Finite Group Databasebeta

← #44559976

Subgroups — #44559976

Order 32128.

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

normal subgroups only

This group has 500 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
1C2×#22548738C211yesmaximal, characteristic, complemented
2C2×M16×D251C211yesmaximal, characteristic, complemented
3#30632200C212yesmaximal, complemented
4#30631942C211yesmaximal, characteristic, complemented
5#30631941C211yesmaximal, characteristic, complemented
6#30631940C211yesmaximal, characteristic, complemented
7#30631936C212yesmaximal, complemented
8#30631935C212yesmaximal, complemented
9#30631722C211yesmaximal, characteristic, complemented
10#30631682C212yesmaximal, complemented
11#470×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2212yes—
13C22×C4×D251C2211yescharacteristic
14#22548738—24no—
15C2×#16860797C2212yescomplemented
16C2×D2008C2211yescharacteristic, complemented
17C2×#16860794C2211yescharacteristic
18C2×#16860793C2211yescharacteristic
19C2×#16860793C2211yescharacteristic
20M16×D251—24no—
21C2×C8×D251C2211yescharacteristic
22C2×C8×D251C2211yescharacteristic
23#22548700—22no—
24#22548691C2212yes—
25#22548690C2212yes—
26#22548689C2212yes—
27#22548685—22no—
28#22548677—22no—
29(C22⋊D251)⋊C4—22no—
30#22548673C2212yes—
31#22548648C2212yescomplemented
32#22548647C2211yescharacteristic, complemented
33#22548639C2212yescomplemented
34#22548637C2212yescomplemented
35#22548635C2212yescomplemented
36C2×Q16×C251C2211yescharacteristic, complemented
37C2×M16×C251C2211yescharacteristic
38C22⋊Q8×C251C2212yes—
39#106×C251C2211yescharacteristic, complemented
40#103×C251C2212yescomplemented
41C23×D251D411yescharacteristic
42#16860811—24no—
43#16860811—24no—
44C2×D1004—22no—
45C2×D1004—22no—
46C2×D1004C2312yes—
47C2×C22⋊D251—22no—
48C22×Dic251D411yescharacteristic
49C2×C4×D251—22no—
50C2×C4×D251—22no—
51C2×C4×D251—22no—
52C2×C4×D251D411yescharacteristic
53C2×C4×D251D411yescharacteristic
54C2×C4×D251C2311yescharacteristic
55C2×C4×D251C2311yescharacteristic
56#16860797—24no—
57D2008—22no—
58#16860794—22no—
59#16860793—22no—
60#16860793—22no—
61C2×C251⋊C8C2311yescharacteristic
62C2×C251⋊C8C2311yescharacteristic
63C8×D251—22no—
64C8×D251D414yescomplemented
65C4⋊Dic251C2312yes—
66Dic251⋊C4—22no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4—22no—
70(C2×D251)⋊C4—22no—
71C4×Dic251—22no—
72C2×Q8×C251C2312yes—
73C22×C4×C251C2311yescharacteristic
74Q16×C251—22no—
75M16×C251—22no—
76C2×C8×C251C2311yescharacteristic
77C2×C8×C251C2311yescharacteristic
78C4⋊C4×C251—22no—
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—22no—
81C22×D251—22no—
82C22×D251—22no—
83C22×D251—42no—
84C22×D251C2×D411yescharacteristic
85C22×D251C2×D411yescharacteristic
86D1004—22no—
87D1004—22no—
88D1004—44no—
89D1004—44no—
90C22⋊D251—44no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C4×D251—22no—
96C4×D251—21no—
97C4×D251—21no—
98C4×D251—22no—
99C4×D251—44no—
100C4×D251C2×D412yes—
101C4×D251#4112yes—
102C251⋊C8—21no—
103C251⋊C8C2×D412yes—
104C23×C251C2×D411yescharacteristic
105Q8×C251—22no—
106Q8×C251—22no—
107C2×C4×C251—21no—
108C2×C4×C251—21no—
109C2×C4×C251—22no—
110C2×C4×C251—22no—
111C2×C4×C251C2×D411yescharacteristic
112C2×C4×C251C2411yescharacteristic
113C8×C251—21no—
114C8×C251C2×D412yes—
115C22×C251—21no—
116C22×C251—22no—
117C22×C251C22×D411yescharacteristic
118C2×D251—22no—
119C2×D251—21no—
120C2×D251—22no—
121C2×D251—44no—
122C2×D251—42no—
123C2×D251—42no—
124C2×D251—42no—
125C2×D251#12112yes—
126C4×C251—21no—
127C4×C251—42no—
128C4×C251—42no—
129C4×C251C22×D411yescharacteristic
130C4×C251C2×#4111yescharacteristic
131Dic251—21no—
132Dic251—42no—
133Dic251#12112yes—
134C2×C251—41no—
135C2×C251C2×#12111yescharacteristic
136C2×C251#57512yes—
137D251—22no—
138D251—41no—
139D251—82no—
140C251#305411yescharacteristic, complemented, Sylow
141#3054—2511nomaximal, Sylow
142C2×#135—2511no—
143C22×M16—2511no—
144#525—2512no—
145#470—2511no—
146#470D25111yescharacteristic, complemented
147#469—2511no—
148#469—2511no—
149#468—2512no—
150#468—2512no—
151#427—2511no—
152#416—2512no—
153C2×#41—2512no—
154C23×C4—2511no—
155#135—5024no—
156C2×Q16C2×D25111yescharacteristic, complemented
157C2×SD16—2512no—
158C2×D8—2511no—
159C2×M16—5024no—
160C2×M16—2511no—
161C2×M16—2511no—
162C2×M16—2511no—
163C2×M16C2×D25111yescharacteristic
164C22×C8—2511no—
165C22×C8—2511no—
166#123—5022no—
167C22⋊Q8—2512no—
168C22⋊Q8C2×D25112yes—
169#121—2512no—
170#121—2512no—
171#120—5022no—
172C4×D4—5022no—
173C4×D4—5022no—
174#117—2512no—
175#107—2512no—
176#106—2511no—
177#106C2×D25111yescharacteristic, complemented
178#103—2512no—
179#103C2×D25112yescomplemented
180#102—2512no—
181#102—2512no—
182C24—2511no—
183#41—5024no—
184#41—5024no—
185C2×Q8C22×D25112yes—
186C2×D4—5022no—
187C2×D4—5022no—
188C2×D4—5022no—
189C2×D4—2512no—
190C22×C4—5022no—
191C22×C4—5022no—
192C22×C4—5022no—
193C22×C4—2511no—
194C22×C4—2511no—
195C22×C4—2511no—
196C22×C4—2511no—
197C22×C4—2511no—
198C22×C4C22×D25111yescharacteristic
199Q16—22no—
200SD16—5024no—
201D8—5022no—
202M16—22no—
203M16—5022no—
204M16—5022no—
205M16—5022no—
206C2×C8—5022no—
207C2×C8—2511no—
208C2×C8—2511no—
209C2×C8—2514no—
210C2×C8C22×D25111yescharacteristic
211C2×C8C22×D25111yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—5022no—
214C4⋊C4—2512no—
215C4⋊C4C22×D25112yes—
216C22⋊C4—22no—
217C22⋊C4—5022no—
218C22⋊C4—5022no—
219C22⋊C4—5022no—
220C22⋊C4—5022no—
221C42—5022no—
222C23—10042no—
223C23—5022no—
224C23—5022no—
225C23—2511no—
226C23—2511no—
227C23D4×D25111yescharacteristic
228Q8—22no—
229Q8—22no—
230D4—10044no—
231D4—10044no—
232D4—10044no—
233D4—5022no—
234D4—5022no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—10044no—
240C2×C4—5022no—
241C2×C4—5022no—
242C2×C4—5022no—
243C2×C4—5022no—
244C2×C4—5021no—
245C2×C4—5021no—
246C2×C4—2511no—
247C2×C4—2511no—
248C2×C4—2512no—
249C2×C4—2512no—
250C2×C4D4×D25111yescharacteristic
251C2×C4C23×D25111yescharacteristic
252C8—21no—
253C8—5021no—
254C8—2512no—
255C8D4×D25112yes—
256C22—22no—
257C22—21no—
258C22—10044no—
259C22—10042no—
260C22—10042no—
261C22—10042no—
262C22—5022no—
263C22—5021no—
264C22—5022no—
265C22—2512no—
266C22C2×D4×D25111yescharacteristic
267C4—21no—
268C4—42no—
269C4—42no—
270C4—10042no—
271C4—5021no—
272C4—2512no—
273C4C2×D4×D25111yescharacteristic
274C4#41×D25111yescharacteristic
275C2—41no—
276C2—20082no—
277C2—10041no—
278C2—5022no—
279C2#121×D25111yescharacteristic
280C2#3063248612yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#470×C251C2
socleC22×C251C22×D4
hypercenter#470D251
hypocenterC251#3054
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#22548738C2semidirect
2. C2×M16×D251C2semidirect
3. #30632200C2semidirect
4. #30631942C2semidirect
5. #30631941C2semidirect
6. #30631940C2semidirect
7. #30631936C2semidirect
8. #30631935C2semidirect
9. #30631722C2semidirect
10. #30631682C2semidirect
11. #470×C251C2semidirect
15. C2×#16860797C22semidirect
16. C2×D2008C22semidirect
31. #22548648C22semidirect
32. #22548647C22semidirect
33. #22548639C22semidirect
34. #22548637C22semidirect
35. #22548635C22semidirect
36. C2×Q16×C251C22semidirect
39. #106×C251C22semidirect
40. #103×C251C22semidirect
64. C8×D251D4semidirect
140. C251#3054semidirect
146. #470D251semidirect
156. C2×Q16C2×D251semidirect
177. #106C2×D251semidirect
179. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487382
C2×M16×D2512
#306322002
#306319422
#306319412
#306319402
#306319362
#306319352
#306317222
#306316822
#470×C2512
#3054251