Finite Group Databasebeta

← #44560295

Subgroups — #44560295

Order 32128.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 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#30632194C211yesmaximal, characteristic
2C2×#22548673C211yesmaximal, characteristic, complemented
3#30632027C212yesmaximal, complemented
4#30632026C212yesmaximal, complemented
5#30632010C212yesmaximal, complemented
6#30632009C212yesmaximal, complemented
7#30631729C211yesmaximal, characteristic, complemented
8#30631727C211yesmaximal, characteristic, complemented
9#30631688C211yesmaximal, characteristic, complemented
10#30631650C211yesmaximal, characteristic, complemented
11#417×C251C211yesmaximal, characteristic, complemented
12C22×C4×D251C2211yescharacteristic
13C2×#16860794C2211yescharacteristic
14#22548696—21no—
15#22548691C2211yescharacteristic
16#22548691C2211yescharacteristic
17#22548686C2211yescharacteristic
18#22548686C2211yescharacteristic
19#22548683—21no—
20C4×C22⋊D251—21no—
21C4×C22⋊D251—21no—
22#22548674C2211yescharacteristic
23#22548673—21no—
24#22548673—21no—
25#22548673C2212yes—
26#22548673C2212yes—
27C2×C4⋊Dic251C2211yescharacteristic
28C2×(C2×D251)⋊C4—21no—
29C2×C4×Dic251—21no—
30#22548649C2212yescomplemented
31#22548648C2212yescomplemented
32#22548647C2212yescomplemented
33#22548646C2212yescomplemented
34#22548639C2212yescomplemented
35#22548636C2212yescomplemented
36(C2×D251)⋊C8C2212yes—
37(C2×D251)⋊C8C2212yes—
38C2×#41×C251C2211yescharacteristic
39C2×M16×C251C2211yescharacteristic
40C2×C4⋊C4×C251C2211yescharacteristic
41#103×C251C2212yescomplemented
42#102×C251C2212yescomplemented
43C23×D251D411yescharacteristic
44C2×C22⋊D251—21no—
45C2×C22⋊D251—21no—
46C22×Dic251—21no—
47C22×Dic251D411yescharacteristic
48C2×C4×D251—22no—
49C2×C4×D251—22no—
50C2×C4×D251D411yescharacteristic
51C2×C4×D251D411yescharacteristic
52C2×C4×D251C2311yescharacteristic
53C2×C4×D251C2311yescharacteristic
54#16860794—22no—
55C2×C251⋊C8C2312yes—
56C4⋊Dic251—21no—
57C4⋊Dic251C2312yes—
58C4⋊Dic251C2311yescharacteristic
59C4⋊Dic251C2311yescharacteristic
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62C22⋊Dic251—21no—
63C22⋊Dic251—21no—
64C22⋊Dic251—21no—
65(C2×D251)⋊C4—22no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—21no—
68(C2×D251)⋊C4—21no—
69C4×Dic251—22no—
70C4×Dic251—22no—
71C4×Dic251—21no—
72#41×C251—22no—
73#41×C251—22no—
74C2×Q8×C251C2311yescharacteristic
75C2×D4×C251—21no—
76C2×D4×C251C2311yescharacteristic
77C22×C4×C251—21no—
78C22×C4×C251—21no—
79C22×C4×C251C2311yescharacteristic
80M16×C251—22no—
81C2×C8×C251C2312yes—
82C4⋊C4×C251—21no—
83C4⋊C4×C251C2312yes—
84C22×D251—22no—
85C22×D251—41no—
86C22×D251—41no—
87C22×D251C2×D411yescharacteristic
88C22×D251C2×D411yescharacteristic
89C22⋊D251—42no—
90C22⋊D251—42no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—42no—
96C2×Dic251C2×D411yescharacteristic
97C2×Dic251C2×D411yescharacteristic
98C4×D251—22no—
99C4×D251—22no—
100C4×D251—21no—
101C4×D251—21no—
102C4×D251—21no—
103C4×D251—21no—
104C251⋊C8—22no—
105C23×C251—21no—
106C23×C251C2×D411yescharacteristic
107Q8×C251—21no—
108Q8×C251—21no—
109D4×C251—21no—
110D4×C251—21no—
111D4×C251—42no—
112C2×C4×C251—22no—
113C2×C4×C251—21no—
114C2×C4×C251—42no—
115C2×C4×C251—42no—
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251#4112yes—
118C2×C4×C251#4112yes—
119C2×C4×C251C2411yescharacteristic
120C8×C251—22no—
121C22×C251—21no—
122C22×C251—21no—
123C22×C251—41no—
124C22×C251—41no—
125C22×C251—41no—
126C22×C251#12312yes—
127C22×C251C22×D411yescharacteristic
128C2×D251—21no—
129C2×D251—21no—
130C2×D251—21no—
131C2×D251—21no—
132C2×D251—21no—
133C2×D251—21no—
134C2×D251—44no—
135C4×C251—41no—
136C4×C251—42no—
137C4×C251#12312yes—
138C4×C251C2×#4111yescharacteristic
139C4×C251C2×#4111yescharacteristic
140Dic251—21no—
141Dic251—21no—
142Dic251—42no—
143Dic251—41no—
144Dic251—41no—
145C2×C251—22no—
146C2×C251—81no—
147C2×C251C2×#12311yescharacteristic
148C2×C251#57511yescharacteristic
149C2×C251#57511yescharacteristic
150D251—41no—
151D251—41no—
152C251#309311yescharacteristic, complemented, Sylow
153#3093—2511nomaximal, Sylow
154#525—2511no—
155C2×#117—2511no—
156#485—2512no—
157#485—2512no—
158#482—2512no—
159#482—2512no—
160#428—2511no—
161#428—2511no—
162#417—2511no—
163#417D25111yescharacteristic, complemented
164#407—2511no—
165C2×#41C2×D25111yescharacteristic
166C23×C4—2511no—
167C2×M16—2511no—
168C2×M16C2×D25111yescharacteristic
169#123—5021no—
170C22⋊Q8—2511no—
171C22⋊Q8—2511no—
172#121—2511no—
173#121—2511no—
174#120—5021no—
175C4×D4—5021no—
176C4×D4—5021no—
177#117—5021no—
178#117—5021no—
179#117—2511no—
180#117—2512no—
181#117—2512no—
182C2×C4⋊C4—2511no—
183C2×C4⋊C4C2×D25111yescharacteristic
184C2×C22⋊C4—5021no—
185C2×C42—5021no—
186#107—2512no—
187#107—2512no—
188#106—2512no—
189#106—2512no—
190#103—2512no—
191#103C2×D25112yescomplemented
192#102—2512no—
193#102C2×D25112yescomplemented
194C22⋊C8—2512no—
195C22⋊C8—2512no—
196C24—2511no—
197#41—22no—
198#41—22no—
199C2×Q8C22×D25111yescharacteristic
200C2×D4—21no—
201C2×D4—5021no—
202C2×D4—5021no—
203C2×D4C22×D25111yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—5021no—
207C22×C4—5022no—
208C22×C4—5022no—
209C22×C4—2511no—
210C22×C4—2511no—
211C22×C4—2511no—
212C22×C4—2511no—
213C22×C4—2511no—
214C22×C4C22×D25111yescharacteristic
215M16—22no—
216M16—5022no—
217C2×C8—2512no—
218C2×C8C22×D25112yes—
219C4⋊C4—21no—
220C4⋊C4—5021no—
221C4⋊C4—5021no—
222C4⋊C4—5021no—
223C4⋊C4—2512no—
224C4⋊C4—2511no—
225C4⋊C4—2511no—
226C4⋊C4C22×D25112yes—
227C22⋊C4—5022no—
228C22⋊C4—5022no—
229C22⋊C4—5021no—
230C22⋊C4—5021no—
231C22⋊C4—5021no—
232C22⋊C4—5021no—
233C22⋊C4—5021no—
234C42—5022no—
235C42—5022no—
236C42—5021no—
237C23—21no—
238C23—10041no—
239C23—10041no—
240C23—5022no—
241C23—2511no—
242C23—2511no—
243C23D4×D25111yescharacteristic
244Q8—21no—
245Q8—21no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—10042no—
250D4—10042no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—10042no—
256C2×C4—5022no—
257C2×C4—5021no—
258C2×C4—5021no—
259C2×C4—5022no—
260C2×C4—5021no—
261C2×C4—5021no—
262C2×C4—5021no—
263C2×C4—5021no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—2511no—
267C2×C4—2511no—
268C2×C4D4×D25111yescharacteristic
269C2×C4#1686080912yes—
270C2×C4#1686081012yes—
271C2×C4C23×D25111yescharacteristic
272C8—22no—
273C8—5022no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—10044no—
280C22—5021no—
281C22—5021no—
282C22—5021no—
283C22—5021no—
284C22—5021no—
285C22—5021no—
286C22#2254869712yes—
287C22C2×D4×D25111yescharacteristic
288C4—42no—
289C4—41no—
290C4—10041no—
291C4—10042no—
292C4—10041no—
293C4—5021no—
294C4—5021no—
295C4#2254869712yes—
296C4C2×#1686080911yescharacteristic
297C4C2×#1686081011yescharacteristic
298C2—22no—
299C2—81no—
300C2—10041no—
301C2—10041no—
302C2C2×#2254869711yescharacteristic
303C2#3063248311yescharacteristic
304C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#417×C251C2
socleC22×C251C22×D4
prime radical#30632194C2
hypercenter#417D251
hypocenterC251#3093
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#22548673C2semidirect
3. #30632027C2semidirect
4. #30632026C2semidirect
5. #30632010C2semidirect
6. #30632009C2semidirect
7. #30631729C2semidirect
8. #30631727C2semidirect
9. #30631688C2semidirect
10. #30631650C2semidirect
11. #417×C251C2semidirect
30. #22548649C22semidirect
31. #22548648C22semidirect
32. #22548647C22semidirect
33. #22548646C22semidirect
34. #22548639C22semidirect
35. #22548636C22semidirect
41. #103×C251C22semidirect
42. #102×C251C22semidirect
152. C251#3093semidirect
163. #417D251semidirect
191. #103C2×D251semidirect
193. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306321942
C2×#225486732
#306320272
#306320262
#306320102
#306320092
#306317292
#306317272
#306316882
#306316502
#417×C2512
#3093251