Finite Group Databasebeta

← #44562204

Subgroups — #44562204

Order 32128.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632379C211yesmaximal, characteristic, complemented
2#30632331C211yesmaximal, characteristic
3#30632040C211yesmaximal, characteristic, complemented
4#30632018C211yesmaximal, characteristic, complemented
5#30632010C211yesmaximal, characteristic, complemented
6#30631989C211yesmaximal, characteristic, complemented
7#30631926C211yesmaximal, characteristic, complemented
8#30631913C211yesmaximal, characteristic, complemented
9#30631907C211yesmaximal, characteristic, complemented
10#30631846C211yesmaximal, characteristic, complemented
11SD16×Dic251C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic, complemented
13#30631680C211yesmaximal, characteristic, complemented
14C2×#22548636C211yesmaximal, characteristic, complemented
15#450×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×#16860800C2211yescharacteristic, complemented
18C22×C251⋊C8C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic, complemented
20#22548710—21no—
21#22548706—21no—
22#22548706C2211yescharacteristic, complemented
23#22548702C2211yescharacteristic
24#22548699—21no—
25#22548696—21no—
26#22548694C2211yescharacteristic
27#22548691C2211yescharacteristic, complemented
28#22548687—21no—
29#22548686C2211yescharacteristic
30#22548684—21no—
31Q8×Dic251C2211yescharacteristic, complemented
32D4×Dic251C2211yescharacteristic
33C4×C22⋊D251C2211yescharacteristic
34#22548673C2211yescharacteristic, complemented
35#22548671—21no—
36C2×C4⋊Dic251C2211yescharacteristic
37C2×C22⋊Dic251—21no—
38#22548648C2211yescharacteristic, complemented
39#22548647C2211yescharacteristic, complemented
40#22548647C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548640C2211yescharacteristic, complemented
44#22548639C2211yescharacteristic, complemented
45#22548638C2211yescharacteristic, complemented
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548636C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C22×D4×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic
55C22⋊Q8×C251C2211yescharacteristic
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59#16860809—22no—
60#16860809—41no—
61C2×Dic502C2311yescharacteristic
62C2×C22⋊D251—21no—
63C2×C22⋊D251—21no—
64C2×C22⋊D251D411yescharacteristic
65C22×Dic251—21no—
66C22×Dic251—21no—
67C22×Dic251—21no—
68C2×C4×D251C2311yescharacteristic
69#16860800—41no—
70C2×C251⋊C8—41no—
71C2×C251⋊C8C2311yescharacteristic
72C2×C251⋊C8C2311yescharacteristic
73C8×D251—41no—
74C4⋊Dic251—21no—
75C4⋊Dic251—21no—
76C4⋊Dic251—21no—
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84C22⋊Dic251—21no—
85C22⋊Dic251—21no—
86C22⋊Dic251—21no—
87C22⋊Dic251—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251D411yescharacteristic
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4D411yescharacteristic
93C4×Dic251—21no—
94C4×Dic251—21no—
95C4×Dic251C2311yescharacteristic
96C24×C251—21no—
97C2×Q8×C251C2311yescharacteristic
98C2×D4×C251—22no—
99C2×D4×C251—21no—
100C2×D4×C251—41no—
101C2×D4×C251C2311yescharacteristic
102C2×D4×C251C2311yescharacteristic
103C22×C4×C251C2311yescharacteristic
104SD16×C251—22no—
105SD16×C251—41no—
106C2×C8×C251C2311yescharacteristic
107C2×C8×C251C2311yescharacteristic
108C4⋊C4×C251—21no—
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C22×D251C2×D411yescharacteristic
112Dic502—21no—
113Dic502—21no—
114C22⋊D251—22no—
115C22⋊D251—42no—
116C22⋊D251—81no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—42no—
122C2×Dic251—42no—
123C2×Dic251—81no—
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C4×D251—21no—
128C4×D251—21no—
129C251⋊C8—21no—
130C251⋊C8—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—42no—
134C23×C251—42no—
135C23×C251—81no—
136C23×C251C2×D411yescharacteristic
137Q8×C251—21no—
138Q8×C251—21no—
139D4×C251—21no—
140D4×C251—21no—
141D4×C251—21no—
142D4×C251—42no—
143D4×C251#4112yes—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—22no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—84no—
161C22×C251#13512yes—
162C22×C251C22×D411yescharacteristic
163C2×D251—21no—
164C2×D251—21no—
165C2×D251—21no—
166C4×C251—21no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251C2×#4111yescharacteristic
170C4×C251H(5,𝔽2)11yescharacteristic
171Dic251—21no—
172Dic251—21no—
173Dic251—21no—
174Dic251—41no—
175Dic251—41no—
176Dic251—41no—
177Dic251—41no—
178C2×C251—22no—
179C2×C251—42no—
180C2×C251—81no—
181C2×C251#54611yescharacteristic
182C2×C251C2×#13511yescharacteristic
183C2×C251#57711yescharacteristic
184D251—41no—
185C251#329611yescharacteristic, complemented, Sylow
186#3296—2511nomaximal, Sylow
187#552—2511no—
188#546—2511no—
189#488—2511no—
190#483—2511no—
191#482—2511no—
192#478—2511no—
193#467—2511no—
194#465—2511no—
195#464—2511no—
196#452—2511no—
197#450D25111yescharacteristic, complemented
198C4×SD16—2511no—
199C8×D4—2511no—
200#416—2511no—
201C2×#102—2511no—
202C2×#41—2511no—
203C22×D4C2×D25111yescharacteristic
204C2×Q16—2511no—
205C2×SD16C2×D25111yescharacteristic
206C2×SD16C2×D25111yescharacteristic
207C22×C8—2511no—
208C22×C8—2511no—
209#126—5021no—
210#125—5021no—
211#125—2511no—
212#124—2511no—
213#123—5021no—
214#123—5021no—
215C22⋊Q8—2511no—
216C22⋊Q8—2511no—
217C22⋊Q8C2×D25111yescharacteristic
218#121—5021no—
219#121—2511no—
220#120—5021no—
221C4×Q8—2511no—
222C4×D4—2511no—
223C4×D4—2511no—
224#117—5021no—
225#117—2511no—
226C2×C4⋊C4—2511no—
227C2×C22⋊C4—5021no—
228#107—2511no—
229#106—2511no—
230#106—2511no—
231C4⋊C8—2511no—
232#103—2511no—
233#103—2511no—
234#103—2511no—
235#103—2511no—
236#102—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102C2×D25111yescharacteristic
241#102C2×D25111yescharacteristic
242C22⋊C8—2511no—
243C22⋊C8C2×D25111yescharacteristic
244C4×C8—2511no—
245C24—21no—
246#41—10041no—
247#41—5022no—
248C2×Q8—2511no—
249C2×Q8C22×D25111yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4C22×D25111yescharacteristic
257C2×D4C22×D25111yescharacteristic
258C22×C4—5021no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—2511no—
262C22×C4C22×D25111yescharacteristic
263Q16—10041no—
264SD16—22no—
265SD16—41no—
266C2×C8—10041no—
267C2×C8—10041no—
268C2×C8—2511no—
269C2×C8—2511no—
270C2×C8C22×D25111yescharacteristic
271C2×C8C22×D25111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—5021no—
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—5021no—
279C4⋊C4—2511no—
280C4⋊C4—2511no—
281C4⋊C4—2511no—
282C4⋊C4—2511no—
283C4⋊C4C22×D25111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4—2511no—
293C22⋊C4—2511no—
294C42—5021no—
295C42—5021no—
296C42—2511no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—2511no—
303C23D4×D25111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5021no—
307Q8—5021no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—20081no—
313D4—10042no—
314D4—5022no—
315D4#1686080912yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—20081no—
321C2×C4—10042no—
322C2×C4—10042no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—2511no—
330C2×C4—2511no—
331C2×C4—2511no—
332C2×C4D4×D25111yescharacteristic
333C2×C4C23×D25111yescharacteristic
334C8—21no—
335C8—21no—
336C8—5021no—
337C8—5021no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—5021no—
348C22—5021no—
349C22—5021no—
350C22#2254873612yes—
351C22C2×D4×D25111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—10041no—
356C4—10041no—
357C4—10041no—
358C4—10041no—
359C4—5021no—
360C4—5021no—
361C4—5021no—
362C4C2×#1686080911yescharacteristic
363C4#2254876011yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—10041no—
368C2#3063232411yescharacteristic
369C2C2×#2254873611yescharacteristic
370C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#450×C251C2
socleC22×C251C22×D4
prime radical#30632331C2
hypercenter#450D251
hypocenterC251#3296
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #30632379C2semidirect
3. #30632040C2semidirect
4. #30632018C2semidirect
5. #30632010C2semidirect
6. #30631989C2semidirect
7. #30631926C2semidirect
8. #30631913C2semidirect
9. #30631907C2semidirect
10. #30631846C2semidirect
11. SD16×Dic251C2semidirect
12. (C22⋊D251)⋊C8C2semidirect
13. #30631680C2semidirect
14. C2×#22548636C2semidirect
15. #450×C251C2semidirect
17. C2×#16860800C22semidirect
19. C2×C8×D251C22semidirect
22. #22548706C22semidirect
27. #22548691C22semidirect
31. Q8×Dic251C22semidirect
34. #22548673C22semidirect
38. #22548648C22semidirect
39. #22548647C22semidirect
40. #22548647C22semidirect
41. Dic251⋊C8C22semidirect
42. #22548640C22semidirect
43. #22548640C22semidirect
44. #22548639C22semidirect
45. #22548638C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. C8×Dic251C22semidirect
185. C251#3296semidirect
197. #450D251semidirect

Maximal subgroups

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

subgroupindex
#306323792
#306323312
#306320402
#306320182
#306320102
#306319892
#306319262
#306319132
#306319072
#306318462
SD16×Dic2512
(C22⋊D251)⋊C82
#306316802
C2×#225486362
#450×C2512
#3296251