Finite Group Databasebeta

← #44559654

Subgroups — #44559654

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×#22548740C211yesmaximal, characteristic, complemented
3C22×#16860799C211yesmaximal, characteristic, complemented
4#30632256C211yesmaximal, characteristic, complemented
5#30631833C214yesmaximal, complemented
6#30631827C214yesmaximal, complemented
7#30631695C211yesmaximal, characteristic, complemented
8#30631653C211yesmaximal, characteristic, complemented
9#416×C251C211yesmaximal, characteristic, complemented
10#22548762—24no—
11#22548762—24no—
12C2×#16860810—21no—
13C2×#16860810C2211yescharacteristic
14C2×#16860809—22no—
15C2×#16860809C2212yescomplemented
16C22×Dic502—21no—
17C22×Dic502C2211yescharacteristic
18C2×Q8×D251C2212yescomplemented
19C22×D1004C2211yescharacteristic
20#22548740—24no—
21C2×#16860799—24no—
22C2×#16860799C2214yes—
23C2×#16860797C2212yes—
24C2×#16860795C2212yes—
25C2×#16860794C2211yescharacteristic
26#22548712—21no—
27#22548703—21no—
28#22548690C2214yes—
29#22548685—22no—
30#22548638C2214yescomplemented
31(C2×D251)⋊C8C2214yescomplemented
32C2×#41×C251C2211yescharacteristic
33C22×C8×C251C2211yescharacteristic
34#117×C251C2211yescharacteristic
35#103×C251C2212yescomplemented
36#102×C251C2212yescomplemented
37C23×D251—21no—
38#16860810—28no—
39#16860810—22no—
40#16860810—22no—
41#16860809—216no—
42#16860809—24no—
43#16860809—24no—
44C2×Dic502—28no—
45C2×Dic502—22no—
46C2×Dic502—21no—
47C2×Dic502—22no—
48C2×Dic502D411yescharacteristic
49C2×Dic502D414yescomplemented
50C2×Dic502C2312yes—
51Q8×D251—24no—
52Q8×D251—24no—
53C2×D1004—24no—
54C2×D1004—22no—
55C2×D1004—22no—
56C2×D1004—21no—
57C2×D1004D411yescharacteristic
58C2×D1004C2312yes—
59C2×C22⋊D251—22no—
60C2×C22⋊D251D412yes—
61C22×Dic251—22no—
62C22×Dic251—21no—
63C2×C4×D251—22no—
64C2×C4×D251C2312yes—
65#16860799—28no—
66#16860797—24no—
67#16860795—24no—
68#16860794—22no—
69C2×C251⋊C8C2312yes—
70(C2×D251)⋊C4—24no—
71#41×C251—22no—
72#41×C251D414yescomplemented
73C2×Q8×C251C2311yescharacteristic
74C2×D4×C251—21no—
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C22×C4×C251C2311yescharacteristic
78C2×C8×C251—22no—
79C2×C8×C251C2312yes—
80C4⋊C4×C251C2312yes—
81C22⋊C4×C251—21no—
82C42×C251—21no—
83C22×D251—22no—
84C22×D251—42no—
85C22×D251—82no—
86C22×D251C2×D412yes—
87Dic502—28no—
88Dic502—22no—
89Dic502—24no—
90Dic502—22no—
91Dic502C2×D414yes—
92D1004—22no—
93D1004—22no—
94D1004—22no—
95D1004—42no—
96D1004—84no—
97C22⋊D251—28no—
98C22⋊D251—24no—
99C2×Dic251—28no—
100C2×Dic251—24no—
101C2×Dic251—22no—
102C2×Dic251C2×D412yes—
103C4×D251—28no—
104C4×D251—22no—
105C4×D251—22no—
106C251⋊C8—22no—
107C23×C251—21no—
108C23×C251C2×D411yescharacteristic
109Q8×C251—21no—
110Q8×C251C2×D412yes—
111D4×C251—24no—
112D4×C251—21no—
113D4×C251C2×D412yes—
114C2×C4×C251—21no—
115C2×C4×C251—24no—
116C2×C4×C251—22no—
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2411yescharacteristic
121C8×C251—22no—
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—42no—
125C22×C251#12012yes—
126C22×C251C22×D411yescharacteristic
127C2×D251—22no—
128C2×D251—24no—
129C2×D251—42no—
130C2×D251—44no—
131C2×D251—84no—
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×C251C2×#12011yescharacteristic
142C2×C251#57712yes—
143D251—42no—
144D251—82no—
145C251#301011yescharacteristic, complemented, Sylow
146#3010—2511nomaximal, Sylow
147C2×2−1+4—2511no—
148C2×AGL(1,ℤ/8)—2511no—
149C22×SD16—2511no—
150#535—2511no—
151#449—2514no—
152#448—2514no—
153#419—2511no—
154#416D25111yescharacteristic, complemented
155#408—2511no—
1562−1+4—5024no—
1572−1+4—5024no—
158C2×#41—5021no—
159C2×#41—5022no—
160C2×#41—2512no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C22×Q8—5021no—
164C22×Q8—2512no—
165C22×Q8—2511no—
166C22×D4—2511no—
167AGL(1,ℤ/8)—5024no—
168C2×SD16—5024no—
169C2×SD16—2512no—
170C2×SD16—2514no—
171C2×D8—2512no—
172C2×M16—2511no—
173C22×C8C2×D25111yescharacteristic
174#127—5021no—
175#124—5021no—
176#121—2514no—
177#120—5022no—
178#117C2×D25111yescharacteristic
179#103—2514no—
180#103C2×D25112yescomplemented
181#102C2×D25112yescomplemented
182C22⋊C8—2514no—
183C24—5021no—
184#41—22no—
185#41—5028no—
186#41—50216no—
187#41—5024no—
188#41—5024no—
189#41—5022no—
190#41—5022no—
191#41D100414yescomplemented
192C2×Q8—5028no—
193C2×Q8—5022no—
194C2×Q8—5024no—
195C2×Q8—5024no—
196C2×Q8—5021no—
197C2×Q8—5022no—
198C2×Q8—2514no—
199C2×Q8—2512no—
200C2×Q8—2511no—
201C2×Q8C22×D25111yescharacteristic
202C2×D4—21no—
203C2×D4—5024no—
204C2×D4—5022no—
205C2×D4—5021no—
206C2×D4—5022no—
207C2×D4—5022no—
208C2×D4—2512no—
209C2×D4—2511no—
210C2×D4—2512no—
211C2×D4C22×D25111yescharacteristic
212C22×C4—21no—
213C22×C4—5022no—
214C22×C4—5021no—
215C22×C4—5022no—
216C22×C4—2512no—
217C22×C4C22×D25111yescharacteristic
218SD16—5028no—
219SD16—5024no—
220D8—5024no—
221M16—5022no—
222C2×C8—22no—
223C2×C8—2512no—
224C2×C8C22×D25112yes—
225C4⋊C4C22×D25112yes—
226C22⋊C4—21no—
227C22⋊C4—5024no—
228C42—21no—
229C23—21no—
230C23—20082no—
231C23—10042no—
232C23—5022no—
233C23—2512no—
234C23D4×D25111yescharacteristic
235Q8—21no—
236Q8—5028no—
237Q8—5022no—
238Q8—5024no—
239Q8—5022no—
240Q8—2514no—
241Q8C2×D100412yes—
242D4—24no—
243D4—21no—
244D4—20084no—
245D4—10042no—
246D4—5028no—
247D4—5024no—
248D4—5022no—
249D4—5022no—
250D4—5022no—
251D4C2×D100412yes—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—5028no—
256C2×C4—5028no—
257C2×C4—5022no—
258C2×C4—5024no—
259C2×C4—5022no—
260C2×C4—5022no—
261C2×C4—2512no—
262C2×C4D4×D25112yes—
263C2×C4D4×D25111yescharacteristic
264C2×C4C2×D100412yes—
265C2×C4C23×D25111yescharacteristic
266C8—22no—
267C8—5022no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—20084no—
272C22—10044no—
273C22—10042no—
274C22—5024no—
275C22—5022no—
276C22#2254868512yes—
277C22C2×D4×D25111yescharacteristic
278C4—22no—
279C4—42no—
280C4—5024no—
281C4—5022no—
282C4#2254868512yes—
283C4C2×D4×D25111yescharacteristic
284C4C22×D100411yescharacteristic
285C2—22no—
286C2—42no—
287C2—20082no—
288C2—10042no—
289C2C2×#2254868511yescharacteristic
290C2#3063249412yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
hypercenter#416D251
hypocenterC251#3010
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#22548762C2semidirect
2. C2×#22548740C2semidirect
3. C22×#16860799C2semidirect
4. #30632256C2semidirect
5. #30631833C2semidirect
6. #30631827C2semidirect
7. #30631695C2semidirect
8. #30631653C2semidirect
9. #416×C251C2semidirect
15. C2×#16860809C22semidirect
18. C2×Q8×D251C22semidirect
30. #22548638C22semidirect
31. (C2×D251)⋊C8C22semidirect
35. #103×C251C22semidirect
36. #102×C251C22semidirect
49. C2×Dic502D4semidirect
72. #41×C251D4semidirect
145. C251#3010semidirect
154. #416D251semidirect
180. #103C2×D251semidirect
181. #102C2×D251semidirect
191. #41D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487622
C2×#225487402
C22×#168607992
#306322562
#306318332
#306318272
#306316952
#306316532
#416×C2512
#3010251