Finite Group Databasebeta

← #44560368

Subgroups — #44560368

Order 32128.

448 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 129 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#30632239C211yesmaximal, characteristic, complemented
2C2×D4×Dic251C211yesmaximal, characteristic
3#30632018C211yesmaximal, characteristic, complemented
4#30632015C211yesmaximal, characteristic, complemented
5#30631975C212yesmaximal, complemented
6#30631890C212yesmaximal, complemented
7#30631838C211yesmaximal, characteristic
8#30631778C212yesmaximal, complemented
9SD16×Dic251C212yesmaximal, complemented
10#30631737C211yesmaximal, characteristic, complemented
11#450×C251C211yesmaximal, characteristic
12C23×Dic251—21no—
13C2×#16860798C2212yes—
14#22548714C2211yescharacteristic, complemented
15#22548713C2211yescharacteristic, complemented
16#22548705—21no—
17#22548694C2211yescharacteristic
18#22548692—21no—
19Q8×Dic251C2211yescharacteristic, complemented
20#22548680—21no—
21#22548680C2211yescharacteristic, complemented
22D4×Dic251—21no—
23D4×Dic251—21no—
24D4×Dic251C2212yes—
25D4×Dic251C2212yes—
26#22548671—21no—
27C2×C4⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251—21no—
29C2×C4×Dic251C2211yescharacteristic
30#22548647C2212yescomplemented
31#22548646C2212yescomplemented
32Dic251⋊C8C2212yescomplemented
33#22548640C2212yescomplemented
34#22548639C2212yescomplemented
35#22548636C2212yes—
36C22⋊(C251⋊C8)C2211yescharacteristic
37C8×Dic251C2212yescomplemented
38C22×D4×C251C2211yescharacteristic
39C2×SD16×C251C2212yes—
40C22⋊Q8×C251C2211yescharacteristic
41#102×C251C2212yes—
42C22⋊C8×C251C2211yescharacteristic
43C2×Dic502—21no—
44C2×Dic502C2311yescharacteristic
45C22×Dic251—22no—
46C22×Dic251—22no—
47C22×Dic251—21no—
48C22×Dic251—44no—
49C22×Dic251D411yescharacteristic
50C22×Dic251D411yescharacteristic
51#16860798—24no—
52C2×C251⋊C8C2312yes—
53C4⋊Dic251—21no—
54C4⋊Dic251—21no—
55C4⋊Dic251C2311yescharacteristic
56C4⋊Dic251C2312yes—
57Dic251⋊C4—21no—
58Dic251⋊C4—21no—
59Dic251⋊C4—21no—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62C22⋊Dic251—22no—
63C22⋊Dic251—22no—
64C22⋊Dic251—21no—
65C4×Dic251—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—
82Dic502—22no—
83Dic502—42no—
84C2×Dic251—21no—
85C2×Dic251—21no—
86C2×Dic251—22no—
87C2×Dic251—41no—
88C2×Dic251—42no—
89C2×Dic251—44no—
90C2×Dic251—44no—
91C2×Dic251SD1614yescomplemented
92C2×Dic251C2×D411yescharacteristic
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C251⋊C8—22no—
97C23×C251—24no—
98C23×C251—22no—
99C23×C251—42no—
100C23×C251—42no—
101C23×C251C2×D411yescharacteristic
102Q8×C251—22no—
103D4×C251—24no—
104D4×C251—22no—
105D4×C251#4114yes—
106C2×C4×C251—22no—
107C2×C4×C251—21no—
108C2×C4×C251—21no—
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2411yescharacteristic
111C8×C251—22no—
112C22×C251—21no—
113C22×C251—21no—
114C22×C251—24no—
115C22×C251—48no—
116C22×C251—42no—
117C22×C251—42no—
118C22×C251C2×SD1612yes—
119C22×C251C22×D411yescharacteristic
120C4×C251—21no—
121C4×C251—41no—
122C4×C251—41no—
123C4×C251C2×#4112yes—
124Dic251—21no—
125Dic251—21no—
126Dic251—41no—
127Dic251—42no—
128Dic251—41no—
129Dic251C2×SD1612yes—
130Dic251C2×SD1612yes—
131C2×C251—22no—
132C2×C251—44no—
133C2×C251#52511yescharacteristic
134C2×C251C22×SD1611yescharacteristic
135C2×C251#57511yescharacteristic
136C251#310111yescharacteristic, complemented, Sylow
137#3101—2511nomaximal, Sylow
138#533—2511no—
139C2×C4×D4—2511no—
140#483—2511no—
141#483—2511no—
142#476—2512no—
143#461—2512no—
144#450—2511no—
145#450D25111yescharacteristic
146C4×SD16—2512no—
147C4×SD16—2512no—
148#431—2511no—
149C22×D4C2×D25111yescharacteristic
150C23×C4—5021no—
151C2×SD16—2512no—
152C2×SD16C2×D25112yes—
153C4⋊Q8—2511no—
154C4⋊Q8—2511no—
155#125—5021no—
156C22⋊Q8—5021no—
157C22⋊Q8—2511no—
158C22⋊Q8C2×D25111yescharacteristic
159C4×Q8—5021no—
160C4×Q8—2511no—
161C4×Q8—2511no—
162C4×D4—5021no—
163C4×D4—5021no—
164C4×D4—2512no—
165C4×D4—2512no—
166#117—5021no—
167C2×C4⋊C4—2511no—
168C2×C22⋊C4—5021no—
169C2×C42—2511no—
170#106—2512no—
171#106—2512no—
172C4⋊C8—2512no—
173#103—2512no—
174#103—2512no—
175#102—2512no—
176#102C2×D25112yes—
177C22⋊C8—2511no—
178C22⋊C8C2×D25111yescharacteristic
179C4×C8—2512no—
180C24—21no—
181C2×Q8—5021no—
182C2×Q8—2511no—
183C2×Q8C22×D25111yescharacteristic
184C2×D4—24no—
185C2×D4—21no—
186C2×D4C22×D25112yes—
187C22×C4—10044no—
188C22×C4—5021no—
189C22×C4—5022no—
190C22×C4—5022no—
191C22×C4—2511no—
192C22×C4—2511no—
193C22×C4C22×D25111yescharacteristic
194SD16—24no—
195SD16—5024no—
196C2×C8—2512no—
197C2×C8C22×D25112yes—
198C4⋊C4—21no—
199C4⋊C4—5021no—
200C4⋊C4—5021no—
201C4⋊C4—5021no—
202C4⋊C4—5021no—
203C4⋊C4—5021no—
204C4⋊C4—5021no—
205C4⋊C4—5021no—
206C4⋊C4—2511no—
207C4⋊C4—2512no—
208C4⋊C4C22×D25111yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—5021no—
211C22⋊C4—5022no—
212C22⋊C4—5022no—
213C42—5021no—
214C42—5021no—
215C42—2511no—
216C42—2511no—
217C42—2511no—
218C42—2511no—
219C23—24no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23D4×D25111yescharacteristic
224Q8—22no—
225Q8—10042no—
226Q8—5022no—
227D4—24no—
228D4—22no—
229D4#1686080914yes—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—10042no—
234C2×C4—10044no—
235C2×C4—10044no—
236C2×C4—10041no—
237C2×C4—5021no—
238C2×C4—5021no—
239C2×C4—5022no—
240C2×C4—2511no—
241C2×C4—2511no—
242C2×C4—2511no—
243C2×C4—2511no—
244C2×C4—2514no—
245C2×C4D4×D25111yescharacteristic
246C2×C4C23×D25111yescharacteristic
247C8—22no—
248C8—5022no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22SD16×D25112yes—
256C22C2×D4×D25111yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—10041no—
261C4—10042no—
262C4—10041no—
263C4—5021no—
264C4—5021no—
265C4—2512no—
266C4—2512no—
267C4C2×#1686080912yes—
268C2—22no—
269C2—44no—
270C2#3063219311yescharacteristic
271C2C2×SD16×D25111yescharacteristic
272C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#450×C251C2
socleC22×C251C22×D4
prime radicalC22×D4×C251C22
hypercenter#450D251
hypocenterC251#3101
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. #30632239C2semidirect
3. #30632018C2semidirect
4. #30632015C2semidirect
5. #30631975C2semidirect
6. #30631890C2semidirect
8. #30631778C2semidirect
9. SD16×Dic251C2semidirect
10. #30631737C2semidirect
14. #22548714C22semidirect
15. #22548713C22semidirect
19. Q8×Dic251C22semidirect
21. #22548680C22semidirect
30. #22548647C22semidirect
31. #22548646C22semidirect
32. Dic251⋊C8C22semidirect
33. #22548640C22semidirect
34. #22548639C22semidirect
37. C8×Dic251C22semidirect
91. C2×Dic251SD16semidirect
136. C251#3101semidirect

Maximal subgroups

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

subgroupindex
#306322392
C2×D4×Dic2512
#306320182
#306320152
#306319752
#306318902
#306318382
#306317782
SD16×Dic2512
#306317372
#450×C2512
#3101251