Finite Group Databasebeta

← #44560360

Subgroups — #44560360

Order 32128.

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

normal subgroups only

This group has 492 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#30632232C211yesmaximal, characteristic, complemented
2C2×D4×Dic251C211yesmaximal, characteristic, complemented
3#30632001C211yesmaximal, characteristic, complemented
4#30631998C211yesmaximal, characteristic, complemented
5#30631965C212yesmaximal, complemented
6#30631880C212yesmaximal, complemented
7#30631821C211yesmaximal, characteristic, complemented
8D8×Dic251C212yesmaximal, complemented
9#30631767C212yesmaximal, complemented
10#30631737C211yesmaximal, characteristic, complemented
11#447×C251C211yesmaximal, characteristic, complemented
12C2×#16860809—21no—
13C23×Dic251—21no—
14C2×#16860795C2212yes—
15#22548714C2211yescharacteristic, complemented
16#22548711C2211yescharacteristic, complemented
17#22548701—21no—
18#22548689C2211yescharacteristic
19#22548687—21no—
20D4×Dic251—21no—
21D4×Dic251—21no—
22D4×Dic251C2212yes—
23D4×Dic251C2212yes—
24D4×Dic251C2211yescharacteristic, complemented
25#22548677C2211yescharacteristic, complemented
26(C22⋊D251)⋊C4—21no—
27C2×C4⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251—21no—
29C2×C4×Dic251C2211yescharacteristic
30#22548649C2212yescomplemented
31#22548648C2212yescomplemented
32Dic251⋊C8C2212yescomplemented
33#22548637C2212yescomplemented
34#22548636C2212yes—
35#22548636C2212yescomplemented
36C22⋊(C251⋊C8)C2211yescharacteristic
37C8×Dic251C2212yescomplemented
38C22×D4×C251C2211yescharacteristic
39C2×D8×C251C2212yes—
40#121×C251C2211yescharacteristic
41#102×C251C2212yes—
42C22⋊C8×C251C2211yescharacteristic
43#16860809—44no—
44C2×Dic502—21no—
45C2×D1004C2311yescharacteristic
46C2×C22⋊D251—21no—
47C2×C22⋊D251—21no—
48C2×C22⋊D251—21no—
49C22×Dic251—22no—
50C22×Dic251—22no—
51C22×Dic251—21no—
52C22×Dic251—21no—
53C22×Dic251—44no—
54C22×Dic251D411yescharacteristic
55C22×Dic251D411yescharacteristic
56C2×C4×D251—21no—
57#16860795—24no—
58C2×C251⋊C8C2312yes—
59C4⋊Dic251—21no—
60C4⋊Dic251C2311yescharacteristic
61C4⋊Dic251C2312yes—
62Dic251⋊C4—21no—
63C22⋊Dic251—22no—
64C22⋊Dic251—22no—
65C22⋊Dic251—21no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4—21no—
68C4×Dic251D411yescharacteristic
69C4×Dic251D411yescharacteristic
70C4×Dic251C2311yescharacteristic
71C4×Dic251C2311yescharacteristic
72C24×C251—21no—
73C2×D4×C251—24no—
74C2×D4×C251—21no—
75C2×D4×C251—21no—
76C2×D4×C251C2311yescharacteristic
77C2×D4×C251C2312yes—
78C22×C4×C251C2311yescharacteristic
79D8×C251—24no—
80C2×C8×C251C2312yes—
81C4⋊C4×C251C2311yescharacteristic
82C22⋊C4×C251—21no—
83C22×D251—21no—
84Dic502—42no—
85D1004—22no—
86C22⋊D251—42no—
87C22⋊D251—42no—
88C22⋊D251—42no—
89C2×Dic251—21no—
90C2×Dic251—22no—
91C2×Dic251—41no—
92C2×Dic251—42no—
93C2×Dic251—44no—
94C2×Dic251—44no—
95C2×Dic251—42no—
96C2×Dic251D814yescomplemented
97C2×Dic251C2×D411yescharacteristic
98C2×Dic251C2×D411yescharacteristic
99C2×Dic251C2×D411yescharacteristic
100C2×Dic251C2×D411yescharacteristic
101C4×D251—42no—
102C251⋊C8—22no—
103C23×C251—24no—
104C23×C251—22no—
105C23×C251—21no—
106C23×C251—42no—
107C23×C251—42no—
108C23×C251C2×D411yescharacteristic
109D4×C251—24no—
110D4×C251—22no—
111D4×C251—22no—
112D4×C251—42no—
113D4×C251#4114yescomplemented
114C2×C4×C251—22no—
115C2×C4×C251—21no—
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251C2411yescharacteristic
118C8×C251—22no—
119C22×C251—21no—
120C22×C251—21no—
121C22×C251—24no—
122C22×C251—48no—
123C22×C251—42no—
124C22×C251—42no—
125C22×C251—41no—
126C22×C251—41no—
127C22×C251—41no—
128C22×C251C2×D812yes—
129C22×C251C22×D411yescharacteristic
130C2×D251—41no—
131C2×D251—41no—
132C2×D251—41no—
133C4×C251—21no—
134C4×C251—41no—
135C4×C251C2×#4112yes—
136Dic251—21no—
137Dic251—21no—
138Dic251—42no—
139Dic251—41no—
140Dic251C2×D812yes—
141Dic251C2×D812yes—
142C2×C251—22no—
143C2×C251—44no—
144C2×C251—81no—
145C2×C251#52511yescharacteristic
146C2×C251C22×D811yescharacteristic
147C2×C251#57511yescharacteristic
148D251—81no—
149C251#310011yescharacteristic, complemented, Sylow
150#3100—2511nomaximal, Sylow
151#532—2511no—
152C2×C4×D4—2511no—
153#480—2511no—
154#480—2511no—
155#474—2512no—
156#459—2512no—
157#447—2511no—
158#447D25111yescharacteristic, complemented
159C4×D8—2512no—
160C4×D8—2512no—
161#431—2511no—
162C2×#41—5021no—
163C22×D4C2×D25111yescharacteristic
164C23×C4—5021no—
165C2×D8—2512no—
166C2×D8C2×D25112yes—
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—2512no—
177C4×D4—2512no—
178C4×D4—2511no—
179C4×D4—2511no—
180C2×C4⋊C4—2511no—
181C2×C22⋊C4—5021no—
182C2×C42—2511no—
183#107—2512no—
184#107—2512no—
185C4⋊C8—2512no—
186#102—2512no—
187#102—2512no—
188#102—2512no—
189#102C2×D25112yes—
190C22⋊C8—2511no—
191C22⋊C8C2×D25111yescharacteristic
192C4×C8—2512no—
193C24—21no—
194#41—10044no—
195C2×Q8—5021no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—5021no—
200C2×D4—5021no—
201C2×D4—5021no—
202C2×D4—2511no—
203C2×D4C22×D25111yescharacteristic
204C2×D4C22×D25112yes—
205C22×C4—10044no—
206C22×C4—5021no—
207C22×C4—5021no—
208C22×C4—5022no—
209C22×C4—5022no—
210C22×C4—5021no—
211C22×C4—2511no—
212C22×C4—2511no—
213C22×C4C22×D25111yescharacteristic
214D8—24no—
215D8—5024no—
216C2×C8—2512no—
217C2×C8C22×D25112yes—
218C4⋊C4—5021no—
219C4⋊C4—5021no—
220C4⋊C4—2512no—
221C4⋊C4—2511no—
222C4⋊C4C22×D25111yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—5021no—
225C22⋊C4—5022no—
226C22⋊C4—5022no—
227C22⋊C4—5021no—
228C22⋊C4—5021no—
229C42—2511no—
230C42—2511no—
231C42—2511no—
232C42—2511no—
233C23—24no—
234C23—22no—
235C23—21no—
236C23—42no—
237C23—42no—
238C23—5021no—
239C23D4×D25111yescharacteristic
240Q8—10042no—
241D4—24no—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—10042no—
246D4—10042no—
247D4—10042no—
248D4—5022no—
249D4#1686080914yescomplemented
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—10042no—
253C2×C4—10042no—
254C2×C4—10044no—
255C2×C4—10044no—
256C2×C4—10042no—
257C2×C4—10041no—
258C2×C4—5022no—
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—24no—
272C22—48no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—10041no—
279C22—10041no—
280C22—10041no—
281C22D8×D25112yes—
282C22C2×D4×D25111yescharacteristic
283C4—21no—
284C4—41no—
285C4—10042no—
286C4—10041no—
287C4—5021no—
288C4—5021no—
289C4—2512no—
290C4—2512no—
291C4C2×#1686080912yes—
292C2—22no—
293C2—44no—
294C2—81no—
295C2—20081no—
296C2#3063219311yescharacteristic
297C2C2×D8×D25111yescharacteristic
298C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251#3100
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #30632232C2semidirect
2. C2×D4×Dic251C2semidirect
3. #30632001C2semidirect
4. #30631998C2semidirect
5. #30631965C2semidirect
6. #30631880C2semidirect
7. #30631821C2semidirect
8. D8×Dic251C2semidirect
9. #30631767C2semidirect
10. #30631737C2semidirect
11. #447×C251C2semidirect
15. #22548714C22semidirect
16. #22548711C22semidirect
24. D4×Dic251C22semidirect
25. #22548677C22semidirect
30. #22548649C22semidirect
31. #22548648C22semidirect
32. Dic251⋊C8C22semidirect
33. #22548637C22semidirect
35. #22548636C22semidirect
37. C8×Dic251C22semidirect
96. C2×Dic251D8semidirect
113. D4×C251#41semidirect
149. C251#3100semidirect
158. #447D251semidirect
249. D4#16860809semidirect

Maximal subgroups

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

subgroupindex
#306322322
C2×D4×Dic2512
#306320012
#306319982
#306319652
#306318802
#306318212
D8×Dic2512
#306317672
#306317372
#447×C2512
#3100251