Finite Group Databasebeta

← #44562203

Subgroups — #44562203

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#30632376C211yesmaximal, characteristic, complemented
2#30632334C211yesmaximal, characteristic
3#30632040C211yesmaximal, characteristic, complemented
4#30632019C211yesmaximal, characteristic, complemented
5#30631990C211yesmaximal, characteristic, complemented
6#30631929C211yesmaximal, characteristic, complemented
7#30631914C211yesmaximal, characteristic, complemented
8#30631908C211yesmaximal, characteristic, complemented
9#30631851C211yesmaximal, characteristic, complemented
10#30631839C211yesmaximal, characteristic, complemented
11#30631774C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic, complemented
13#30631681C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#482×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×Dic1004C2211yescharacteristic, complemented
19C2×#16860799C2211yescharacteristic
20C2×#16860798C2211yescharacteristic
21C22×C251⋊C8C2211yescharacteristic, complemented
22C2×C8×D251C2211yescharacteristic
23#22548708—21no—
24#22548707—21no—
25#22548706C2211yescharacteristic, complemented
26#22548702C2211yescharacteristic
27#22548697—21no—
28#22548697—21no—
29#22548695C2211yescharacteristic
30#22548695C2211yescharacteristic
31#22548694C2211yescharacteristic, complemented
32#22548686—21no—
33#22548683—21no—
34#22548680C2211yescharacteristic, complemented
35#22548677C2211yescharacteristic
36C4×C22⋊D251C2211yescharacteristic
37#22548672—21no—
38C4⋊C4×D251C2211yescharacteristic
39C22⋊C4×D251—21no—
40#22548646C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548638C2211yescharacteristic, complemented
44#22548638C2211yescharacteristic, complemented
45#22548638C2211yescharacteristic, complemented
46#22548637C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548635C2211yescharacteristic
49#22548635C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic, complemented
52#121×C251C2211yescharacteristic
53#117×C251C2211yescharacteristic, complemented
54#107×C251C2211yescharacteristic, complemented
55#106×C251C2211yescharacteristic, complemented
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860810—22no—
61#16860810—41no—
62C2×Dic502C2311yescharacteristic
63C2×Dic502C2311yescharacteristic
64C2×D1004C2311yescharacteristic
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67C2×C22⋊D251D411yescharacteristic
68D4×D251—22no—
69D4×D251—41no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251C2311yescharacteristic
74Dic1004—41no—
75#16860799—22no—
76#16860798—41no—
77C2×C251⋊C8—41no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C8×D251—41no—
81C4⋊Dic251—21no—
82C4⋊Dic251—21no—
83C4⋊Dic251—21no—
84C4⋊Dic251C2311yescharacteristic
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4D411yescharacteristic
90C22⋊Dic251—21no—
91C22⋊Dic251—21no—
92C22⋊Dic251—21no—
93C22⋊Dic251D411yescharacteristic
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4—21no—
97(C2×D251)⋊C4D411yescharacteristic
98C4×Dic251—21no—
99C4×Dic251C2311yescharacteristic
100C2×D4×C251—21no—
101C2×D4×C251C2311yescharacteristic
102C22×C4×C251C2311yescharacteristic
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C4⋊C4×C251C2311yescharacteristic
108C22⋊C4×C251—21no—
109C22⋊C4×C251—21no—
110C42×C251—21no—
111C22×D251—21no—
112C22×D251—42no—
113C22×D251—42no—
114C22×D251—81no—
115C22×D251C2×D411yescharacteristic
116Dic502—21no—
117Dic502—21no—
118Dic502—21no—
119Dic502—21no—
120D1004—21no—
121D1004#4112yes—
122C22⋊D251—22no—
123C22⋊D251—42no—
124C22⋊D251—42no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C2×Dic251C2×D411yescharacteristic
131C4×D251—21no—
132C4×D251—21no—
133C4×D251—42no—
134C4×D251—42no—
135C4×D251—81no—
136C251⋊C8—21no—
137C251⋊C8—21no—
138C23×C251—21no—
139C23×C251C2×D411yescharacteristic
140D4×C251—21no—
141D4×C251—21no—
142D4×C251—81no—
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—21no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251C22×D411yescharacteristic
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—22no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—84no—
165C2×D251#13512yes—
166C4×C251—21no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251—41no—
170C4×C251C2×#4111yescharacteristic
171C4×C251H(5,𝔽2)11yescharacteristic
172Dic251—21no—
173Dic251—21no—
174Dic251—21no—
175Dic251—41no—
176Dic251—41no—
177Dic251—41no—
178C2×C251—41no—
179C2×C251—81no—
180C2×C251#54611yescharacteristic
181C2×C251C2×#13511yescharacteristic
182C2×C251#57711yescharacteristic
183D251—22no—
184D251—42no—
185C251#329611yescharacteristic, complemented, Sylow
186#3296—2511nomaximal, Sylow
187#552—2511no—
188#546—2511no—
189#488—2511no—
190#483—2511no—
191#482D25111yescharacteristic, complemented
192#478—2511no—
193#467—2511no—
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×Q16—2511no—
205C2×SD16—2511no—
206C2×SD16—2511no—
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⋊Q8—2511no—
218#121—5021no—
219#121C2×D25111yescharacteristic
220#120—5021no—
221C4×Q8—2511no—
222C4×D4—2511no—
223C4×D4—2511no—
224#117—5021no—
225#117C2×D25111yescharacteristic, complemented
226C2×C4⋊C4—2511no—
227C2×C22⋊C4—5021no—
228#107C2×D25111yescharacteristic, complemented
229#106—2511no—
230#106C2×D25111yescharacteristic, complemented
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, complemented
244C4×C8—2511no—
245C24—5021no—
246#41—10041no—
247#41—5022no—
248C2×Q8—2511no—
249C2×Q8—2511no—
250C2×D4—21no—
251C2×D4—10041no—
252C2×D4—5022no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4C22×D25111yescharacteristic
258C22×C4—5021no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—2511no—
262C22×C4C22×D25111yescharacteristic
263Q16—10041no—
264SD16—10041no—
265SD16—5022no—
266C2×C8—10041no—
267C2×C8—10041no—
268C2×C8—2511no—
269C2×C8—2511no—
270C2×C8C22×D25111yescharacteristic
271C2×C8C22×D25111yescharacteristic
272C4⋊C4—5021no—
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⋊C4C22×D25111yescharacteristic
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—21no—
295C42—5021no—
296C42—2511no—
297C23—21no—
298C23—20081no—
299C23—10042no—
300C23—10042no—
301C23—5021no—
302C23—2511no—
303C23D4×D25111yescharacteristic
304Q8—5021no—
305Q8—5021no—
306Q8—5021no—
307Q8—5021no—
308D4—21no—
309D4—21no—
310D4—81no—
311D4—10042no—
312D4—10042no—
313D4—5022no—
314D4—5021no—
315D4—2512no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—20081no—
322C2×C4—10042no—
323C2×C4—10042no—
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—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—20084no—
345C22—10041no—
346C22—10041no—
347C22—5022no—
348C22—5021no—
349C22—5021no—
350C22—2512no—
351C22C2×D4×D25111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—10041no—
357C4—10041no—
358C4—10041no—
359C4—5021no—
360C4—5021no—
361C4—5021no—
362C4#41×D25111yescharacteristic
363C4#2254876111yescharacteristic
364C2—41no—
365C2—81no—
366C2—10042no—
367C2—5022no—
368C2#3063232911yescharacteristic
369C2#135×D25111yescharacteristic
370C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#482×C251C2
socleC22×C251C22×D4
prime radical#30632334C2
hypercenter#482D251
hypocenterC251#3296
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #30632376C2semidirect
3. #30632040C2semidirect
4. #30632019C2semidirect
5. #30631990C2semidirect
6. #30631929C2semidirect
7. #30631914C2semidirect
8. #30631908C2semidirect
9. #30631851C2semidirect
10. #30631839C2semidirect
11. #30631774C2semidirect
12. (C22⋊D251)⋊C8C2semidirect
13. #30631681C2semidirect
14. #102×D251C2semidirect
15. #482×C251C2semidirect
18. C2×Dic1004C22semidirect
21. C22×C251⋊C8C22semidirect
25. #22548706C22semidirect
31. #22548694C22semidirect
34. #22548680C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. #22548640C22semidirect
43. #22548638C22semidirect
44. #22548638C22semidirect
45. #22548638C22semidirect
51. C8×Dic251C22semidirect
53. #117×C251C22semidirect
54. #107×C251C22semidirect
55. #106×C251C22semidirect
58. C22⋊C8×C251C22semidirect
185. C251#3296semidirect
191. #482D251semidirect
225. #117C2×D251semidirect
228. #107C2×D251semidirect
230. #106C2×D251semidirect
243. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323762
#306323342
#306320402
#306320192
#306319902
#306319292
#306319142
#306319082
#306318512
#306318392
#306317742
(C22⋊D251)⋊C82
#306316812
#102×D2512
#482×C2512
#3296251