Finite Group Databasebeta

← #44560362

Subgroups — #44560362

Order 32128.

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

normal subgroups only

This group has 492 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#30632232C211yesmaximal, characteristic, complemented
2C2×#22548677C211yesmaximal, characteristic, complemented
3#30632000C211yesmaximal, characteristic, complemented
4#30631966C212yesmaximal, complemented
5#30631881C212yesmaximal, complemented
6#30631823C211yesmaximal, characteristic, complemented
7#30631820C211yesmaximal, characteristic, complemented
8#30631769C212yesmaximal, complemented
9#30631767C212yesmaximal, complemented
10#30631737C211yesmaximal, characteristic, complemented
11#480×C251C211yesmaximal, characteristic, complemented
12C2×#16860809—21no—
13C22×D1004C2211yescharacteristic
14C22×C4×D251—21no—
15C2×D2008C2212yes—
16C2×#16860795C2212yes—
17#22548714C2211yescharacteristic, complemented
18#22548711C2211yescharacteristic, complemented
19#22548701—21no—
20#22548689C2211yescharacteristic
21#22548687—21no—
22D4×Dic251C2211yescharacteristic, complemented
23#22548677—21no—
24#22548677—21no—
25#22548677C2212yes—
26#22548677C2212yes—
27#22548677C2211yescharacteristic, complemented
28(C22⋊D251)⋊C4—21no—
29C2×(C2×D251)⋊C4—21no—
30C2×C4×Dic251C2211yescharacteristic
31#22548648C2212yescomplemented
32Dic251⋊C8C2212yescomplemented
33#22548637C2212yes—
34#22548635C2212yes—
35#22548635C2212yescomplemented
36C22⋊(C251⋊C8)C2211yescharacteristic
37C8×Dic251C2212yescomplemented
38#121×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic
40#107×C251C2212yescomplemented
41#102×C251C2212yescomplemented
42C22⋊C8×C251C2211yescharacteristic
43C23×D251—21no—
44#16860809—44no—
45C2×Dic502—21no—
46C2×D1004—24no—
47C2×D1004—21no—
48C2×D1004C2311yescharacteristic
49C2×D1004C2312yes—
50C2×C22⋊D251—21no—
51C2×C22⋊D251—21no—
52C2×C22⋊D251—21no—
53C22×Dic251—21no—
54C22×Dic251D411yescharacteristic
55C22×Dic251D411yescharacteristic
56C2×C4×D251—22no—
57C2×C4×D251—22no—
58C2×C4×D251—21no—
59C2×C4×D251—44no—
60D2008—24no—
61#16860795—24no—
62C2×C251⋊C8C2312yes—
63C4⋊Dic251C2311yescharacteristic
64Dic251⋊C4—21no—
65C22⋊Dic251—21no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—21no—
69(C2×D251)⋊C4—21no—
70C4×Dic251D411yescharacteristic
71C4×Dic251D411yescharacteristic
72C4×Dic251C2311yescharacteristic
73C4×Dic251C2311yescharacteristic
74C2×D4×C251—21no—
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C22×C4×C251C2311yescharacteristic
78C2×C8×C251C2312yes—
79C4⋊C4×C251—21no—
80C4⋊C4×C251C2312yes—
81C4⋊C4×C251C2311yescharacteristic
82C22⋊C4×C251—21no—
83C22×D251—24no—
84C22×D251—22no—
85C22×D251—21no—
86C22×D251—42no—
87C22×D251—42no—
88Dic502—42no—
89D1004—24no—
90D1004—22no—
91D1004—22no—
92D1004#4114yescomplemented
93C22⋊D251—42no—
94C22⋊D251—42no—
95C22⋊D251—42no—
96C2×Dic251—21no—
97C2×Dic251—41no—
98C2×Dic251—42no—
99C2×Dic251D814yescomplemented
100C2×Dic251C2×D411yescharacteristic
101C2×Dic251C2×D411yescharacteristic
102C2×Dic251C2×D411yescharacteristic
103C2×Dic251C2×D411yescharacteristic
104C4×D251—44no—
105C4×D251—44no—
106C4×D251—42no—
107C251⋊C8—22no—
108C23×C251—21no—
109C23×C251C2×D411yescharacteristic
110D4×C251—22no—
111D4×C251—42no—
112C2×C4×C251—22no—
113C2×C4×C251—21no—
114C2×C4×C251—22no—
115C2×C4×C251—42no—
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251C2411yescharacteristic
118C8×C251—22no—
119C22×C251—21no—
120C22×C251—21no—
121C22×C251—41no—
122C22×C251—41no—
123C22×C251—41no—
124C22×C251C2×D812yes—
125C22×C251C22×D411yescharacteristic
126C2×D251—24no—
127C2×D251—48no—
128C2×D251—42no—
129C2×D251—42no—
130C2×D251—41no—
131C2×D251—41no—
132C2×D251—41no—
133C4×C251—21no—
134C4×C251—42no—
135C4×C251—41no—
136C4×C251C2×#4112yes—
137Dic251—21no—
138Dic251—21no—
139Dic251—41no—
140Dic251C2×D812yes—
141Dic251C2×D812yes—
142C2×C251—22no—
143C2×C251—81no—
144C2×C251#52511yescharacteristic
145C2×C251C22×D811yescharacteristic
146C2×C251#57511yescharacteristic
147D251—44no—
148D251—81no—
149C251#310011yescharacteristic, complemented, Sylow
150#3100—2511nomaximal, Sylow
151#532—2511no—
152C2×C4×D4—2511no—
153#480—2511no—
154#480D25111yescharacteristic, complemented
155#474—2512no—
156#459—2512no—
157#447—2511no—
158#447—2511no—
159C4×D8—2512no—
160C4×D8—2512no—
161#431—2511no—
162C2×#41—5021no—
163C22×D4—2511no—
164C23×C4—5021no—
165C2×D8—2512no—
166C2×D8—2512no—
167C4⋊Q8—2511no—
168#127—2511no—
169#124—5021no—
170#121—5021no—
171#121—2511no—
172#121C2×D25111yescharacteristic
173C4×D4—5021no—
174C4×D4—5021no—
175C4×D4—5021no—
176C4×D4—2511no—
177C4×D4—2512no—
178C4×D4—2512no—
179C4×D4—2511no—
180C2×C4⋊C4C2×D25111yescharacteristic
181C2×C22⋊C4—5021no—
182C2×C42—2511no—
183#107—2512no—
184#107C2×D25112yescomplemented
185C4⋊C8—2512no—
186#102—2512no—
187#102—2512no—
188#102—2512no—
189#102C2×D25112yescomplemented
190C22⋊C8—2511no—
191C22⋊C8C2×D25111yescharacteristic
192C4×C8—2512no—
193C24—5021no—
194#41—10044no—
195C2×Q8—5021no—
196C2×D4—21no—
197C2×D4—5024no—
198C2×D4—5021no—
199C2×D4—5021no—
200C2×D4—5021no—
201C2×D4—5021no—
202C2×D4—2511no—
203C2×D4—2512no—
204C2×D4C22×D25111yescharacteristic
205C22×C4—21no—
206C22×C4—10044no—
207C22×C4—5022no—
208C22×C4—5022no—
209C22×C4—5021no—
210C22×C4—5021no—
211C22×C4—2511no—
212C22×C4—2511no—
213C22×C4C22×D25111yescharacteristic
214D8—5024no—
215D8—5024no—
216C2×C8—2512no—
217C2×C8C22×D25112yes—
218C4⋊C4—21no—
219C4⋊C4—5021no—
220C4⋊C4—2511no—
221C4⋊C4C22×D25111yescharacteristic
222C4⋊C4C22×D25112yes—
223C22⋊C4—21no—
224C22⋊C4—5022no—
225C22⋊C4—5022no—
226C22⋊C4—5021no—
227C22⋊C4—5021no—
228C22⋊C4—5021no—
229C42—2511no—
230C42—2511no—
231C42—2511no—
232C42—2511no—
233C23—21no—
234C23—10042no—
235C23—10042no—
236C23—5024no—
237C23—5021no—
238C23—5022no—
239C23D4×D25111yescharacteristic
240Q8—10042no—
241D4—22no—
242D4—42no—
243D4—10042no—
244D4—10042no—
245D4—10042no—
246D4—5024no—
247D4—5022no—
248D4—5022no—
249D4—2514no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—10044no—
255C2×C4—10044no—
256C2×C4—10042no—
257C2×C4—10042no—
258C2×C4—10041no—
259C2×C4—5021no—
260C2×C4—2511no—
261C2×C4—2511no—
262C2×C4—2511no—
263C2×C4—2511no—
264C2×C4—2514no—
265C2×C4D4×D25111yescharacteristic
266C2×C4C23×D25111yescharacteristic
267C8—22no—
268C8—5022no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—10048no—
275C22—10041no—
276C22—10041no—
277C22—10041no—
278C22—10042no—
279C22—10042no—
280C22—5024no—
281C22D8×D25112yes—
282C22C2×D4×D25111yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—10041no—
287C4—5021no—
288C4—5021no—
289C4—2512no—
290C4—2512no—
291C4#41×D25112yes—
292C2—22no—
293C2—81no—
294C2—20081no—
295C2—10044no—
296C2#3063219511yescharacteristic
297C2C2×D8×D25111yescharacteristic
298C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#480×C251C2
socleC22×C251C22×D4
hypercenter#480D251
hypocenterC251#3100
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. #30632232C2semidirect
2. C2×#22548677C2semidirect
3. #30632000C2semidirect
4. #30631966C2semidirect
5. #30631881C2semidirect
6. #30631823C2semidirect
7. #30631820C2semidirect
8. #30631769C2semidirect
9. #30631767C2semidirect
10. #30631737C2semidirect
11. #480×C251C2semidirect
17. #22548714C22semidirect
18. #22548711C22semidirect
22. D4×Dic251C22semidirect
27. #22548677C22semidirect
31. #22548648C22semidirect
32. Dic251⋊C8C22semidirect
35. #22548635C22semidirect
37. C8×Dic251C22semidirect
40. #107×C251C22semidirect
41. #102×C251C22semidirect
92. D1004#41semidirect
99. C2×Dic251D8semidirect
149. C251#3100semidirect
154. #480D251semidirect
184. #107C2×D251semidirect
189. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322322
C2×#225486772
#306320002
#306319662
#306318812
#306318232
#306318202
#306317692
#306317672
#306317372
#480×C2512
#3100251