Finite Group Databasebeta

← #44560714

Subgroups — #44560714

Order 32128.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632314C211yesmaximal, characteristic, complemented
2#30632250C211yesmaximal, characteristic
3#30632051C211yesmaximal, characteristic, complemented
4#30632050C211yesmaximal, characteristic, complemented
5#30632049C211yesmaximal, characteristic, complemented
6#30632045C211yesmaximal, characteristic, complemented
7#30632042C211yesmaximal, characteristic, complemented
8#30632040C211yesmaximal, characteristic, complemented
9#30632032C211yesmaximal, characteristic, complemented
10#30631690C211yesmaximal, characteristic, complemented
11#30631685C211yesmaximal, characteristic, complemented
12#30631680C211yesmaximal, characteristic, complemented
13#102×D251C211yesmaximal, characteristic, complemented
14#30631641C211yesmaximal, characteristic, complemented
15#486×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×#16860793C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20#22548714C2211yescharacteristic, complemented
21#22548713—21no—
22#22548711C2211yescharacteristic
23#22548707C2211yescharacteristic, complemented
24#22548706C2211yescharacteristic, complemented
25#22548705—21no—
26#22548703C2211yescharacteristic
27#22548702C2211yescharacteristic
28#22548693—21no—
29#22548691—21no—
30#22548687—21no—
31#22548687—21no—
32#22548683—21no—
33#22548683—21no—
34#22548673C2211yescharacteristic
35#22548672C2211yescharacteristic
36C4⋊C4×D251C2211yescharacteristic
37#22548640C2211yescharacteristic, complemented
38#22548640C2211yescharacteristic, complemented
39#22548638C2211yescharacteristic, complemented
40#22548638C2211yescharacteristic, complemented
41#22548637C2211yescharacteristic
42#22548637C2211yescharacteristic
43#22548636C2211yescharacteristic
44#22548636C2211yescharacteristic
45#22548636C2211yescharacteristic, complemented
46#22548636C2211yescharacteristic, complemented
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49#22548626C2211yescharacteristic, complemented
50#22548625C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C4⋊Q8×C251C2211yescharacteristic, complemented
53#127×C251C2211yescharacteristic
54#102×C251C2211yescharacteristic
55#102×C251C2211yescharacteristic
56#102×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic, complemented
58C4×C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860809—22no—
61#16860809—22no—
62C2×Dic502—21no—
63C2×Dic502C2311yescharacteristic
64C2×D1004C2311yescharacteristic
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69D4×D251—22no—
70D4×D251—22no—
71C22×Dic251—21no—
72C2×C4×D251—21no—
73C2×C4×D251C2311yescharacteristic
74#16860793—22no—
75C2×C251⋊C8C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C8×D251—22no—
78C4⋊Dic251—21no—
79C4⋊Dic251C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4D411yescharacteristic
87Dic251⋊C4D411yescharacteristic
88C22⋊Dic251—21no—
89C22⋊Dic251—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4D411yescharacteristic
94(C2×D251)⋊C4D411yescharacteristic
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97C2×Q8×C251—21no—
98C2×D4×C251—21no—
99C2×D4×C251—21no—
100C2×D4×C251C2311yescharacteristic
101C2×D4×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C2×C8×C251C2311yescharacteristic
104C4⋊C4×C251—21no—
105C4⋊C4×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109C22×D251—42no—
110C22×D251—81no—
111C22×D251—81no—
112C22×D251C2×D411yescharacteristic
113Dic502—21no—
114Dic502—21no—
115Dic502—81no—
116D1004—21no—
117D1004—21no—
118C22⋊D251—42no—
119C22⋊D251—42no—
120C22⋊D251—81no—
121C22⋊D251—81no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—42no—
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—81no—
130C4×D251#4112yes—
131C4×D251#4112yes—
132C251⋊C8—21no—
133C251⋊C8—21no—
134C23×C251—21no—
135C23×C251—21no—
136Q8×C251—81no—
137D4×C251—21no—
138D4×C251—21no—
139D4×C251—21no—
140D4×C251—21no—
141D4×C251—81no—
142D4×C251—81no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C8×C251—21no—
149C8×C251—21no—
150C22×C251—41no—
151C22×C251—41no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251—41no—
156C22×C251C22×D411yescharacteristic
157C2×D251—21no—
158C2×D251—21no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—41no—
162C2×D251—84no—
163C2×D251#12412yes—
164C4×C251—21no—
165C4×C251—21no—
166C4×C251—41no—
167C4×C251—41no—
168C4×C251C2×#4111yescharacteristic
169C4×C251C2×#4111yescharacteristic
170Dic251—21no—
171Dic251—21no—
172Dic251—41no—
173Dic251—41no—
174Dic251—41no—
175Dic251#12412yes—
176C2×C251—81no—
177C2×C251—81no—
178C2×C251C2×#12411yescharacteristic
179C2×C251#57611yescharacteristic
180C2×C251#57711yescharacteristic
181D251—22no—
182D251—81no—
183C251#313611yescharacteristic, complemented, Sylow
184#3136—2511nomaximal, Sylow
185#544—2511no—
186#535—2511no—
187#490—2511no—
188#490—2511no—
189#489—2511no—
190#489—2511no—
191#488—2511no—
192#488—2511no—
193#486—2511no—
194#486D25111yescharacteristic, complemented
195#418—2511no—
196#417—2511no—
197#416—2511no—
198C2×#102—2511no—
199#405—2511no—
200C2×#41—2511no—
201C22×D4—2511no—
202C2×M16—2511no—
203C22×C8—2511no—
204C4⋊Q8—5021no—
205C4⋊Q8—2511no—
206C4⋊Q8C2×D25111yescharacteristic, complemented
207#127—2511no—
208#127C2×D25111yescharacteristic
209#125—5021no—
210#125—2511no—
211#125—2511no—
212#124—2511no—
213#124—2511no—
214C22⋊Q8—5021no—
215C22⋊Q8—5021no—
216#121—5021no—
217#121—5021no—
218#120—5021no—
219#120—5021no—
220#117—2511no—
221#117—2511no—
222C2×C4⋊C4—2511no—
223#103—2511no—
224#103—2511no—
225#103—2511no—
226#103—2511no—
227#102—2511no—
228#102—2511no—
229#102—2511no—
230#102—2511no—
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102C2×D25111yescharacteristic
236#102C2×D25111yescharacteristic
237#102C2×D25111yescharacteristic, complemented
238#102C2×D25111yescharacteristic, complemented
239#97—2511no—
240#97—2511no—
241C4×C8—2511no—
242C4×C8C2×D25111yescharacteristic, complemented
243C24—5021no—
244#41—5022no—
245#41—5022no—
246C2×Q8—21no—
247C2×Q8—5021no—
248C2×Q8—2511no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—5021no—
252C2×D4—5022no—
253C2×D4—5022no—
254C2×D4—5021no—
255C2×D4—5021no—
256C2×D4—5021no—
257C2×D4—2511no—
258C2×D4C22×D25111yescharacteristic
259C2×D4C22×D25111yescharacteristic
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4—2511no—
263M16—5022no—
264C2×C8—5022no—
265C2×C8—2511no—
266C2×C8—2511no—
267C2×C8C22×D25111yescharacteristic
268C2×C8C22×D25111yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—5021no—
271C4⋊C4—5021no—
272C4⋊C4—5021no—
273C4⋊C4—5021no—
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—2511no—
277C4⋊C4—2511no—
278C4⋊C4—2511no—
279C4⋊C4—2511no—
280C4⋊C4C22×D25111yescharacteristic
281C4⋊C4C22×D25111yescharacteristic
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—2511no—
288C22⋊C4—2511no—
289C42—5021no—
290C42—2511no—
291C42C22×D25111yescharacteristic
292C23—21no—
293C23—21no—
294C23—20081no—
295C23—20081no—
296C23—10042no—
297C23—5021no—
298C23—2511no—
299Q8—81no—
300Q8—20081no—
301Q8—5021no—
302Q8—5021no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—81no—
309D4—20081no—
310D4—20081no—
311D4—10042no—
312D4—10042no—
313D4—5021no—
314D4—5021no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—20081no—
318C2×C4—10042no—
319C2×C4—5021no—
320C2×C4—5021no—
321C2×C4—5021no—
322C2×C4—2511no—
323C2×C4—2511no—
324C2×C4—2511no—
325C2×C4—2512no—
326C2×C4—2512no—
327C2×C4D4×D25111yescharacteristic
328C2×C4D4×D25111yescharacteristic
329C2×C4C23×D25111yescharacteristic
330C8—21no—
331C8—21no—
332C8—5021no—
333C8—5021no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—20084no—
341C22—10041no—
342C22—10041no—
343C22—10041no—
344C22—5021no—
345C22—5021no—
346C22—2512no—
347C22C2×D4×D25111yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—10041no—
353C4—10041no—
354C4—10041no—
355C4—5021no—
356C4—5021no—
357C4—2512no—
358C4#41×D25111yescharacteristic
359C4#41×D25111yescharacteristic
360C2—81no—
361C2—81no—
362C2—20081no—
363C2—5022no—
364C2#124×D25111yescharacteristic
365C2#3063248811yescharacteristic
366C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#486×C251C2
socleC22×C251C22×D4
prime radical#30632250C2
hypercenter#486D251
hypocenterC251#3136
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #30632314C2semidirect
3. #30632051C2semidirect
4. #30632050C2semidirect
5. #30632049C2semidirect
6. #30632045C2semidirect
7. #30632042C2semidirect
8. #30632040C2semidirect
9. #30632032C2semidirect
10. #30631690C2semidirect
11. #30631685C2semidirect
12. #30631680C2semidirect
13. #102×D251C2semidirect
14. #30631641C2semidirect
15. #486×C251C2semidirect
20. #22548714C22semidirect
23. #22548707C22semidirect
24. #22548706C22semidirect
37. #22548640C22semidirect
38. #22548640C22semidirect
39. #22548638C22semidirect
40. #22548638C22semidirect
45. #22548636C22semidirect
46. #22548636C22semidirect
49. #22548626C22semidirect
50. #22548625C22semidirect
51. C8×Dic251C22semidirect
52. C4⋊Q8×C251C22semidirect
56. #102×C251C22semidirect
57. #102×C251C22semidirect
58. C4×C8×C251C22semidirect
183. C251#3136semidirect
194. #486D251semidirect
206. C4⋊Q8C2×D251semidirect
237. #102C2×D251semidirect
238. #102C2×D251semidirect
242. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323142
#306322502
#306320512
#306320502
#306320492
#306320452
#306320422
#306320402
#306320322
#306316902
#306316852
#306316802
#102×D2512
#306316412
#486×C2512
#3136251