Finite Group Databasebeta

← #44562153

Subgroups — #44562153

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#30632499C212yesmaximal, complemented
3#30632489C211yesmaximal, characteristic, complemented
4#30632071C211yesmaximal, characteristic, complemented
5#30631963C211yesmaximal, characteristic, complemented
6#30631958C211yesmaximal, characteristic, complemented
7#30631950C212yesmaximal, complemented
8#30631866C212yesmaximal, complemented
9#30631861C212yesmaximal, complemented
10#30631805C211yesmaximal, characteristic, complemented
11#443×C251C211yesmaximal, characteristic, complemented
12#22548762C2212yescomplemented
13#22548761C2212yes—
14#22548748—22no—
15#22548748C2212yes—
16#22548745C2212yescomplemented
17#22548744—22no—
18#22548743—22no—
19#22548743C2212yescomplemented
20#22548742—22no—
21#22548740C2212yes—
22#22548738—21no—
23#22548737—22no—
24#22548737C2211yescharacteristic
25#22548734—21no—
26C2×Dic1004C2211yescharacteristic, complemented
27Q16×D251—21no—
28C2×#16860799—22no—
29C2×#16860799C2212yescomplemented
30C2×D2008C2211yescharacteristic
31D8×D251—21no—
32#22548721C2211yescharacteristic
33#22548721C2211yescharacteristic, complemented
34#22548703C2212yescomplemented
35#22548651C2211yescharacteristic, complemented
36#22548643C2211yescharacteristic
37#22548643C2211yescharacteristic, complemented
38#22548633C2212yescomplemented
39#22548631C2212yescomplemented
40#135×C251C2211yescharacteristic
41#131×C251C2212yescomplemented
42#108×C251C2211yescharacteristic, complemented
43#104×C251C2212yescomplemented
44C4×C8×C251C2211yescharacteristic, complemented
45#16860811—22no—
46#16860810—22no—
47#16860810—22no—
48#16860810C2311yescharacteristic
49#16860810C2311yescharacteristic
50#16860809—22no—
51#16860809—42no—
52C2×Dic502—22no—
53C2×Dic502C2312yes—
54Q8×D251—22no—
55C2×D1004—22no—
56C2×D1004C2312yes—
57D4×D251—22no—
58D4×D251—42no—
59Dic1004—22no—
60Dic1004—21no—
61Dic1004—21no—
62Dic1004D411yescharacteristic
63#16860800—22no—
64#16860799—22no—
65#16860799—22no—
66#16860799—22no—
67#16860799—22no—
68#16860799D411yescharacteristic
69#16860799D411yescharacteristic
70#16860798—22no—
71#16860797—22no—
72D2008—22no—
73D2008—21no—
74D2008—21no—
75D2008D411yescharacteristic
76#16860795—22no—
77#16860794C2311yescharacteristic
78#16860794C2311yescharacteristic
79#16860793—21no—
80#16860793—21no—
81C8×D251—21no—
82C8×D251—21no—
83(C2×D251)⋊C4—42no—
84#41×C251C2312yes—
85Q16×C251D411yescharacteristic
86SD16×C251D412yes—
87D8×C251D411yescharacteristic
88M16×C251—22no—
89M16×C251C2312yes—
90C2×C8×C251—22no—
91C2×C8×C251C2311yescharacteristic
92C2×C8×C251C2311yescharacteristic
93C42×C251C2311yescharacteristic
94C22×D251—22no—
95C22×D251—42no—
96Dic502—22no—
97Dic502—22no—
98Dic502—42no—
99Dic502C2×D411yescharacteristic
100Dic502C2×D411yescharacteristic
101D1004—22no—
102D1004—22no—
103D1004—42no—
104D1004C2×D411yescharacteristic
105D1004C2×D411yescharacteristic
106C22⋊D251—21no—
107C22⋊D251—21no—
108C22⋊D251—42no—
109C22⋊D251—42no—
110C2×Dic251—22no—
111C2×Dic251—42no—
112C4×D251—21no—
113C4×D251—21no—
114C4×D251—42no—
115C4×D251—42no—
116C251⋊C8—21no—
117C251⋊C8—21no—
118Q8×C251C2×D412yes—
119D4×C251—22no—
120D4×C251C2×D412yes—
121C2×C4×C251—22no—
122C2×C4×C251—21no—
123C2×C4×C251C2411yescharacteristic
124C8×C251—22no—
125C8×C251C2×D411yescharacteristic
126C8×C251C2×D411yescharacteristic
127C8×C251C2×D411yescharacteristic
128C8×C251C2×D411yescharacteristic
129C22×C251—22no—
130C22×C251H(5,𝔽2)11yescharacteristic
131C2×D251—21no—
132C2×D251—21no—
133C2×D251—42no—
134C2×D251—82no—
135C2×D251—162no—
136C4×C251—22no—
137C4×C251—41no—
138C4×C251C22×D411yescharacteristic
139C4×C251C22×D411yescharacteristic
140Dic251—21no—
141Dic251—21no—
142Dic251—42no—
143C2×C251—21no—
144C2×C251—42no—
145C2×C251D4211yescharacteristic
146D251—41no—
147D251—41no—
148D251—82no—
149C251#329011yescharacteristic, complemented, Sylow
150#3290—2511nomaximal, Sylow
151#578—2511no—
152#577—2512no—
153#576—2511no—
154#495—2511no—
155#473—2511no—
156#472—2511no—
157#471—2512no—
158#455—2512no—
159#454—2512no—
160#443—2511no—
161#443D25111yescharacteristic, complemented
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—5022no—
173#135—5021no—
174#135—5021no—
175#135—2511no—
176#135C2×D25111yescharacteristic
177C2×Q16—5021no—
178C2×Q16—2511no—
179C2×SD16—5022no—
180C2×SD16—2512no—
181C2×D8—5021no—
182C2×D8—2511no—
183#131—2511no—
184#131—2511no—
185#131C2×D25112yescomplemented
186#124—2512no—
187#108—2511no—
188#108C2×D25111yescharacteristic, complemented
189#104—2511no—
190#104—2511no—
191#104C2×D25112yescomplemented
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—2511no—
202#41C22×D25112yes—
203C2×Q8—5022no—
204C2×Q8—5022no—
205C2×Q8—2512no—
206C2×D4—10042no—
207C2×D4—5022no—
208C2×D4—5022no—
209C2×D4—2512no—
210Q16—5022no—
211Q16—5022no—
212Q16—5021no—
213Q16—5021no—
214Q16—2511no—
215Q16D100411yescharacteristic
216SD16—5022no—
217SD16—5022no—
218SD16—5022no—
219SD16—5022no—
220SD16—5022no—
221SD16—5022no—
222SD16—2511no—
223SD16—2511no—
224SD16D100412yes—
225D8—5022no—
226D8—5022no—
227D8—5021no—
228D8—5021no—
229D8—2511no—
230D8D100411yescharacteristic
231M16—22no—
232M16—5021no—
233M16—5021no—
234M16—2511no—
235M16—2511no—
236M16C22×D25112yes—
237C2×C8—22no—
238C2×C8—5021no—
239C2×C8—5021no—
240C2×C8C22×D25111yescharacteristic
241C2×C8C22×D25111yescharacteristic
242C22⋊C4—10042no—
243C42C22×D25111yescharacteristic
244C23—10042no—
245C23—5022no—
246Q8—10042no—
247Q8—5022no—
248Q8—5022no—
249Q8—2511no—
250Q8—2511no—
251Q8C2×D100412yes—
252D4—22no—
253D4—10042no—
254D4—10042no—
255D4—10042no—
256D4—5021no—
257D4—5021no—
258D4—5022no—
259D4—5022no—
260D4—2511no—
261D4—2511no—
262D4C2×D100412yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—10042no—
266C2×C4—10042no—
267C2×C4—10042no—
268C2×C4—5022no—
269C2×C4—5021no—
270C2×C4—5021no—
271C2×C4C23×D25111yescharacteristic
272C8—22no—
273C8—5021no—
274C8—5021no—
275C8D4×D25111yescharacteristic
276C8D4×D25111yescharacteristic
277C8C2×D100411yescharacteristic
278C8C2×D100411yescharacteristic
279C22—22no—
280C22—40162no—
281C22—20082no—
282C22—10042no—
283C22—5021no—
284C22—5021no—
285C22#2254876111yescharacteristic
286C4—22no—
287C4—41no—
288C4—10042no—
289C4—5021no—
290C4—5021no—
291C4C2×D4×D25111yescharacteristic
292C4C22×D100411yescharacteristic
293C2—21no—
294C2—42no—
295C2—20082no—
296C2—10041no—
297C2—10041no—
298C2D4×D100411yescharacteristic
subgroupG/H
centerC2D4×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#443×C251C2
socleC2×C251D42
hypercenter#443D251
hypocenterC251#3290
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #30632501C2semidirect
2. #30632499C2semidirect
3. #30632489C2semidirect
4. #30632071C2semidirect
5. #30631963C2semidirect
6. #30631958C2semidirect
7. #30631950C2semidirect
8. #30631866C2semidirect
9. #30631861C2semidirect
10. #30631805C2semidirect
11. #443×C251C2semidirect
12. #22548762C22semidirect
16. #22548745C22semidirect
19. #22548743C22semidirect
26. C2×Dic1004C22semidirect
29. C2×#16860799C22semidirect
33. #22548721C22semidirect
34. #22548703C22semidirect
35. #22548651C22semidirect
37. #22548643C22semidirect
38. #22548633C22semidirect
39. #22548631C22semidirect
41. #131×C251C22semidirect
42. #108×C251C22semidirect
43. #104×C251C22semidirect
44. C4×C8×C251C22semidirect
149. C251#3290semidirect
161. #443D251semidirect
185. #131C2×D251semidirect
188. #108C2×D251semidirect
191. #104C2×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
#306325012
#306324992
#306324892
#306320712
#306319632
#306319582
#306319502
#306318662
#306318612
#306318052
#443×C2512
#3290251