Finite Group Databasebeta

← #44561850

Subgroups — #44561850

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×#22548743C211yesmaximal, characteristic, complemented
2C22×D2008C211yesmaximal, characteristic, complemented
3#30632333C211yesmaximal, characteristic, complemented
4D4×D1004C211yesmaximal, characteristic, complemented
5#30632076C211yesmaximal, characteristic, complemented
6#30631940C211yesmaximal, characteristic, complemented
7#30631916C211yesmaximal, characteristic, complemented
8#30631903C211yesmaximal, characteristic, complemented
9#30631883C211yesmaximal, characteristic, complemented
10#30631840C211yesmaximal, characteristic, complemented
11#30631834C211yesmaximal, characteristic, complemented
12#30631823C211yesmaximal, characteristic, complemented
13#30631823C211yesmaximal, characteristic, complemented
14#30631800C211yesmaximal, characteristic, complemented
15#435×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic, complemented
17C22×D1004C2211yescharacteristic
18C22×D1004C2211yescharacteristic, complemented
19C22×D1004C2211yescharacteristic, complemented
20C2×D4×D251—21no—
21#22548743—24no—
22C2×#16860799C2211yescharacteristic, complemented
23C2×#16860799C2211yescharacteristic, complemented
24C2×#16860799C2211yescharacteristic, complemented
25C2×D2008—24no—
26C2×D2008C2211yescharacteristic
27C2×D2008C2211yescharacteristic
28C2×D2008C2211yescharacteristic, complemented
29C2×D2008C2211yescharacteristic, complemented
30C2×D2008C2211yescharacteristic, complemented
31C2×D2008C2211yescharacteristic, complemented
32#22548712C2211yescharacteristic, complemented
33#22548703C2211yescharacteristic, complemented
34#22548700—21no—
35#22548694C2211yescharacteristic, complemented
36#22548690—21no—
37#22548689C2211yescharacteristic
38#22548689C2211yescharacteristic, complemented
39#22548689C2211yescharacteristic, complemented
40#22548688—21no—
41#22548685—21no—
42#22548685—21no—
43#22548683—21no—
44C4×D1004C2211yescharacteristic, complemented
45C4×D1004C2211yescharacteristic, complemented
46C22⋊C4×D251—21no—
47#22548647C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic, complemented
51#22548637C2211yescharacteristic, complemented
52#22548637C2211yescharacteristic, complemented
53#22548637C2211yescharacteristic, complemented
54C2×M16×C251C2211yescharacteristic, complemented
55C22×C8×C251C2211yescharacteristic, complemented
56C4×D4×C251C2211yescharacteristic
57C4⋊C8×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic, complemented
59C22⋊C8×C251C2211yescharacteristic, complemented
60#97×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C23×D251—21no—
63C23×D251—21no—
64#16860810—22no—
65#16860810—41no—
66C2×Dic502C2311yescharacteristic
67C2×D1004—22no—
68C2×D1004—22no—
69C2×D1004—22no—
70C2×D1004—21no—
71C2×D1004—21no—
72C2×D1004—21no—
73C2×D1004—21no—
74C2×D1004—21no—
75C2×D1004—41no—
76C2×D1004—41no—
77C2×D1004—41no—
78C2×D1004C2311yescharacteristic
79C2×D1004C2311yescharacteristic
80C2×D1004C2311yescharacteristic
81C2×D1004C2311yescharacteristic
82C2×D1004C2311yescharacteristic
83C2×C22⋊D251—21no—
84C2×C22⋊D251—21no—
85C2×C22⋊D251—21no—
86C2×C22⋊D251—21no—
87D4×D251—44no—
88C2×C4×D251—21no—
89C2×C4×D251—21no—
90C2×C4×D251—21no—
91#16860799—22no—
92#16860799—22no—
93#16860799—41no—
94D2008—22no—
95D2008—22no—
96D2008—22no—
97D2008—22no—
98D2008—41no—
99D2008D414yescomplemented
100C4⋊Dic251C2311yescharacteristic
101C4⋊Dic251C2311yescharacteristic
102Dic251⋊C4—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)⋊C4—21no—
110(C2×D251)⋊C4—21no—
111C2×D4×C251D411yescharacteristic
112C22×C4×C251C2311yescharacteristic
113C22×C4×C251C2311yescharacteristic
114M16×C251—22no—
115C2×C8×C251—22no—
116C2×C8×C251C2311yescharacteristic
117C2×C8×C251C2311yescharacteristic
118C2×C8×C251C2311yescharacteristic
119C2×C8×C251C2311yescharacteristic
120C4⋊C4×C251D411yescharacteristic
121C22⋊C4×C251D411yescharacteristic
122C22⋊C4×C251D411yescharacteristic
123C42×C251C2311yescharacteristic
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—21no—
127C22×D251—21no—
128C22×D251—21no—
129C22×D251—42no—
130C22×D251—42no—
131C22×D251—42no—
132C22×D251—42no—
133C22×D251—42no—
134C22×D251—42no—
135C22×D251—81no—
136C22×D251—81no—
137C22×D251—81no—
138Dic502—21no—
139Dic502—21no—
140D1004—21no—
141D1004—21no—
142D1004—21no—
143D1004—21no—
144D1004—21no—
145D1004—21no—
146D1004—21no—
147D1004—21no—
148D1004—42no—
149D1004—42no—
150D1004—42no—
151D1004—42no—
152D1004—81no—
153D1004C2×D412yes—
154D1004C2×D412yescomplemented
155C22⋊D251—42no—
156C22⋊D251—42no—
157C22⋊D251—42no—
158C22⋊D251—81no—
159C2×Dic251—21no—
160C2×Dic251—21no—
161C2×Dic251—21no—
162C4×D251—42no—
163C4×D251—42no—
164C4×D251—81no—
165C23×C251C2×D411yescharacteristic
166C23×C251C2×D411yescharacteristic
167D4×C251—22no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251C2×D411yescharacteristic
173C2×C4×C251C2×D411yescharacteristic
174C2×C4×C251C2×D411yescharacteristic
175C2×C4×C251C2×D411yescharacteristic
176C2×C4×C251C2411yescharacteristic
177C8×C251—21no—
178C8×C251—21no—
179C8×C251—21no—
180C8×C251C2×D412yescomplemented
181C22×C251—21no—
182C22×C251—21no—
183C22×C251—21no—
184C22×C251—21no—
185C22×C251—21no—
186C22×C251AGL(1,ℤ/8)12yescomplemented
187C22×C251C22×D411yescharacteristic
188C2×D251—22no—
189C2×D251—22no—
190C2×D251—41no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
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—41no—
203C2×D251—84no—
204C2×D251—84no—
205C2×D251—84no—
206C4×C251—21no—
207C4×C251—21no—
208C4×C251—21no—
209C4×C251—21no—
210C4×C251C22×D411yescharacteristic
211C4×C251H(5,𝔽2)11yescharacteristic
212Dic251—41no—
213Dic251—41no—
214Dic251—41no—
215C2×C251—22no—
216C2×C251—41no—
217C2×C251D4211yescharacteristic
218C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C251#57611yescharacteristic
220D251—42no—
221D251—42no—
222D251—81no—
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#463—2511no—
235#459—2511no—
236#450—2511no—
237#449—2511no—
238#447—2511no—
239#447—2511no—
240#442—2511no—
241#435D25111yescharacteristic, complemented
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×SD16—2511no—
251C2×D8—5024no—
252C2×D8—2511no—
253C2×D8—2511no—
254C2×D8—2511no—
255C2×D8—2511no—
256C2×D8—2511no—
257C2×D8—2511no—
258C2×M16C2×D25111yescharacteristic, complemented
259C22×C8C2×D25111yescharacteristic, complemented
260#127—2511no—
261#124—2511no—
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
275C2×C22⋊C4—5021no—
276#106—2511no—
277C4⋊C8C2×D25111yescharacteristic, complemented
278#103—2511no—
279#102—2511no—
280#102—2511no—
281#102—2511no—
282#102—2511no—
283#102—2511no—
284C22⋊C8C2×D25111yescharacteristic, complemented
285C22⋊C8C2×D25111yescharacteristic, complemented
286#97C2×D25111yescharacteristic, complemented
287C24—5021no—
288C24—5021no—
289C24—5021no—
290#41—10041no—
291#41—5022no—
292C2×Q8—2511no—
293C2×D4—10041no—
294C2×D4—10041no—
295C2×D4—10041no—
296C2×D4—10044no—
297C2×D4—5022no—
298C2×D4—5022no—
299C2×D4—5022no—
300C2×D4—5021no—
301C2×D4—5021no—
302C2×D4—5021no—
303C2×D4—5021no—
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×D4—2511no—
314C2×D4D100411yescharacteristic
315C22×C4—5021no—
316C22×C4—5021no—
317C22×C4—5021no—
318C22×C4C22×D25111yescharacteristic
319C22×C4C22×D25111yescharacteristic
320SD16—10041no—
321SD16—5022no—
322SD16—5022no—
323D8—10041no—
324D8—5022no—
325D8—5022no—
326D8—5022no—
327D8—5022no—
328D8—2514no—
329M16—22no—
330C2×C8—22no—
331C2×C8C22×D25111yescharacteristic
332C2×C8C22×D25111yescharacteristic
333C2×C8C22×D25111yescharacteristic
334C2×C8C22×D25111yescharacteristic
335C4⋊C4—5021no—
336C4⋊C4—2511no—
337C4⋊C4—2511no—
338C4⋊C4D100411yescharacteristic
339C22⋊C4—5021no—
340C22⋊C4—5021no—
341C22⋊C4—5021no—
342C22⋊C4—5021no—
343C22⋊C4—5021no—
344C22⋊C4—5021no—
345C22⋊C4—5021no—
346C22⋊C4—5021no—
347C22⋊C4D100411yescharacteristic
348C22⋊C4D100411yescharacteristic
349C42C22×D25111yescharacteristic
350C23—20081no—
351C23—20081no—
352C23—20081no—
353C23—10042no—
354C23—10042no—
355C23—10042no—
356C23—10042no—
357C23—10042no—
358C23—10042no—
359C23—5021no—
360C23—5021no—
361C23—5021no—
362C23—5021no—
363C23—5021no—
364C23C2×D100411yescharacteristic
365C23C2×D100411yescharacteristic
366Q8—5021no—
367Q8—5021no—
368D4—22no—
369D4—20081no—
370D4—20081no—
371D4—10042no—
372D4—10042no—
373D4—10042no—
374D4—10042no—
375D4—10042no—
376D4—10042no—
377D4—10042no—
378D4—5021no—
379D4—5021no—
380D4—5021no—
381D4—5021no—
382D4—5021no—
383D4—5021no—
384D4—5021no—
385D4—5021no—
386D4—2512no—
387D4—2512no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—20081no—
393C2×C4—10042no—
394C2×C4—10042no—
395C2×C4—5021no—
396C2×C4—5021no—
397C2×C4—5021no—
398C2×C4C2×D100411yescharacteristic
399C2×C4C2×D100411yescharacteristic
400C2×C4C2×D100411yescharacteristic
401C2×C4C2×D100411yescharacteristic
402C2×C4C23×D25111yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D25112yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—20084no—
413C22—20084no—
414C22—20084no—
415C22—10041no—
416C22—10041no—
417C22—10041no—
418C22—10041no—
419C22—10041no—
420C22—10041no—
421C22—10041no—
422C22—10041no—
423C22—10041no—
424C22—10041no—
425C22—10041no—
426C22—10041no—
427C22—10041no—
428C22—5022no—
429C22—5022no—
430C22#2254874312yescomplemented
431C22C22×D100411yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—10041no—
437C4—10041no—
438C4—10041no—
439C4C2×D4×D25111yescharacteristic
440C4#2254876111yescharacteristic
441C2—22no—
442C2—41no—
443C2—20081no—
444C2—20081no—
445C2—20081no—
446C2—10042no—
447C2—10042no—
448C2D4×D100411yescharacteristic
449C2C2×#2254874311yescharacteristic
450C2#3063249211yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#435×C251C2
socleC22×C251C22×D4
hypercenter#435D251
hypocenterC251#3264
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C22×D2008C2semidirect
3. #30632333C2semidirect
4. D4×D1004C2semidirect
5. #30632076C2semidirect
6. #30631940C2semidirect
7. #30631916C2semidirect
8. #30631903C2semidirect
9. #30631883C2semidirect
10. #30631840C2semidirect
11. #30631834C2semidirect
12. #30631823C2semidirect
13. #30631823C2semidirect
14. #30631800C2semidirect
15. #435×C251C2semidirect
16. C2×#16860810C22semidirect
18. C22×D1004C22semidirect
19. C22×D1004C22semidirect
22. C2×#16860799C22semidirect
23. C2×#16860799C22semidirect
24. C2×#16860799C22semidirect
28. C2×D2008C22semidirect
29. C2×D2008C22semidirect
30. C2×D2008C22semidirect
31. C2×D2008C22semidirect
32. #22548712C22semidirect
33. #22548703C22semidirect
35. #22548694C22semidirect
38. #22548689C22semidirect
39. #22548689C22semidirect
44. C4×D1004C22semidirect
45. C4×D1004C22semidirect
47. #22548647C22semidirect
48. #22548640C22semidirect
49. #22548637C22semidirect
50. #22548637C22semidirect
51. #22548637C22semidirect
52. #22548637C22semidirect
53. #22548637C22semidirect
54. C2×M16×C251C22semidirect
55. C22×C8×C251C22semidirect
57. C4⋊C8×C251C22semidirect
58. C22⋊C8×C251C22semidirect
59. C22⋊C8×C251C22semidirect
60. #97×C251C22semidirect
99. D2008D4semidirect
154. D1004C2×D4semidirect
180. C8×C251C2×D4semidirect
186. C22×C251AGL(1,ℤ/8)semidirect
225. C251#3264semidirect
241. #435D251semidirect
258. C2×M16C2×D251semidirect
259. C22×C8C2×D251semidirect
277. C4⋊C8C2×D251semidirect
284. C22⋊C8C2×D251semidirect
285. C22⋊C8C2×D251semidirect
286. #97C2×D251semidirect
406. C8D4×D251semidirect
430. C22#22548743semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C22×D20082
#306323332
D4×D10042
#306320762
#306319402
#306319162
#306319032
#306318832
#306318402
#306318342
#306318232
#306318002
#435×C2512
#3264251