Finite Group Databasebeta

← #44560375

Subgroups — #44560375

Order 32128.

448 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 127 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#30632242C211yesmaximal, characteristic, complemented
2C4×D4×D251C211yesmaximal, characteristic
3#30632019C212yesmaximal, complemented
4#30631974C211yesmaximal, characteristic, complemented
5#30631893C211yesmaximal, characteristic, complemented
6#30631892C211yesmaximal, characteristic, complemented
7#30631839C212yesmaximal
8#30631778C212yesmaximal, complemented
9#30631774C212yesmaximal, complemented
10#30631739C211yesmaximal, characteristic, complemented
11#476×C251C211yesmaximal, characteristic, complemented
12C2×D4×D251C2211yescharacteristic
13C22×C4×D251—21no—
14C2×#16860799C2212yes—
15C2×#16860798C2212yes—
16#22548715C2211yescharacteristic, complemented
17#22548706—21no—
18#22548695C2212yes—
19#22548691—21no—
20Q8×Dic251—21no—
21#22548680C2212yescomplemented
22C4×D1004C2211yescharacteristic
23D4×Dic251C2211yescharacteristic
24#22548677C2211yescharacteristic
25C4×C22⋊D251—21no—
26(C22⋊D251)⋊C4—21no—
27#22548673—21no—
28C4⋊C4×D251C2211yescharacteristic
29C22⋊C4×D251—21no—
30C42×D251C2211yescharacteristic
31#22548646C2212yescomplemented
32C4⋊(C251⋊C8)C2211yescharacteristic, complemented
33#22548640C2212yescomplemented
34#22548638C2212yescomplemented
35#22548635C2212yes—
36(C2×D251)⋊C8C2212yes—
37C8×Dic251C2212yescomplemented
38C4⋊Q8×C251C2211yescharacteristic, complemented
39C4×D4×C251C2211yescharacteristic
40#106×C251C2212yescomplemented
41C4⋊C8×C251C2211yescharacteristic, complemented
42#102×C251C2212yes—
43C23×D251—21no—
44C2×Dic502C2312yes—
45C2×D1004C2311yescharacteristic
46C2×C22⋊D251—21no—
47D4×D251—22no—
48D4×D251—22no—
49C22×Dic251—21no—
50C2×C4×D251—21no—
51C2×C4×D251—21no—
52C2×C4×D251—21no—
53C2×C4×D251—44no—
54C2×C4×D251D411yescharacteristic
55C2×C4×D251D411yescharacteristic
56C2×C4×D251C2311yescharacteristic
57#16860799—24no—
58#16860798—24no—
59C2×C251⋊C8C2312yes—
60C4⋊Dic251—22no—
61C4⋊Dic251—21no—
62C4⋊Dic251—21no—
63C4⋊Dic251C2311yescharacteristic
64Dic251⋊C4—21no—
65Dic251⋊C4—22no—
66C22⋊Dic251—21no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—21no—
69(C2×D251)⋊C4—21no—
70C4×Dic251—22no—
71C4×Dic251D411yescharacteristic
72C4×Dic251D411yescharacteristic
73C4×Dic251C2311yescharacteristic
74C2×Q8×C251—21no—
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C2×C8×C251C2312yes—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2312yes—
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C42×C251C2311yescharacteristic
83C22×D251—24no—
84C22×D251—21no—
85C22×D251—42no—
86C22×D251—42no—
87C22×D251C2×D411yescharacteristic
88Dic502—22no—
89Dic502—22no—
90D1004—21no—
91D1004#4112yes—
92C22⋊D251—24no—
93C2×Dic251—22no—
94C2×Dic251—21no—
95C2×Dic251—42no—
96C2×Dic251C2×D411yescharacteristic
97C2×Dic251C2×D411yescharacteristic
98C2×Dic251C2×D411yescharacteristic
99C4×D251—21no—
100C4×D251—21no—
101C4×D251—41no—
102C4×D251—42no—
103C4×D251—42no—
104C4×D251—42no—
105C4×D251SD1614yescomplemented
106C251⋊C8—22no—
107C23×C251—21no—
108Q8×C251—42no—
109D4×C251—21no—
110D4×C251#4112yes—
111C2×C4×C251—21no—
112C2×C4×C251—22no—
113C2×C4×C251—42no—
114C2×C4×C251C2×D411yescharacteristic
115C2×C4×C251C2×D411yescharacteristic
116C2×C4×C251C2411yescharacteristic
117C8×C251—22no—
118C22×C251—22no—
119C22×C251—41no—
120C22×C251—41no—
121C22×C251C22×D411yescharacteristic
122C2×D251—21no—
123C2×D251—21no—
124C2×D251—22no—
125C2×D251—48no—
126C2×D251—41no—
127C2×D251—41no—
128C2×D251C2×SD1612yes—
129C4×C251—21no—
130C4×C251—41no—
131C4×C251—42no—
132C4×C251C2×SD1612yes—
133C4×C251C2×#4111yescharacteristic
134C4×C251C2×#4111yescharacteristic
135Dic251—21no—
136Dic251—21no—
137Dic251—42no—
138Dic251—41no—
139Dic251C2×SD1612yes—
140C2×C251—42no—
141C2×C251#52511yescharacteristic
142C2×C251C22×SD1611yescharacteristic
143C2×C251#57511yescharacteristic
144D251—22no—
145D251—42no—
146C251#310111yescharacteristic, complemented, Sylow
147#3101—2511nomaximal, Sylow
148#533—2511no—
149C2×C4×D4—2511no—
150#483—2512no—
151#476—2511no—
152#476D25111yescharacteristic, complemented
153#461—2511no—
154#461—2511no—
155#450—2512no—
156C4×SD16—2512no—
157C4×SD16—2512no—
158#431—2511no—
159C22×D4—2511no—
160C23×C4—5021no—
161C2×SD16—2512no—
162C2×SD16—2512no—
163C4⋊Q8—2511no—
164C4⋊Q8C2×D25111yescharacteristic, complemented
165#125—5021no—
166C22⋊Q8—5021no—
167C22⋊Q8—2512no—
168C4×Q8—5021no—
169C4×Q8—2512no—
170C4×D4—5021no—
171C4×D4—5021no—
172C4×D4—2511no—
173C4×D4—2511no—
174C4×D4—2511no—
175C4×D4C2×D25111yescharacteristic
176#117—5021no—
177C2×C4⋊C4—2511no—
178C2×C22⋊C4—5021no—
179C2×C42—2511no—
180#106—2512no—
181#106C2×D25112yescomplemented
182C4⋊C8—2511no—
183C4⋊C8C2×D25111yescharacteristic, complemented
184#103—2512no—
185#103—2512no—
186#102—2512no—
187#102C2×D25112yes—
188C22⋊C8—2512no—
189C4×C8—2512no—
190C24—5021no—
191C2×Q8—21no—
192C2×Q8—2512no—
193C2×D4—5022no—
194C2×D4—5022no—
195C2×D4—5021no—
196C2×D4—2511no—
197C2×D4C22×D25111yescharacteristic
198C22×C4—21no—
199C22×C4—10044no—
200C22×C4—5021no—
201C22×C4—5021no—
202C22×C4—5021no—
203C22×C4—5021no—
204C22×C4—2511no—
205C22×C4—2511no—
206C22×C4—2511no—
207SD16—5024no—
208SD16—5024no—
209C2×C8—2512no—
210C2×C8C22×D25112yes—
211C4⋊C4—21no—
212C4⋊C4—5021no—
213C4⋊C4—5022no—
214C4⋊C4—5022no—
215C4⋊C4—5021no—
216C4⋊C4—5021no—
217C4⋊C4—2511no—
218C4⋊C4C22×D25112yes—
219C4⋊C4C22×D25111yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—5022no—
222C22⋊C4—5021no—
223C22⋊C4—5021no—
224C22⋊C4—5021no—
225C42—5022no—
226C42—2511no—
227C42—2511no—
228C42—2511no—
229C42C22×D25111yescharacteristic
230C23—21no—
231C23—10042no—
232C23—10042no—
233C23—5024no—
234C23—5021no—
235C23—2511no—
236Q8—42no—
237Q8—5022no—
238Q8—5022no—
239D4—21no—
240D4—5024no—
241D4—5021no—
242D4—2512no—
243D4#1686081112yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—10041no—
248C2×C4—10042no—
249C2×C4—10042no—
250C2×C4—10042no—
251C2×C4—10042no—
252C2×C4—5022no—
253C2×C4—5021no—
254C2×C4—5021no—
255C2×C4—5021no—
256C2×C4—2511no—
257C2×C4—2511no—
258C2×C4—2511no—
259C2×C4—2514no—
260C2×C4D4×D25111yescharacteristic
261C2×C4D4×D25111yescharacteristic
262C2×C4C23×D25111yescharacteristic
263C8—22no—
264C8—5022no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—10048no—
269C22—10041no—
270C22—10041no—
271C22—5022no—
272C22—5021no—
273C22—5021no—
274C22—2512no—
275C22C2×D4×D25111yescharacteristic
276C4—21no—
277C4—41no—
278C4—42no—
279C4—10042no—
280C4—10041no—
281C4—5021no—
282C4—5021no—
283C4—2512no—
284C4SD16×D25112yes—
285C4#41×D25111yescharacteristic
286C4C2×#1686081111yescharacteristic
287C2—42no—
288C2—10042no—
289C2—5022no—
290C2#3063220011yescharacteristic
291C2C2×SD16×D25111yescharacteristic
292C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#476×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
hypercenter#476D251
hypocenterC251#3101
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #30632242C2semidirect
3. #30632019C2semidirect
4. #30631974C2semidirect
5. #30631893C2semidirect
6. #30631892C2semidirect
8. #30631778C2semidirect
9. #30631774C2semidirect
10. #30631739C2semidirect
11. #476×C251C2semidirect
16. #22548715C22semidirect
21. #22548680C22semidirect
31. #22548646C22semidirect
32. C4⋊(C251⋊C8)C22semidirect
33. #22548640C22semidirect
34. #22548638C22semidirect
37. C8×Dic251C22semidirect
38. C4⋊Q8×C251C22semidirect
40. #106×C251C22semidirect
41. C4⋊C8×C251C22semidirect
105. C4×D251SD16semidirect
146. C251#3101semidirect
152. #476D251semidirect
164. C4⋊Q8C2×D251semidirect
181. #106C2×D251semidirect
183. C4⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322422
C4×D4×D2512
#306320192
#306319742
#306318932
#306318922
#306318392
#306317782
#306317742
#306317392
#476×C2512
#3101251