Finite Group Databasebeta

← #44562206

Subgroups — #44562206

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#30632328C211yesmaximal, characteristic
3#30632043C211yesmaximal, characteristic, complemented
4#30632015C211yesmaximal, characteristic, complemented
5#30632014C211yesmaximal, characteristic, complemented
6#30631990C211yesmaximal, characteristic, complemented
7#30631919C211yesmaximal, characteristic, complemented
8#30631911C211yesmaximal, characteristic, complemented
9#30631848C211yesmaximal, characteristic, complemented
10#30631837C211yesmaximal, characteristic, complemented
11#30631772C211yesmaximal, characteristic, complemented
12C8×C22⋊D251C211yesmaximal, characteristic, complemented
13#30631682C211yesmaximal, characteristic, complemented
14C2×#22548637C211yesmaximal, characteristic, complemented
15#467×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×#16860797C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic, complemented
21#22548710—21no—
22#22548706—21no—
23#22548706C2211yescharacteristic, complemented
24#22548704C2211yescharacteristic
25#22548700—21no—
26#22548699—21no—
27#22548691C2211yescharacteristic, complemented
28#22548690—21no—
29#22548690C2211yescharacteristic
30#22548685—21no—
31Q8×Dic251C2211yescharacteristic, complemented
32#22548677C2211yescharacteristic
33C4×C22⋊D251C2211yescharacteristic
34#22548673C2211yescharacteristic, complemented
35#22548671—21no—
36C2×C4⋊Dic251C2211yescharacteristic
37C22⋊C4×D251—21no—
38#22548646C2211yescharacteristic, complemented
39#22548646C2211yescharacteristic, complemented
40Dic251⋊C8C2211yescharacteristic, complemented
41#22548639C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic
44#22548637C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C2×Q16×C251C2211yescharacteristic, complemented
53C22×C8×C251C2211yescharacteristic
54C22⋊Q8×C251C2211yescharacteristic
55C22⋊Q8×C251C2211yescharacteristic
56#107×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58#103×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860811—22no—
61#16860811—41no—
62C2×D1004—22no—
63C2×D1004—21no—
64C2×D1004—21no—
65C2×D1004—21no—
66C2×D1004—41no—
67C2×D1004C2311yescharacteristic
68C2×D1004C2311yescharacteristic
69C2×C22⋊D251D411yescharacteristic
70C22×Dic251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251C2311yescharacteristic
74#16860797—22no—
75#16860797—41no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C8×D251—41no—
79C4⋊Dic251—21no—
80C4⋊Dic251—21no—
81C4⋊Dic251—21no—
82C4⋊Dic251C2311yescharacteristic
83C4⋊Dic251C2311yescharacteristic
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4D411yescharacteristic
87C22⋊Dic251D411yescharacteristic
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4D411yescharacteristic
95C4×Dic251—21no—
96C4×Dic251—21no—
97C4×Dic251C2311yescharacteristic
98C2×Q8×C251C2311yescharacteristic
99C2×Q8×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101Q16×C251—41no—
102C2×C8×C251—41no—
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251—21no—
106C4⋊C4×C251—21no—
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C22⋊C4×C251—21no—
110C22⋊C4×C251—21no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—42no—
114C22×D251—42no—
115C22×D251—81no—
116C22×D251C2×D411yescharacteristic
117D1004—21no—
118D1004—21no—
119D1004—21no—
120D1004—42no—
121D1004—42no—
122D1004—81no—
123D1004#4112yes—
124C22⋊D251—22no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—42no—
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C2×Dic251C2×D411yescharacteristic
131C4×D251—21no—
132C4×D251—21no—
133C4×D251—42no—
134C4×D251—81no—
135C251⋊C8—21no—
136C251⋊C8—21no—
137C23×C251C2×D411yescharacteristic
138Q8×C251—21no—
139Q8×C251—21no—
140Q8×C251—21no—
141Q8×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
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#13512yes—
155C22×C251C22×D411yescharacteristic
156C2×D251—21no—
157C2×D251—22no—
158C2×D251—21no—
159C2×D251—21no—
160C2×D251—41no—
161C2×D251—41no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—41no—
165C2×D251—84no—
166C4×C251—21no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251—41no—
170C4×C251—41no—
171C4×C251C2×#4111yescharacteristic
172C4×C251H(5,𝔽2)11yescharacteristic
173Dic251—21no—
174Dic251—21no—
175Dic251—21no—
176Dic251—41no—
177Dic251—41no—
178C2×C251—22no—
179C2×C251#54611yescharacteristic
180C2×C251C2×#13511yescharacteristic
181C2×C251#57711yescharacteristic
182D251—41no—
183D251—42no—
184D251—81no—
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#467D25111yescharacteristic, complemented
194#465—2511no—
195#464—2511no—
196#452—2511no—
197#450—2511no—
198C4×SD16—2511no—
199C8×D4—2511no—
200#416—2511no—
201C2×#102—2511no—
202C2×#41—2511no—
203C22×D4—2511no—
204C2×Q16C2×D25111yescharacteristic, complemented
205C2×SD16—2511no—
206C2×SD16—2511no—
207C22×C8—2511no—
208C22×C8C2×D25111yescharacteristic
209#126—5021no—
210#125—5021no—
211#125—2511no—
212#124—2511no—
213#123—5021no—
214#123—5021no—
215C22⋊Q8—2511no—
216C22⋊Q8C2×D25111yescharacteristic
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#107C2×D25111yescharacteristic, complemented
229#106—2511no—
230#106—2511no—
231C4⋊C8—2511no—
232#103—2511no—
233#103—2511no—
234#103C2×D25111yescharacteristic, complemented
235#103C2×D25111yescharacteristic, complemented
236#102—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241#102—2511no—
242C22⋊C8—2511no—
243C22⋊C8—2511no—
244C4×C8—2511no—
245C24—5021no—
246#41—10041no—
247#41—5022no—
248C2×Q8C22×D25111yescharacteristic
249C2×Q8C22×D25111yescharacteristic
250C2×D4—10041no—
251C2×D4—5021no—
252C2×D4—5021no—
253C2×D4—5022no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4—2511no—
258C22×C4—5021no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—2511no—
262C22×C4C22×D25111yescharacteristic
263Q16—41no—
264SD16—10041no—
265SD16—5022no—
266C2×C8—41no—
267C2×C8—10041no—
268C2×C8—2511no—
269C2×C8—2511no—
270C2×C8C22×D25111yescharacteristic
271C2×C8C22×D25111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
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⋊C4C22×D25111yescharacteristic
283C4⋊C4C22×D25111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
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—20081no—
298C23—10042no—
299C23—10042no—
300C23—5021no—
301C23—5021no—
302C23—2511no—
303C23D4×D25111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—20081no—
309D4—10042no—
310D4—10042no—
311D4—5022no—
312D4—5021no—
313D4—5021no—
314D4—5021no—
315D4—2512no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—20081no—
323C2×C4—10042no—
324C2×C4—10042no—
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—20084no—
341C22—10041no—
342C22—10041no—
343C22—10041no—
344C22—10041no—
345C22—10041no—
346C22—5021no—
347C22—5021no—
348C22—5022no—
349C22—5021no—
350C22#2254873812yes—
351C22C2×D4×D25111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—10041no—
358C4—10041no—
359C4—5021no—
360C4—5021no—
361C4—5021no—
362C4#41×D25111yescharacteristic
363C4#2254876011yescharacteristic
364C2—22no—
365C2—20081no—
366C2—10042no—
367C2—10041no—
368C2#3063232611yescharacteristic
369C2C2×#2254873811yescharacteristic
370C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#467×C251C2
socleC22×C251C22×D4
prime radical#30632328C2
hypercenter#467D251
hypocenterC251#3296
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #30632379C2semidirect
3. #30632043C2semidirect
4. #30632015C2semidirect
5. #30632014C2semidirect
6. #30631990C2semidirect
7. #30631919C2semidirect
8. #30631911C2semidirect
9. #30631848C2semidirect
10. #30631837C2semidirect
11. #30631772C2semidirect
12. C8×C22⋊D251C2semidirect
13. #30631682C2semidirect
14. C2×#22548637C2semidirect
15. #467×C251C2semidirect
20. C2×C8×D251C22semidirect
23. #22548706C22semidirect
27. #22548691C22semidirect
31. Q8×Dic251C22semidirect
34. #22548673C22semidirect
38. #22548646C22semidirect
39. #22548646C22semidirect
40. Dic251⋊C8C22semidirect
41. #22548639C22semidirect
42. #22548639C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. C8×Dic251C22semidirect
52. C2×Q16×C251C22semidirect
56. #107×C251C22semidirect
57. #103×C251C22semidirect
58. #103×C251C22semidirect
185. C251#3296semidirect
193. #467D251semidirect
204. C2×Q16C2×D251semidirect
228. #107C2×D251semidirect
234. #103C2×D251semidirect
235. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323792
#306323282
#306320432
#306320152
#306320142
#306319902
#306319192
#306319112
#306318482
#306318372
#306317722
C8×C22⋊D2512
#306316822
C2×#225486372
#467×C2512
#3296251