
226#3264—2511nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2511no—
228C22×D8—2511no—
229#546—2511no—
230D42—2511no—
231#496—2511no—
232#469—2511no—
233#465—2511no—
234#463D25111yescharacteristic, complemented
235#459—2511no—
236#450—2511no—
237#449—2511no—
238#447—2511no—
239#447—2511no—
240#442—2511no—
241#435—2511no—
242C2×#41—2511no—
243C22×D4—5021no—
244C22×D4—2511no—
245C22×D4—2511no—
246C22×D4—2511no—
247AGL(1,ℤ/8)—5024no—
248C2×SD16—2511no—
249C2×SD16—2511no—
250C2×SD16C2×D25111yescharacteristic, complemented
251C2×D8—5024no—
252C2×D8—2511no—
253C2×D8—2511no—
254C2×D8—2511no—
255C2×D8—2511no—
256C2×D8—2511no—
257C2×D8C2×D25111yescharacteristic
258C2×M16—2511no—
259C22×C8—2511no—
260#127—2511no—
261#124C2×D25111yescharacteristic, complemented
262#123—5021no—
263C22⋊Q8—2511no—
264#121—5021no—
265#121—5021no—
266#121—2511no—
267#121—2511no—
268#121—2511no—
269#120—5021no—
270#120—5021no—
271#120—5021no—
272C4×D4—2511no—
273C4×D4—2511no—
274C4×D4C2×D25111yescharacteristic, complemented
275C2×C22⋊C4—5021no—
276#106—2511no—
277C4⋊C8C2×D25111yescharacteristic, complemented
278#103C2×D25111yescharacteristic, complemented
279#102—2511no—
280#102—2511no—
281#102—2511no—
282#102—2511no—
283#102C2×D25111yescharacteristic, complemented
284C22⋊C8—2511no—
285C22⋊C8—2511no—
286#97—2511no—
287C24—5021no—
288C24—5021no—
289C24—5021no—
290#41—10041no—
291#41—5022no—
292C2×Q8C22×D25111yescharacteristic
293C2×D4—10044no—
294C2×D4—10041no—
295C2×D4—10041no—
296C2×D4—10041no—
297C2×D4—5021no—
298C2×D4—5021no—
299C2×D4—5021no—
300C2×D4—5022no—
301C2×D4—5021no—
302C2×D4—5022no—
303C2×D4—5022no—
304C2×D4—5021no—
305C2×D4—5021no—
306C2×D4—5021no—
307C2×D4—5021no—
308C2×D4—5021no—
309C2×D4—2511no—
310C2×D4—2511no—
311C2×D4—2511no—
312C2×D4—2511no—
313C2×D4C22×D25111yescharacteristic
314C2×D4C22×D25111yescharacteristic
315C22×C4—21no—
316C22×C4—5021no—
317C22×C4—5021no—
318C22×C4—2511no—
319C22×C4—2511no—
320SD16—22no—
321SD16—10041no—
322SD16—5022no—
323D8—22no—
324D8—10041no—
325D8—5022no—
326D8—5022no—
327D8—5022no—
328D8—2514no—
329M16—5022no—
330C2×C8—5022no—
331C2×C8—2511no—
332C2×C8—2511no—
333C2×C8C22×D25111yescharacteristic
334C2×C8C22×D25111yescharacteristic
335C4⋊C4—5021no—
336C4⋊C4—2511no—
337C4⋊C4—2511no—
338C4⋊C4C22×D25111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—5021no—
343C22⋊C4—5021no—
344C22⋊C4—5021no—
345C22⋊C4—5021no—
346C22⋊C4—5021no—
347C22⋊C4—2511no—
348C22⋊C4—2511no—
349C42C22×D25111yescharacteristic
350C23—21no—
351C23—21no—
352C23—20081no—
353C23—20081no—
354C23—20081no—
355C23—10042no—
356C23—10042no—
357C23—10042no—
358C23—10042no—
359C23—10042no—
360C23—10042no—
361C23—5021no—
362C23—5021no—
363C23—5021no—
364C23—2511no—
365C23—2511no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—20081no—
372D4—20081no—
373D4—10042no—
374D4—10042no—
375D4—10042no—
376D4—10042no—
377D4—10042no—
378D4—10042no—
379D4—10042no—
380D4—5022no—
381D4—5021no—
382D4—5021no—
383D4—5021no—
384D4—5021no—
385D4—5021no—
386D4—2512no—
387D4D4×D25112yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—20081no—
392C2×C4—10042no—
393C2×C4—5021no—
394C2×C4—5021no—
395C2×C4—5021no—
396C2×C4—5021no—
397C2×C4—5021no—
398C2×C4—2511no—
399C2×C4—2511no—
400C2×C4D4×D25111yescharacteristic
401C2×C4D4×D25111yescharacteristic
402C2×C4C23×D25111yescharacteristic
403C8—21no—
404C8—21no—
405C8—5021no—
406C8—2512no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—20084no—
414C22—20084no—
415C22—20084no—
416C22—10041no—
417C22—10041no—
418C22—10041no—
419C22—10041no—
420C22—10041no—
421C22—10041no—
422C22—10041no—
423C22—10041no—
424C22—5021no—
425C22—5021no—
426C22—5022no—
427C22—5021no—
428C22—5021no—
429C22—5021no—
430C22—2512no—
431C22C2×D4×D25111yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—10041no—
437C4—5021no—
438C4—5021no—
439C4C2×D4×D25111yescharacteristic
440C4#2254876111yescharacteristic
441C2—42no—
442C2—81no—
443C2—20081no—
444C2—20081no—
445C2—10041no—
446C2—10042no—
447C2—5022no—
448C2#3063232111yescharacteristic
449C2AGL(1,ℤ/8)×D25111yescharacteristic
450C2#3063248911yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#463×C251C2
socleC22×C251C22×D4
hypercenter#463D251
hypocenterC251#3264
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22548742C2semidirect
2. C2×D8×D251C2semidirect
3. #30632330C2semidirect
4. D4×D1004C2semidirect
5. #30632077C2semidirect
6. #30631939C2semidirect
7. #30631919C2semidirect
8. #30631883C2semidirect
9. #30631836C2semidirect
10. #30631835C2semidirect
11. #30631822C2semidirect
12. #30631819C2semidirect
13. #30631797C2semidirect
14. #30631757C2semidirect
15. #463×C251C2semidirect
16. C2×#16860811C22semidirect
19. C2×D4×D251C22semidirect
20. C2×D4×D251C22semidirect
22. C2×#16860797C22semidirect
23. C2×#16860797C22semidirect
24. C2×D2008C22semidirect
25. C2×D2008C22semidirect
27. C2×#16860795C22semidirect
28. C2×#16860795C22semidirect
30. C2×#16860793C22semidirect
31. C2×C8×D251C22semidirect
32. #22548712C22semidirect
34. #22548691C22semidirect
36. #22548690C22semidirect
39. #22548686C22semidirect
44. C4×D1004C22semidirect
46. #22548646C22semidirect
47. #22548637C22semidirect
48. #22548637C22semidirect
49. #22548636C22semidirect
50. #22548635C22semidirect
51. (C2×D251)⋊C8C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. #22548626C22semidirect
54. C2×SD16×C251C22semidirect
56. #124×C251C22semidirect
57. C4×D4×C251C22semidirect
58. C4⋊C8×C251C22semidirect
59. #103×C251C22semidirect
60. #102×C251C22semidirect
96. #16860795D4semidirect
149. D1004C2×D4semidirect
164. C251⋊C8C2×D4semidirect
205. C2×D251AGL(1,ℤ/8)semidirect
225. C251#3264semidirect
234. #463D251semidirect
250. C2×SD16C2×D251semidirect
261. #124C2×D251semidirect
274. C4×D4C2×D251semidirect
277. C4⋊C8C2×D251semidirect
278. #103C2×D251semidirect
283. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487422
C2×D8×D2512
#306323302
D4×D10042
#306320772
#306319392
#306319192
#306318832
#306318362
#306318352
#306318222
#306318192
#306317972
#306317572
#463×C2512
#3264251