Finite Group Databasebeta

← #44562147

Subgroups — #44562147

Order 32128.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632501C211yesmaximal, characteristic, complemented
2#30632498C212yesmaximal, complemented
3#30632070C211yesmaximal, characteristic, complemented
4#30631962C211yesmaximal, characteristic, complemented
5#30631954C211yesmaximal, characteristic, complemented
6#30631948C212yesmaximal, complemented
7#30631865C212yesmaximal, complemented
8#30631858C212yesmaximal, complemented
9#30631804C211yesmaximal, characteristic, complemented
10#30631802C211yesmaximal, characteristic, complemented
11#576×C251C211yesmaximal, characteristic, complemented
12#22548762C2212yescomplemented
13#22548745—22no—
14#22548745C2212yes—
15#22548745C2212yescomplemented
16#22548744—22no—
17#22548741—22no—
18#22548741C2212yescomplemented
19#22548735—22no—
20#22548735C2211yescharacteristic
21#22548734—21no—
22C2×Dic1004C2211yescharacteristic, complemented
23Q16×D251—21no—
24C2×#16860798—22no—
25C2×#16860798C2212yescomplemented
26#22548722C2212yescomplemented
27#22548721C2211yescharacteristic, complemented
28#22548702C2212yescomplemented
29#22548651C2211yescharacteristic, complemented
30#22548650C2211yescharacteristic, complemented
31#22548642C2211yescharacteristic
32#22548642C2212yescomplemented
33#22548641C2211yescharacteristic, complemented
34#22548633C2212yescomplemented
35#22548632C2212yescomplemented
36C8×Dic251C2211yescharacteristic, complemented
37H(5,𝔽2)×C251C2212yes—
38AGL(1,ℤ/8)×C251—22no—
39AGL(1,ℤ/8)×C251C2212yes—
40#135×C251—21no—
41#135×C251C2211yescharacteristic
42C2×D8×C251—21no—
43C2×D8×C251C2211yescharacteristic
44#131×C251C2211yescharacteristic
45#16860810—22no—
46#16860810C2311yescharacteristic
47#16860809—22no—
48#16860809—42no—
49C2×Dic502—22no—
50C2×Dic502C2312yes—
51Q8×D251—22no—
52Dic1004—21no—
53Dic1004—21no—
54#16860800—22no—
55#16860800—22no—
56#16860800D411yescharacteristic
57#16860798—22no—
58#16860798—22no—
59#16860798—22no—
60#16860798—22no—
61#16860798—22no—
62#16860798D411yescharacteristic
63#16860797D411yescharacteristic
64#16860795—22no—
65#16860795D411yescharacteristic
66#16860794—22no—
67#16860794C2312yes—
68#16860794C2311yescharacteristic
69#16860793—21no—
70C2×C251⋊C8—22no—
71C2×C251⋊C8C2311yescharacteristic
72C8×D251—21no—
73C22⋊Dic251—42no—
74C4×Dic251C2311yescharacteristic
75#41×C251—22no—
76#41×C251—22no—
77#41×C251C2311yescharacteristic
78#41×C251C2312yes—
79C2×D4×C251—22no—
80C2×D4×C251—22no—
81C2×D4×C251—42no—
82C2×D4×C251C2312yes—
83Q16×C251D411yescharacteristic
84SD16×C251—22no—
85SD16×C251D412yes—
86D8×C251—22no—
87D8×C251—21no—
88D8×C251—21no—
89D8×C251D411yescharacteristic
90M16×C251—21no—
91M16×C251C2311yescharacteristic
92C2×C8×C251—21no—
93C2×C8×C251C2311yescharacteristic
94Dic502—22no—
95Dic502—22no—
96Dic502—42no—
97Dic502C2×D411yescharacteristic
98D1004C2×D411yescharacteristic
99C22⋊D251—21no—
100C22⋊D251—42no—
101C2×Dic251—21no—
102C2×Dic251—22no—
103C2×Dic251—42no—
104C4×D251—21no—
105C4×D251—42no—
106C251⋊C8—21no—
107C251⋊C8—22no—
108C251⋊C8C2×D411yescharacteristic
109C251⋊C8C2×D411yescharacteristic
110C23×C251—22no—
111C23×C251—42no—
112Q8×C251C2×D411yescharacteristic
113Q8×C251C2×D412yes—
114D4×C251—21no—
115D4×C251—22no—
116D4×C251—22no—
117D4×C251—22no—
118D4×C251—42no—
119D4×C251—42no—
120D4×C251C2×D411yescharacteristic
121D4×C251C2×D412yes—
122C2×C4×C251—21no—
123C2×C4×C251—22no—
124C2×C4×C251—42no—
125C2×C4×C251C2411yescharacteristic
126C8×C251—21no—
127C8×C251C2×D411yescharacteristic
128C8×C251C2×D411yescharacteristic
129C22×C251—21no—
130C22×C251—22no—
131C22×C251—42no—
132C22×C251—82no—
133C22×C251—162no—
134C22×C251H(5,𝔽2)11yescharacteristic
135C2×D251—21no—
136C4×C251—21no—
137C4×C251—22no—
138C4×C251C22×D411yescharacteristic
139C4×C251C22×D411yescharacteristic
140Dic251—21no—
141Dic251—41no—
142Dic251—42no—
143C2×C251—21no—
144C2×C251—42no—
145C2×C251—41no—
146C2×C251—82no—
147C2×C251D4211yescharacteristic
148D251—41no—
149C251#329011yescharacteristic, complemented, Sylow
150#3290—2511nomaximal, Sylow
151#578—2511no—
152#577—2512no—
153#576D25111yescharacteristic, complemented
154#495—2511no—
155#473—2511no—
156#472—2511no—
157#471—2512no—
158#455—2512no—
159#454—2512no—
160#443—2511no—
161#443—2511no—
1622−1+4—2512no—
163H(5,𝔽2)C2×D25112yes—
164#137—5022no—
165#137—5022no—
166#137—2512no—
167#137—2512no—
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—5022no—
170AGL(1,ℤ/8)—2512no—
171AGL(1,ℤ/8)C2×D25112yes—
172#135—21no—
173#135—5022no—
174#135—5021no—
175#135—2511no—
176#135C2×D25111yescharacteristic
177C2×Q16—5021no—
178C2×Q16—2511no—
179C2×SD16—5022no—
180C2×SD16—2512no—
181C2×D8—21no—
182C2×D8C2×D25111yescharacteristic
183#131—2512no—
184#131—2511no—
185#131C2×D25111yescharacteristic
186#124—2512no—
187#108—2511no—
188#108—2511no—
189#104—2511no—
190#104—2512no—
191#104—2511no—
192#101—2512no—
193#100—2512no—
194C4×C8—2511no—
195#41—22no—
196#41—22no—
197#41—10042no—
198#41—5022no—
199#41—5022no—
200#41—2511no—
201#41C22×D25111yescharacteristic
202#41C22×D25112yes—
203C2×Q8—5022no—
204C2×Q8—5022no—
205C2×Q8—2512no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4C22×D25112yes—
210Q16—5022no—
211Q16—5022no—
212Q16—5021no—
213Q16—5021no—
214Q16—2511no—
215Q16C22⋊D25111yescharacteristic
216SD16—22no—
217SD16—5022no—
218SD16—5022no—
219SD16—5022no—
220SD16—5022no—
221SD16—5022no—
222SD16—2511no—
223SD16—2511no—
224SD16C22⋊D25112yes—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—5022no—
229D8—2511no—
230D8C22⋊D25111yescharacteristic
231M16—21no—
232M16—5022no—
233M16—5021no—
234M16—2511no—
235M16—2512no—
236M16C22×D25111yescharacteristic
237C2×C8—21no—
238C2×C8—5022no—
239C2×C8—5021no—
240C2×C8—2511no—
241C2×C8C22×D25111yescharacteristic
242C22⋊C4—10042no—
243C42—2511no—
244C23—22no—
245C23—42no—
246Q8—10042no—
247Q8—5022no—
248Q8—5022no—
249Q8—2511no—
250Q8D4×D25111yescharacteristic
251Q8C2×C22⋊D25112yes—
252D4—21no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—10042no—
259D4—5021no—
260D4—2511no—
261D4D4×D25111yescharacteristic
262D4C2×C22⋊D25112yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—10042no—
267C2×C4—10042no—
268C2×C4—5021no—
269C2×C4—5022no—
270C2×C4—5021no—
271C2×C4C23×D25111yescharacteristic
272C8—21no—
273C8—5022no—
274C8—5021no—
275C8—2511no—
276C8—2511no—
277C8C2×C22⋊D25111yescharacteristic
278C8C2×C22⋊D25111yescharacteristic
279C22—22no—
280C22—21no—
281C22—42no—
282C22—82no—
283C22—162no—
284C22—5021no—
285C22#2254876011yescharacteristic
286C4—21no—
287C4—22no—
288C4—10041no—
289C4—10042no—
290C4—5021no—
291C4C2×D4×D25111yescharacteristic
292C4C22×C22⋊D25111yescharacteristic
293C2—21no—
294C2—41no—
295C2—42no—
296C2—82no—
297C2—10041no—
298C2D4×C22⋊D25111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#576×C251C2
socleC2×C251D42
hypercenter#576D251
hypocenterC251#3290
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #30632501C2semidirect
2. #30632498C2semidirect
3. #30632070C2semidirect
4. #30631962C2semidirect
5. #30631954C2semidirect
6. #30631948C2semidirect
7. #30631865C2semidirect
8. #30631858C2semidirect
9. #30631804C2semidirect
10. #30631802C2semidirect
11. #576×C251C2semidirect
12. #22548762C22semidirect
15. #22548745C22semidirect
18. #22548741C22semidirect
22. C2×Dic1004C22semidirect
25. C2×#16860798C22semidirect
26. #22548722C22semidirect
27. #22548721C22semidirect
28. #22548702C22semidirect
29. #22548651C22semidirect
30. #22548650C22semidirect
32. #22548642C22semidirect
33. #22548641C22semidirect
34. #22548633C22semidirect
35. #22548632C22semidirect
36. C8×Dic251C22semidirect
149. C251#3290semidirect
153. #576D251semidirect

Maximal subgroups

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

subgroupindex
#306325012
#306324982
#306320702
#306319622
#306319542
#306319482
#306318652
#306318582
#306318042
#306318022
#576×C2512
#3290251