Finite Group Databasebeta

← #44560500

Subgroups — #44560500

Order 32128.

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

normal subgroups only

This group has 450 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#30632218C211yesmaximal, characteristic, complemented
2C2×#22548677C211yesmaximal, characteristic, complemented
3#30632017C211yesmaximal, characteristic, complemented
4#30631990C212yesmaximal, complemented
5#30631901C212yesmaximal, complemented
6#30631840C211yesmaximal, characteristic, complemented
7#30631820C211yesmaximal, characteristic
8#30631796C212yesmaximal, complemented
9#30631786C212yesmaximal, complemented
10#30631741C211yesmaximal, characteristic, complemented
11#480×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C4×D251—21no—
14C2×#16860799C2212yes—
15C2×#16860795C2212yes—
16#22548708—21no—
17#22548706C2211yescharacteristic, complemented
18#22548702C2211yescharacteristic, complemented
19#22548696—21no—
20#22548694C2211yescharacteristic
21#22548680C2211yescharacteristic, complemented
22D4×Dic251—21no—
23D4×Dic251C2211yescharacteristic, complemented
24#22548677—21no—
25#22548677—21no—
26#22548677C2212yes—
27#22548677C2212yes—
28#22548671—21no—
29C2×(C2×D251)⋊C4—21no—
30C2×C4×Dic251C2211yescharacteristic
31#22548646C2212yescomplemented
32Dic251⋊C8C2212yescomplemented
33#22548638C2212yescomplemented
34#22548637C2212yes—
35#22548635C2212yes—
36C22⋊(C251⋊C8)C2211yescharacteristic
37#22548625C2212yescomplemented
38#121×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic
40#107×C251C2212yescomplemented
41#102×C251C2212yescomplemented
42C22⋊C8×C251C2211yescharacteristic
43C23×D251—21no—
44C2×Dic502C2311yescharacteristic
45C2×D1004—24no—
46C2×D1004—21no—
47C2×D1004C2312yes—
48C22×Dic251—21no—
49C22×Dic251D411yescharacteristic
50C22×Dic251D411yescharacteristic
51C2×C4×D251—22no—
52C2×C4×D251—22no—
53C2×C4×D251—44no—
54#16860799—24no—
55#16860795—24no—
56C2×C251⋊C8C2312yes—
57C4⋊Dic251—21no—
58C4⋊Dic251C2311yescharacteristic
59Dic251⋊C4—21no—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62C22⋊Dic251—21no—
63C22⋊Dic251—21no—
64C22⋊Dic251—21no—
65C22⋊Dic251—21no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—22no—
68C4×Dic251—21no—
69C4×Dic251D411yescharacteristic
70C4×Dic251D411yescharacteristic
71C4×Dic251C2311yescharacteristic
72C4×Dic251C2311yescharacteristic
73C2×D4×C251—21no—
74C2×D4×C251C2311yescharacteristic
75C22×C4×C251—21no—
76C22×C4×C251C2311yescharacteristic
77C2×C8×C251C2312yes—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2312yes—
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C22×D251—24no—
83C22×D251—22no—
84C22×D251—42no—
85C22×D251—42no—
86Dic502—22no—
87D1004—24no—
88D1004—22no—
89D1004#4114yes—
90C2×Dic251—22no—
91C2×Dic251—21no—
92C2×Dic251—21no—
93C2×Dic251—41no—
94C2×Dic251—42no—
95C2×Dic251C2×D411yescharacteristic
96C2×Dic251C2×D411yescharacteristic
97C2×Dic251C2×D411yescharacteristic
98C2×Dic251C2×D411yescharacteristic
99C4×D251—44no—
100C4×D251—44no—
101C251⋊C8—22no—
102C23×C251—21no—
103C23×C251C2×D411yescharacteristic
104D4×C251—22no—
105D4×C251—42no—
106C2×C4×C251—22no—
107C2×C4×C251—21no—
108C2×C4×C251—22no—
109C2×C4×C251—42no—
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2411yescharacteristic
112C8×C251—22no—
113C22×C251—21no—
114C22×C251—21no—
115C22×C251—41no—
116C22×C251—41no—
117C22×C251—41no—
118C22×C251AGL(1,ℤ/8)12yes—
119C22×C251C22×D411yescharacteristic
120C2×D251—24no—
121C2×D251—48no—
122C2×D251—42no—
123C2×D251—42no—
124C4×C251—21no—
125C4×C251—42no—
126C4×C251—41no—
127C4×C251C2×#4112yes—
128Dic251—21no—
129Dic251—21no—
130Dic251—21no—
131Dic251—21no—
132Dic251—41no—
133Dic251—41no—
134C2×C251—22no—
135C2×C251—81no—
136C2×C251#52511yescharacteristic
137C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C251#57511yescharacteristic
139D251—44no—
140C251#311411yescharacteristic, complemented, Sylow
141#3114—2511nomaximal, Sylow
142#529—2511no—
143C2×C4×D4—2511no—
144#483—2511no—
145#480D25111yescharacteristic, complemented
146#478—2512no—
147#463—2512no—
148#450—2511no—
149#447—2511no—
150#442—2512no—
151#440—2512no—
152#432—2511no—
153C22×D4—2511no—
154C23×C4—5021no—
155C2×SD16—2512no—
156C2×D8—2512no—
157#126—5021no—
158#125—2511no—
159#124—2511no—
160#123—5021no—
161C22⋊Q8—2511no—
162#121C2×D25111yescharacteristic
163C4×Q8—2511no—
164C4×D4—5021no—
165C4×D4—5021no—
166C4×D4—5021no—
167C4×D4—2511no—
168C4×D4—2512no—
169C4×D4—2512no—
170#117—5021no—
171C2×C4⋊C4C2×D25111yescharacteristic
172C2×C22⋊C4—5021no—
173C2×C42—2511no—
174#107C2×D25112yescomplemented
175#106—2512no—
176C4⋊C8—2512no—
177#103—2512no—
178#102—2512no—
179#102—2512no—
180#102C2×D25112yescomplemented
181C22⋊C8—2511no—
182C22⋊C8C2×D25111yescharacteristic
183#97—2512no—
184C24—5021no—
185C2×Q8—2511no—
186C2×D4—21no—
187C2×D4—5024no—
188C2×D4—5021no—
189C2×D4—2512no—
190C2×D4C22×D25111yescharacteristic
191C22×C4—21no—
192C22×C4—10044no—
193C22×C4—5022no—
194C22×C4—5022no—
195C22×C4—5021no—
196C22×C4—2511no—
197C22×C4—2511no—
198C22×C4C22×D25111yescharacteristic
199SD16—5024no—
200D8—5024no—
201C2×C8—2512no—
202C2×C8C22×D25112yes—
203C4⋊C4—21no—
204C4⋊C4—5021no—
205C4⋊C4—5021no—
206C4⋊C4—5021no—
207C4⋊C4—5021no—
208C4⋊C4—2511no—
209C4⋊C4C22×D25111yescharacteristic
210C4⋊C4C22×D25112yes—
211C22⋊C4—21no—
212C22⋊C4—5022no—
213C22⋊C4—5022no—
214C22⋊C4—5021no—
215C22⋊C4—5021no—
216C22⋊C4—5021no—
217C22⋊C4—5021no—
218C42—5021no—
219C42—2511no—
220C42—2511no—
221C42—2511no—
222C42—2511no—
223C23—21no—
224C23—10042no—
225C23—10042no—
226C23—5024no—
227C23—5022no—
228C23D4×D25111yescharacteristic
229Q8—5022no—
230D4—22no—
231D4—42no—
232D4—5024no—
233D4—5022no—
234D4—2514no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—10044no—
240C2×C4—10044no—
241C2×C4—10042no—
242C2×C4—10041no—
243C2×C4—5021no—
244C2×C4—5021no—
245C2×C4—5022no—
246C2×C4—2511no—
247C2×C4—2511no—
248C2×C4—2511no—
249C2×C4—2511no—
250C2×C4D4×D25111yescharacteristic
251C2×C4C23×D25111yescharacteristic
252C8—22no—
253C8—5022no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—10048no—
260C22—10042no—
261C22—10042no—
262C22—5024no—
263C22#2254873912yes—
264C22C2×D4×D25111yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—10041no—
269C4—10041no—
270C4—5021no—
271C4—5021no—
272C4—5021no—
273C4—5021no—
274C4#41×D25112yes—
275C2—22no—
276C2—81no—
277C2—10044no—
278C2#3063219511yescharacteristic
279C2C2×#2254873911yescharacteristic
280C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#480×C251C2
socleC22×C251C22×D4
prime radical#30631820C2
hypercenter#480D251
hypocenterC251#3114
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #30632218C2semidirect
2. C2×#22548677C2semidirect
3. #30632017C2semidirect
4. #30631990C2semidirect
5. #30631901C2semidirect
6. #30631840C2semidirect
8. #30631796C2semidirect
9. #30631786C2semidirect
10. #30631741C2semidirect
11. #480×C251C2semidirect
17. #22548706C22semidirect
18. #22548702C22semidirect
21. #22548680C22semidirect
23. D4×Dic251C22semidirect
31. #22548646C22semidirect
32. Dic251⋊C8C22semidirect
33. #22548638C22semidirect
37. #22548625C22semidirect
40. #107×C251C22semidirect
41. #102×C251C22semidirect
140. C251#3114semidirect
145. #480D251semidirect
174. #107C2×D251semidirect
180. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322182
C2×#225486772
#306320172
#306319902
#306319012
#306318402
#306318202
#306317962
#306317862
#306317412
#480×C2512
#3114251