Finite Group Databasebeta

← #44559649

Subgroups — #44559649

Order 32128.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22548762C211yesmaximal, characteristic, complemented
2C2×#22548741C211yesmaximal, characteristic, complemented
3C22×#16860798C211yesmaximal, characteristic, complemented
4#30631830C214yesmaximal, complemented
5#30631825C214yesmaximal, complemented
6#30631695C211yesmaximal, characteristic, complemented
7#30631681C211yesmaximal, characteristic, complemented
8#30631654C211yesmaximal, characteristic, complemented
9#535×C251C211yesmaximal, characteristic, complemented
10#22548762—24no—
11#22548762—24no—
12C2×#16860810C2211yescharacteristic
13C2×#16860810C2212yescomplemented
14C2×#16860809—22no—
15C2×#16860809—21no—
16C22×Dic502C2211yescharacteristic
17C22×Dic502C2212yescomplemented
18C2×Q8×D251—21no—
19#22548741—24no—
20C2×#16860798—24no—
21C2×#16860798C2214yes—
22C2×#16860798C2212yes—
23C2×#16860795C2212yes—
24C2×#16860794C2211yescharacteristic
25C22×C251⋊C8C2211yescharacteristic
26#22548638C2212yescomplemented
27#22548638C2214yescomplemented
28#22548635C2212yescomplemented
29C22⋊(C251⋊C8)C2214yescomplemented
30C2×#41×C251C2211yescharacteristic
31C22×D4×C251C2211yescharacteristic
32#127×C251—21no—
33#124×C251—21no—
34#121×C251C2214yes—
35#120×C251—22no—
36#117×C251C2211yescharacteristic
37#16860810—24no—
38#16860810—24no—
39#16860810—22no—
40#16860810D414yescomplemented
41#16860809—216no—
42#16860809—28no—
43C2×Dic502—24no—
44C2×Dic502—24no—
45C2×Dic502—21no—
46C2×Dic502—22no—
47C2×Dic502—22no—
48C2×Dic502D414yescomplemented
49C2×Dic502C2312yes—
50C2×Dic502C2311yescharacteristic
51Q8×D251—28no—
52C2×D1004C2311yescharacteristic
53C2×C22⋊D251—21no—
54C2×C22⋊D251—22no—
55C22×Dic251—21no—
56C22×Dic251—22no—
57C2×C4×D251—21no—
58C2×C4×D251—22no—
59#16860798—24no—
60#16860798—28no—
61#16860795—24no—
62#16860794—22no—
63C2×C251⋊C8—22no—
64C2×C251⋊C8C2312yes—
65C2×C251⋊C8C2312yes—
66C24×C251—21no—
67#41×C251—22no—
68#41×C251—22no—
69C2×Q8×C251D411yescharacteristic
70C2×D4×C251—24no—
71C2×D4×C251—22no—
72C2×D4×C251—22no—
73C2×D4×C251—21no—
74C2×D4×C251D412yes—
75C2×D4×C251D411yescharacteristic
76C2×D4×C251C2312yes—
77C22×C4×C251C2312yes—
78C22×C4×C251C2311yescharacteristic
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—21no—
81C22⋊C4×C251—24no—
82C42×C251—21no—
83C22×D251—21no—
84Dic502—28no—
85Dic502—24no—
86Dic502—21no—
87Dic502—22no—
88Dic502C2×D412yes—
89Dic502C2×D414yes—
90D1004—21no—
91D1004C2×D412yes—
92C22⋊D251—28no—
93C22⋊D251—24no—
94C2×Dic251—28no—
95C2×Dic251—24no—
96C2×Dic251—21no—
97C2×Dic251—22no—
98C4×D251—28no—
99C4×D251—24no—
100C251⋊C8—22no—
101C251⋊C8—22no—
102C23×C251—22no—
103C23×C251—42no—
104C23×C251—82no—
105C23×C251C2×D412yes—
106C23×C251C2×D411yescharacteristic
107Q8×C251—22no—
108D4×C251—24no—
109D4×C251—22no—
110D4×C251—22no—
111D4×C251—22no—
112D4×C251—42no—
113D4×C251—84no—
114C2×C4×C251—22no—
115C2×C4×C251—22no—
116C2×C4×C251—22no—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2411yescharacteristic
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—24no—
125C22×C251—42no—
126C22×C251—44no—
127C22×C251—84no—
128C22×C251#12012yes—
129C22×C251C22×D411yescharacteristic
130C2×D251—22no—
131C2×D251—42no—
132C4×C251—22no—
133C4×C251—42no—
134C4×C251#12012yes—
135C4×C251C22×D411yescharacteristic
136C4×C251C22×D411yescharacteristic
137Dic251—24no—
138Dic251—22no—
139C2×C251—22no—
140C2×C251—42no—
141C2×C251—82no—
142C2×C251C2×#12011yescharacteristic
143C2×C251#57712yes—
144D251—42no—
145C251#301011yescharacteristic, complemented, Sylow
146#3010—2511nomaximal, Sylow
147C2×2−1+4—2511no—
148C2×AGL(1,ℤ/8)—2511no—
149C22×SD16—2511no—
150#535D25111yescharacteristic, complemented
151#449—2514no—
152#448—2514no—
153#419—2511no—
154#416—2511no—
155#408—2511no—
1562−1+4—5024no—
1572−1+4—5024no—
158C2×#41—5022no—
159C2×#41—5021no—
160C2×#41—2512no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C22×Q8—5021no—
164C22×Q8—2512no—
165C22×Q8—2511no—
166C22×D4C2×D25111yescharacteristic
167AGL(1,ℤ/8)—5024no—
168C2×SD16—5024no—
169C2×SD16—2514no—
170C2×SD16—2512no—
171C2×D8—2512no—
172C2×M16—2511no—
173C22×C8—2511no—
174#127—21no—
175#124—21no—
176#121C2×D25114yes—
177#120—22no—
178#117C2×D25111yescharacteristic
179#103—2512no—
180#103—2514no—
181#102—2512no—
182C22⋊C8—2514no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—5024no—
187#41—5024no—
188#41—50216no—
189#41—5028no—
190#41—5022no—
191#41—2514no—
192C2×Q8—5024no—
193C2×Q8—5024no—
194C2×Q8—5028no—
195C2×Q8—5022no—
196C2×Q8—5022no—
197C2×Q8—5021no—
198C2×Q8—2514no—
199C2×Q8—2511no—
200C2×Q8—2512no—
201C2×Q8C22⋊D25111yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—5022no—
207C2×D4—5021no—
208C2×D4—2511no—
209C2×D4C22⋊D25112yes—
210C2×D4C22⋊D25111yescharacteristic
211C2×D4C22×D25112yes—
212C22×C4—5022no—
213C22×C4—5021no—
214C22×C4—5022no—
215C22×C4—5021no—
216C22×C4C22×D25112yes—
217C22×C4C22×D25111yescharacteristic
218SD16—5028no—
219SD16—5024no—
220D8—5024no—
221M16—5022no—
222C2×C8—5022no—
223C2×C8—2512no—
224C2×C8—2512no—
225C4⋊C4C22×D25112yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—5021no—
233C23C2×C22⋊D25112yes—
234C23C2×C22⋊D25111yescharacteristic
235Q8—22no—
236Q8—5028no—
237Q8—5021no—
238Q8—5022no—
239Q8—5024no—
240Q8—2512no—
241Q8—2514no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—5028no—
249D4—5024no—
250D4—5021no—
251D4—2512no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—5028no—
256C2×C4—5028no—
257C2×C4—5022no—
258C2×C4—5021no—
259C2×C4—5024no—
260C2×C4—5024no—
261C2×C4D4×D25112yes—
262C2×C4D4×D25112yes—
263C2×C4C2×C22⋊D25112yes—
264C2×C4C2×C22⋊D25111yescharacteristic
265C2×C4C23×D25111yescharacteristic
266C8—5022no—
267C8—5022no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—10042no—
275C22—5022no—
276C22#2254868312yes—
277C22C22×C22⋊D25111yescharacteristic
278C4—22no—
279C4—42no—
280C4—5024no—
281C4—5022no—
282C4#2254868312yes—
283C4C2×D4×D25111yescharacteristic
284C4C2×D4×D25111yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—10042no—
289C2C2×#2254868311yescharacteristic
290C2#3063249812yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#535×C251C2
socleC22×C251C22×D4
hypercenter#535D251
hypocenterC251#3010
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#22548762C2semidirect
2. C2×#22548741C2semidirect
3. C22×#16860798C2semidirect
4. #30631830C2semidirect
5. #30631825C2semidirect
6. #30631695C2semidirect
7. #30631681C2semidirect
8. #30631654C2semidirect
9. #535×C251C2semidirect
13. C2×#16860810C22semidirect
17. C22×Dic502C22semidirect
26. #22548638C22semidirect
27. #22548638C22semidirect
28. #22548635C22semidirect
29. C22⋊(C251⋊C8)C22semidirect
40. #16860810D4semidirect
48. C2×Dic502D4semidirect
145. C251#3010semidirect
150. #535D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487622
C2×#225487412
C22×#168607982
#306318302
#306318252
#306316952
#306316812
#306316542
#535×C2512
#3010251