Finite Group Databasebeta

← #44560415

Subgroups — #44560415

Order 32128.

454 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 119 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×#22548677C211yesmaximal, characteristic, complemented
3#30632005C211yesmaximal, characteristic, complemented
4#30631990C212yesmaximal, complemented
5#30631905C212yesmaximal, complemented
6#30631837C211yesmaximal, characteristic, complemented
7#30631823C211yesmaximal, characteristic
8#30631772C212yesmaximal, complemented
9#30631769C212yesmaximal, complemented
10#30631737C211yesmaximal, characteristic, complemented
11#484×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C4×D251—21no—
14C2×#16860797C2212yes—
15C2×D2008C2212yes—
16#22548710—21no—
17#22548706C2211yescharacteristic, complemented
18#22548704C2211yescharacteristic, complemented
19#22548699—21no—
20#22548689C2211yescharacteristic
21Q8×Dic251C2211yescharacteristic, complemented
22#22548677—21no—
23#22548677—21no—
24#22548677C2212yes—
25#22548677C2212yes—
26#22548677C2211yescharacteristic, complemented
27(C22⋊D251)⋊C4—21no—
28#22548673—21no—
29C2×(C2×D251)⋊C4—21no—
30C2×C4×Dic251C2211yescharacteristic
31#22548646C2212yescomplemented
32Dic251⋊C8C2212yescomplemented
33#22548637C2212yes—
34#22548635C2212yes—
35#22548635C2212yescomplemented
36C22⋊(C251⋊C8)C2211yescharacteristic
37C8×Dic251C2212yescomplemented
38C22⋊Q8×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic
40#107×C251C2212yescomplemented
41#103×C251C2212yescomplemented
42C22⋊C8×C251C2211yescharacteristic
43C23×D251—21no—
44C2×D1004—24no—
45C2×D1004—21no—
46C2×D1004C2311yescharacteristic
47C2×D1004C2312yes—
48C2×C22⋊D251—21no—
49C22×Dic251D411yescharacteristic
50C22×Dic251D411yescharacteristic
51C2×C4×D251—22no—
52C2×C4×D251—22no—
53C2×C4×D251—21no—
54C2×C4×D251—44no—
55#16860797—24no—
56D2008—24no—
57C2×C251⋊C8C2312yes—
58C4⋊Dic251—21no—
59C4⋊Dic251—21no—
60C4⋊Dic251C2311yescharacteristic
61Dic251⋊C4—21no—
62(C2×D251)⋊C4—22no—
63(C2×D251)⋊C4—22no—
64(C2×D251)⋊C4—21no—
65(C2×D251)⋊C4—21no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4—21no—
68C4×Dic251—21no—
69C4×Dic251D411yescharacteristic
70C4×Dic251D411yescharacteristic
71C4×Dic251C2311yescharacteristic
72C4×Dic251C2311yescharacteristic
73C2×Q8×C251C2311yescharacteristic
74C22×C4×C251—21no—
75C22×C4×C251C2311yescharacteristic
76C2×C8×C251C2312yes—
77C4⋊C4×C251—21no—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2312yes—
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C22×D251—24no—
83C22×D251—22no—
84C22×D251—21no—
85C22×D251—42no—
86C22×D251—42no—
87D1004—24no—
88D1004—22no—
89D1004—22no—
90D1004#4114yes—
91C22⋊D251—42no—
92C2×Dic251—22no—
93C2×Dic251—22no—
94C2×Dic251—21no—
95C2×Dic251C2×D411yescharacteristic
96C2×Dic251C2×D411yescharacteristic
97C2×Dic251C2×D411yescharacteristic
98C2×Dic251C2×D411yescharacteristic
99C4×D251—44no—
100C4×D251—42no—
101C4×D251—44no—
102C251⋊C8—22no—
103C23×C251C2×D411yescharacteristic
104Q8×C251—22no—
105C2×C4×C251—22no—
106C2×C4×C251—21no—
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#13512yes—
116C22×C251C22×D411yescharacteristic
117C2×D251—24no—
118C2×D251—48no—
119C2×D251—42no—
120C2×D251—42no—
121C2×D251—41no—
122C2×D251—41no—
123C2×D251—41no—
124C4×C251—21no—
125C4×C251—41no—
126C4×C251—42no—
127C4×C251—41no—
128C4×C251C2×#4112yes—
129Dic251—21no—
130Dic251—21no—
131Dic251—21no—
132Dic251—41no—
133Dic251AGL(1,ℤ/8)12yes—
134C2×C251—22no—
135C2×C251#52511yescharacteristic
136C2×C251C2×#13511yescharacteristic
137C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
138D251—44no—
139D251—81no—
140C251#310511yescharacteristic, complemented, Sylow
141#3105—2511nomaximal, Sylow
142#529—2511no—
143C2×C4×D4—2511no—
144#484D25111yescharacteristic, complemented
145#481—2511no—
146#478—2512no—
147#463—2512no—
148#450—2511no—
149#447—2511no—
150C4×SD16—2512no—
151C4×D8—2512no—
152#431—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⋊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⋊C4C2×D25111yescharacteristic
172C2×C22⋊C4—5021no—
173C2×C42—2511no—
174#107C2×D25112yescomplemented
175#106—2512no—
176C4⋊C8—2512no—
177#103C2×D25112yescomplemented
178#102—2512no—
179#102—2512no—
180#102—2512no—
181C22⋊C8—2511no—
182C22⋊C8C2×D25111yescharacteristic
183C4×C8—2512no—
184C24—5021no—
185C2×Q8C22×D25111yescharacteristic
186C2×D4—5024no—
187C2×D4—5021no—
188C2×D4—5021no—
189C2×D4—2511no—
190C2×D4—2512no—
191C22×C4—21no—
192C22×C4—10044no—
193C22×C4—5021no—
194C22×C4—5022no—
195C22×C4—5022no—
196C22×C4—2511no—
197C22×C4—2511no—
198C22×C4C22×D25111yescharacteristic
199SD16—5024no—
200D8—5024no—
201C2×C8—2512no—
202C2×C8C22×D25112yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
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—10042no—
224C23—10042no—
225C23—5024no—
226C23—5021no—
227C23—5022no—
228C23D4×D25111yescharacteristic
229Q8—22no—
230D4—10042no—
231D4—5024no—
232D4—5022no—
233D4—5022no—
234D4—2514no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—10044no—
241C2×C4—10044no—
242C2×C4—10042no—
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—10048no—
257C22—10041no—
258C22—10041no—
259C22—10041no—
260C22—10042no—
261C22—10042no—
262C22—5024no—
263C22#2254873812yes—
264C22C2×D4×D25111yescharacteristic
265C4—21no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—10041no—
270C4—5021no—
271C4—5021no—
272C4—5021no—
273C4—2512no—
274C4#41×D25112yes—
275C2—22no—
276C2—20081no—
277C2—10044no—
278C2#3063219511yescharacteristic
279C2C2×#2254873811yescharacteristic
280C2AGL(1,ℤ/8)×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#484×C251C2
socleC22×C251C22×D4
prime radical#30631823C2
hypercenter#484D251
hypocenterC251#3105
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #30632223C2semidirect
2. C2×#22548677C2semidirect
3. #30632005C2semidirect
4. #30631990C2semidirect
5. #30631905C2semidirect
6. #30631837C2semidirect
8. #30631772C2semidirect
9. #30631769C2semidirect
10. #30631737C2semidirect
11. #484×C251C2semidirect
17. #22548706C22semidirect
18. #22548704C22semidirect
21. Q8×Dic251C22semidirect
26. #22548677C22semidirect
31. #22548646C22semidirect
32. Dic251⋊C8C22semidirect
35. #22548635C22semidirect
37. C8×Dic251C22semidirect
40. #107×C251C22semidirect
41. #103×C251C22semidirect
140. C251#3105semidirect
144. #484D251semidirect
174. #107C2×D251semidirect
177. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322232
C2×#225486772
#306320052
#306319902
#306319052
#306318372
#306318232
#306317722
#306317692
#306317372
#484×C2512
#3105251