Finite Group Databasebeta

← #44560504

Subgroups — #44560504

Order 32128.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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#30632223C211yesmaximal, characteristic, complemented
2C2×D4×Dic251C211yesmaximal, characteristic, complemented
3#30632015C211yesmaximal, characteristic, complemented
4#30632001C211yesmaximal, characteristic, complemented
5#30631985C212yesmaximal, complemented
6#30631904C212yesmaximal, complemented
7#30631821C211yesmaximal, characteristic
8#30631796C212yesmaximal, complemented
9#30631788C212yesmaximal, complemented
10#30631741C211yesmaximal, characteristic, complemented
11#450×C251C211yesmaximal, characteristic, complemented
12C23×Dic251—21no—
13C2×#16860795C2212yes—
14#22548710—21no—
15#22548706C2211yescharacteristic, complemented
16#22548704C2211yescharacteristic, complemented
17#22548699—21no—
18#22548689C2211yescharacteristic
19Q8×Dic251C2211yescharacteristic, complemented
20D4×Dic251—21no—
21D4×Dic251—21no—
22D4×Dic251C2212yes—
23D4×Dic251C2212yes—
24#22548677C2211yescharacteristic, complemented
25(C22⋊D251)⋊C4—21no—
26#22548673—21no—
27C2×C4⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251—21no—
29C2×C4×Dic251C2211yescharacteristic
30#22548649C2212yescomplemented
31#22548646C2212yescomplemented
32Dic251⋊C8C2212yescomplemented
33#22548639C2212yescomplemented
34#22548637C2212yescomplemented
35#22548636C2212yes—
36C22⋊(C251⋊C8)C2211yescharacteristic
37#22548625C2212yescomplemented
38C22×D4×C251C2211yescharacteristic
39C2×SD16×C251C2212yes—
40C22⋊Q8×C251C2211yescharacteristic
41#102×C251C2212yes—
42C22⋊C8×C251C2211yescharacteristic
43C2×D1004C2311yescharacteristic
44C2×C22⋊D251—21no—
45C22×Dic251—22no—
46C22×Dic251—22no—
47C22×Dic251—21no—
48C22×Dic251—44no—
49C22×Dic251D411yescharacteristic
50C22×Dic251D411yescharacteristic
51C2×C4×D251—21no—
52#16860795—24no—
53C2×C251⋊C8C2312yes—
54C4⋊Dic251—21no—
55C4⋊Dic251—21no—
56C4⋊Dic251—21no—
57C4⋊Dic251C2311yescharacteristic
58C4⋊Dic251C2312yes—
59Dic251⋊C4—21no—
60C22⋊Dic251—22no—
61C22⋊Dic251—22no—
62(C2×D251)⋊C4—21no—
63(C2×D251)⋊C4—21no—
64(C2×D251)⋊C4—21no—
65(C2×D251)⋊C4—21no—
66C4×Dic251—21no—
67C4×Dic251D411yescharacteristic
68C4×Dic251D411yescharacteristic
69C4×Dic251C2311yescharacteristic
70C4×Dic251C2311yescharacteristic
71C24×C251—21no—
72C2×Q8×C251C2311yescharacteristic
73C2×D4×C251—24no—
74C2×D4×C251—21no—
75C2×D4×C251C2312yes—
76C22×C4×C251C2311yescharacteristic
77SD16×C251—24no—
78C2×C8×C251C2312yes—
79C4⋊C4×C251—21no—
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C22×D251—21no—
83D1004—22no—
84C22⋊D251—42no—
85C2×Dic251—22no—
86C2×Dic251—21no—
87C2×Dic251—22no—
88C2×Dic251—41no—
89C2×Dic251—44no—
90C2×Dic251—44no—
91C2×Dic251—42no—
92C2×Dic251C2×D411yescharacteristic
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C4×D251—42no—
97C251⋊C8—22no—
98C23×C251—24no—
99C23×C251—22no—
100C23×C251—42no—
101C23×C251—42no—
102C23×C251C2×D411yescharacteristic
103Q8×C251—22no—
104D4×C251—24no—
105D4×C251—22no—
106D4×C251#4114yes—
107C2×C4×C251—22no—
108C2×C4×C251—21no—
109C2×C4×C251—21no—
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2411yescharacteristic
112C8×C251—22no—
113C22×C251—21no—
114C22×C251—21no—
115C22×C251—24no—
116C22×C251—48no—
117C22×C251—42no—
118C22×C251—42no—
119C22×C251AGL(1,ℤ/8)12yes—
120C22×C251C22×D411yescharacteristic
121C2×D251—41no—
122C2×D251—41no—
123C2×D251—41no—
124C4×C251—21no—
125C4×C251—41no—
126C4×C251—41no—
127C4×C251C2×#4112yes—
128Dic251—21no—
129Dic251—21no—
130Dic251—21no—
131Dic251—21no—
132Dic251—42no—
133Dic251—41no—
134C2×C251—22no—
135C2×C251—44no—
136C2×C251#52511yescharacteristic
137C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C251#57511yescharacteristic
139D251—81no—
140C251#311411yescharacteristic, complemented, Sylow
141#3114—2511nomaximal, Sylow
142#529—2511no—
143C2×C4×D4—2511no—
144#483—2511no—
145#480—2511no—
146#478—2512no—
147#463—2512no—
148#450D25111yescharacteristic, complemented
149#447—2511no—
150#442—2512no—
151#440—2512no—
152#432—2511no—
153C22×D4C2×D25111yescharacteristic
154C23×C4—5021no—
155C2×SD16C2×D25112yes—
156C2×D8—2512no—
157#126—5021no—
158#125—2511no—
159#124—2511no—
160#123—5021no—
161C22⋊Q8C2×D25111yescharacteristic
162#121—2511no—
163C4×Q8—2511no—
164C4×D4—5021no—
165C4×D4—5021no—
166C4×D4—5021no—
167C4×D4—2512no—
168C4×D4—2512no—
169C4×D4—2511no—
170#117—5021no—
171C2×C4⋊C4—2511no—
172C2×C22⋊C4—5021no—
173C2×C42—2511no—
174#107—2512no—
175#106—2512no—
176C4⋊C8—2512no—
177#103—2512no—
178#102—2512no—
179#102—2512no—
180#102C2×D25112yes—
181C22⋊C8—2511no—
182C22⋊C8C2×D25111yescharacteristic
183#97—2512no—
184C24—21no—
185C2×Q8C22×D25111yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—5021no—
189C2×D4—2511no—
190C2×D4C22×D25112yes—
191C22×C4—10044no—
192C22×C4—5021no—
193C22×C4—5021no—
194C22×C4—5022no—
195C22×C4—5022no—
196C22×C4—2511no—
197C22×C4—2511no—
198C22×C4C22×D25111yescharacteristic
199SD16—24no—
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⋊C4—2512no—
210C4⋊C4C22×D25111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—5021no—
213C22⋊C4—5021no—
214C22⋊C4—5021no—
215C22⋊C4—5022no—
216C22⋊C4—5022no—
217C22⋊C4—5021no—
218C42—5021no—
219C42—2511no—
220C42—2511no—
221C42—2511no—
222C42—2511no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—5021no—
228C23D4×D25111yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—10042no—
233D4—5022no—
234D4#1686080914yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—10042no—
239C2×C4—10042no—
240C2×C4—10044no—
241C2×C4—10044no—
242C2×C4—10041no—
243C2×C4—5021no—
244C2×C4—5022no—
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—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—10041no—
261C22—10041no—
262C22—10041no—
263C22#2254874212yes—
264C22C2×D4×D25111yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—10042no—
269C4—10041no—
270C4—5021no—
271C4—5021no—
272C4—5021no—
273C4—5021no—
274C4C2×#1686080912yes—
275C2—22no—
276C2—44no—
277C2—20081no—
278C2#3063219311yescharacteristic
279C2C2×#2254874211yescharacteristic
280C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#450×C251C2
socleC22×C251C22×D4
prime radical#30631821C2
hypercenter#450D251
hypocenterC251#3114
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #30632223C2semidirect
2. C2×D4×Dic251C2semidirect
3. #30632015C2semidirect
4. #30632001C2semidirect
5. #30631985C2semidirect
6. #30631904C2semidirect
8. #30631796C2semidirect
9. #30631788C2semidirect
10. #30631741C2semidirect
11. #450×C251C2semidirect
15. #22548706C22semidirect
16. #22548704C22semidirect
19. Q8×Dic251C22semidirect
24. #22548677C22semidirect
30. #22548649C22semidirect
31. #22548646C22semidirect
32. Dic251⋊C8C22semidirect
33. #22548639C22semidirect
34. #22548637C22semidirect
37. #22548625C22semidirect
140. C251#3114semidirect
148. #450D251semidirect

Maximal subgroups

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

subgroupindex
#306322232
C2×D4×Dic2512
#306320152
#306320012
#306319852
#306319042
#306318212
#306317962
#306317882
#306317412
#450×C2512
#3114251