Finite Group Databasebeta

← #44562150

Subgroups — #44562150

Order 32128.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#30632500C211yesmaximal, characteristic, complemented
2#30632493C212yesmaximal, complemented
3#30632488C211yesmaximal, characteristic, complemented
4#30631959C211yesmaximal, characteristic, complemented
5#30631954C211yesmaximal, characteristic, complemented
6#30631947C212yesmaximal, complemented
7#30631864C212yesmaximal, complemented
8#30631859C212yesmaximal, complemented
9#30631805C211yesmaximal, characteristic, complemented
10#30631805C211yesmaximal, characteristic, complemented
11#495×C251C211yesmaximal, characteristic, complemented
12#22548763C2212yescomplemented
13#22548760C2212yes—
14#22548747—22no—
15#22548746C2212yes—
16#22548746C2212yescomplemented
17#22548744—22no—
18#22548742—22no—
19#22548741C2212yes—
20#22548741C2212yescomplemented
21#22548739—22no—
22#22548738—21no—
23#22548737C2211yescharacteristic
24#22548737C2211yescharacteristic
25#22548736—22no—
26#22548734—21no—
27Q16×D251—21no—
28SD16×D251—22no—
29D8×D251—21no—
30#22548721C2211yescharacteristic
31#22548721C2212yescomplemented
32#22548721C2211yescharacteristic, complemented
33#22548651C2211yescharacteristic, complemented
34#22548651C2211yescharacteristic, complemented
35#22548643C2211yescharacteristic
36#22548643C2211yescharacteristic, complemented
37#22548643C2212yescomplemented
38#22548634C2212yescomplemented
39#22548632C2212yescomplemented
40C2×Q16×C251C2211yescharacteristic, complemented
41C2×SD16×C251C2212yescomplemented
42C2×D8×C251C2211yescharacteristic
43#124×C251C2212yescomplemented
44C4×C8×C251C2211yescharacteristic, complemented
45#16860811—22no—
46#16860811—22no—
47#16860810—42no—
48#16860810C2312yes—
49#16860810C2311yescharacteristic
50#16860810C2311yescharacteristic
51#16860809—22no—
52#16860809—22no—
53Q8×D251—22no—
54Q8×D251—22no—
55C2×C22⋊D251—42no—
56D4×D251—22no—
57D4×D251—22no—
58Dic1004D411yescharacteristic
59Dic1004D411yescharacteristic
60#16860800—22no—
61#16860800—22no—
62#16860799D411yescharacteristic
63#16860799D412yes—
64#16860799D411yescharacteristic
65#16860798—22no—
66#16860798—22no—
67#16860797—22no—
68#16860797—22no—
69D2008D411yescharacteristic
70D2008D411yescharacteristic
71#16860795—22no—
72#16860795—22no—
73#16860794C2311yescharacteristic
74#16860794C2312yes—
75#16860794C2311yescharacteristic
76#16860793—21no—
77#16860793—22no—
78#16860793—21no—
79C8×D251—22no—
80C8×D251—21no—
81C8×D251—21no—
82C2×Q8×C251C2312yes—
83C2×D4×C251C2312yes—
84Q16×C251—21no—
85Q16×C251—21no—
86SD16×C251—22no—
87SD16×C251—22no—
88D8×C251—21no—
89D8×C251—21no—
90C2×C8×C251C2311yescharacteristic
91C2×C8×C251C2311yescharacteristic
92C22⋊C4×C251—42no—
93C42×C251C2311yescharacteristic
94C22×D251—42no—
95Dic502—42no—
96Dic502C2×D411yescharacteristic
97Dic502C2×D412yes—
98Dic502C2×D411yescharacteristic
99D1004—42no—
100D1004C2×D411yescharacteristic
101D1004C2×D412yes—
102D1004C2×D411yescharacteristic
103C22⋊D251—21no—
104C22⋊D251—22no—
105C22⋊D251—21no—
106C22⋊D251—42no—
107C22⋊D251—42no—
108C2×Dic251—42no—
109C4×D251—21no—
110C4×D251—22no—
111C4×D251—21no—
112C4×D251—42no—
113C4×D251—42no—
114C251⋊C8—21no—
115C251⋊C8—22no—
116C251⋊C8—21no—
117C23×C251—22no—
118Q8×C251—22no—
119Q8×C251—22no—
120D4×C251—22no—
121D4×C251—22no—
122C2×C4×C251—21no—
123C2×C4×C251—22no—
124C2×C4×C251C2411yescharacteristic
125C8×C251C2×D411yescharacteristic
126C8×C251C2×D411yescharacteristic
127C8×C251C2×D411yescharacteristic
128C8×C251C2×D411yescharacteristic
129C22×C251—42no—
130C22×C251—82no—
131C22×C251H(5,𝔽2)11yescharacteristic
132C2×D251—21no—
133C2×D251—22no—
134C2×D251—21no—
135C2×D251—162no—
136C4×C251—41no—
137C4×C251—42no—
138C4×C251C22×D411yescharacteristic
139C4×C251C22×D411yescharacteristic
140Dic251—21no—
141Dic251—22no—
142Dic251—21no—
143C2×C251—21no—
144C2×C251—82no—
145C2×C251D4211yescharacteristic
146D251—41no—
147D251—42no—
148D251—41no—
149C251#329011yescharacteristic, complemented, Sylow
150#3290—2511nomaximal, Sylow
151#578—2511no—
152#577—2512no—
153#576—2511no—
154#495D25111yescharacteristic, complemented
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)—2512no—
164#137—5022no—
165#137—5022no—
166#137—2512no—
167#137—2512no—
168AGL(1,ℤ/8)—5022no—
169AGL(1,ℤ/8)—5022no—
170AGL(1,ℤ/8)—2512no—
171AGL(1,ℤ/8)—2512no—
172#135—5021no—
173#135—5022no—
174#135—5021no—
175#135—2511no—
176#135—2511no—
177C2×Q16—5021no—
178C2×Q16C2×D25111yescharacteristic, complemented
179C2×SD16—5022no—
180C2×SD16C2×D25112yescomplemented
181C2×D8—5021no—
182C2×D8C2×D25111yescharacteristic
183#131—2511no—
184#131—2512no—
185#131—2511no—
186#124C2×D25112yescomplemented
187#108—2511no—
188#108—2511no—
189#104—2511no—
190#104—2511no—
191#104—2512no—
192#101—2512no—
193#100—2512no—
194C4×C8C2×D25111yescharacteristic, complemented
195#41—10042no—
196#41—5022no—
197#41—5022no—
198#41—5022no—
199#41—5022no—
200#41—2511no—
201#41—2512no—
202#41—2511no—
203C2×Q8—5022no—
204C2×Q8—5022no—
205C2×Q8C22×D25112yes—
206C2×D4—10042no—
207C2×D4—5022no—
208C2×D4—5022no—
209C2×D4C22×D25112yes—
210Q16—21no—
211Q16—21no—
212Q16—5022no—
213Q16—5022no—
214Q16—2511no—
215Q16—2511no—
216SD16—22no—
217SD16—22no—
218SD16—5022no—
219SD16—5022no—
220SD16—5022no—
221SD16—5022no—
222SD16—2511no—
223SD16—2511no—
224SD16—2512no—
225D8—21no—
226D8—21no—
227D8—5022no—
228D8—5022no—
229D8—2511no—
230D8—2511no—
231M16—5021no—
232M16—5022no—
233M16—5021no—
234M16—2511no—
235M16—2512no—
236M16—2511no—
237C2×C8—5021no—
238C2×C8—5022no—
239C2×C8—5021no—
240C2×C8C22×D25111yescharacteristic
241C2×C8C22×D25111yescharacteristic
242C22⋊C4—42no—
243C42C22×D25111yescharacteristic
244C23—22no—
245C23—10042no—
246Q8—22no—
247Q8—22no—
248Q8—10042no—
249Q8—2511no—
250Q8—2512no—
251Q8—2511no—
252D4—22no—
253D4—22no—
254D4—10042no—
255D4—10042no—
256D4—10042no—
257D4—5021no—
258D4—5022no—
259D4—5021no—
260D4—2511no—
261D4—2512no—
262D4—2511no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—10042no—
266C2×C4—10042no—
267C2×C4—10042no—
268C2×C4—5021no—
269C2×C4—5022no—
270C2×C4—5021no—
271C2×C4C23×D25111yescharacteristic
272C8—5021no—
273C8—5022no—
274C8—5021no—
275C8D4×D25111yescharacteristic
276C8D4×D25111yescharacteristic
277C8D4×D25111yescharacteristic
278C8D4×D25111yescharacteristic
279C22—42no—
280C22—82no—
281C22—40162no—
282C22—5021no—
283C22—5022no—
284C22—5021no—
285C22#2254876011yescharacteristic
286C4—41no—
287C4—42no—
288C4—5021no—
289C4—5022no—
290C4—5021no—
291C4C2×D4×D25111yescharacteristic
292C4C2×D4×D25111yescharacteristic
293C2—21no—
294C2—82no—
295C2—10041no—
296C2—10042no—
297C2—10041no—
298C2#3063232211yescharacteristic
subgroupG/H
centerC2#30632322
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#495×C251C2
socleC2×C251D42
hypercenter#495D251
hypocenterC251#3290
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #30632500C2semidirect
2. #30632493C2semidirect
3. #30632488C2semidirect
4. #30631959C2semidirect
5. #30631954C2semidirect
6. #30631947C2semidirect
7. #30631864C2semidirect
8. #30631859C2semidirect
9. #30631805C2semidirect
10. #30631805C2semidirect
11. #495×C251C2semidirect
12. #22548763C22semidirect
16. #22548746C22semidirect
20. #22548741C22semidirect
31. #22548721C22semidirect
32. #22548721C22semidirect
33. #22548651C22semidirect
34. #22548651C22semidirect
36. #22548643C22semidirect
37. #22548643C22semidirect
38. #22548634C22semidirect
39. #22548632C22semidirect
40. C2×Q16×C251C22semidirect
41. C2×SD16×C251C22semidirect
43. #124×C251C22semidirect
44. C4×C8×C251C22semidirect
149. C251#3290semidirect
154. #495D251semidirect
178. C2×Q16C2×D251semidirect
180. C2×SD16C2×D251semidirect
186. #124C2×D251semidirect
194. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306325002
#306324932
#306324882
#306319592
#306319542
#306319472
#306318642
#306318592
#306318052
#495×C2512
#3290251