Finite Group Databasebeta

← #44559945

Subgroups — #44559945

Order 32128.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22548735C211yesmaximal, characteristic, complemented
2C22×C8×D251C211yesmaximal, characteristic, complemented
3#30632199C211yesmaximal, characteristic, complemented
4#30632194C211yesmaximal, characteristic, complemented
5#30631927C212yesmaximal, complemented
6#30631922C212yesmaximal, complemented
7#30631919C212yesmaximal, complemented
8#30631915C212yesmaximal, complemented
9#30631723C211yesmaximal, characteristic, complemented
10#30631684C211yesmaximal, characteristic, complemented
11#416×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×C4×D251C2211yescharacteristic
14#22548735—24no—
15C2×#16860800C2211yescharacteristic, complemented
16C2×#16860798C2211yescharacteristic, complemented
17C2×#16860797C2211yescharacteristic, complemented
18C2×#16860795C2211yescharacteristic, complemented
19C22×C251⋊C8C2211yescharacteristic
20C2×C8×D251—24no—
21C2×C8×D251C2212yes—
22C2×C8×D251C2212yes—
23#22548699—21no—
24#22548696—21no—
25#22548695C2212yes—
26#22548691C2212yes—
27#22548690C2212yes—
28#22548686C2212yes—
29#22548685—21no—
30#22548683—21no—
31C4×D1004—22no—
32C4×C22⋊D251—22no—
33#22548674C2211yescharacteristic
34#22548648C2212yescomplemented
35#22548646C2212yescomplemented
36#22548640C2212yescomplemented
37#22548637C2212yescomplemented
38C2×#41×C251C2211yescharacteristic
39C22×C8×C251C2211yescharacteristic
40#117×C251C2211yescharacteristic
41#103×C251C2212yescomplemented
42#102×C251C2212yescomplemented
43C23×D251D411yescharacteristic
44#16860810—22no—
45#16860810—22no—
46C2×Dic502C2311yescharacteristic
47C2×D1004—22no—
48C2×D1004C2311yescharacteristic
49C2×C22⋊D251—22no—
50C2×C22⋊D251—21no—
51C22×Dic251D411yescharacteristic
52C2×C4×D251—22no—
53C2×C4×D251—21no—
54C2×C4×D251—22no—
55C2×C4×D251D412yes—
56C2×C4×D251C2312yes—
57#16860800—22no—
58#16860798—22no—
59#16860797—22no—
60#16860795—22no—
61C2×C251⋊C8D414yescomplemented
62C2×C251⋊C8C2311yescharacteristic
63C2×C251⋊C8C2311yescharacteristic
64C8×D251—24no—
65C8×D251—24no—
66C4⋊Dic251—22no—
67C4⋊Dic251C2312yes—
68Dic251⋊C4—22no—
69C22⋊Dic251—22no—
70C22⋊Dic251—21no—
71(C2×D251)⋊C4—22no—
72(C2×D251)⋊C4—22no—
73(C2×D251)⋊C4—22no—
74C4×Dic251—21no—
75#41×C251—22no—
76#41×C251—22no—
77C2×Q8×C251C2311yescharacteristic
78C2×D4×C251—21no—
79C2×D4×C251C2311yescharacteristic
80C22×C4×C251—21no—
81C22×C4×C251C2311yescharacteristic
82C2×C8×C251—22no—
83C2×C8×C251C2312yes—
84C4⋊C4×C251C2312yes—
85C22⋊C4×C251—21no—
86C42×C251—21no—
87C22×D251—22no—
88C22×D251—24no—
89C22×D251—21no—
90C22×D251C2×D412yes—
91Dic502—21no—
92Dic502—21no—
93D1004—21no—
94D1004—21no—
95D1004—44no—
96C22⋊D251—42no—
97C22⋊D251—44no—
98C2×Dic251—22no—
99C2×Dic251—22no—
100C2×Dic251—21no—
101C2×Dic251C2×D412yes—
102C4×D251—24no—
103C4×D251—22no—
104C4×D251—22no—
105C4×D251—42no—
106C251⋊C8C2×D412yes—
107C251⋊C8C2×D412yes—
108C23×C251—21no—
109C23×C251C2×D411yescharacteristic
110Q8×C251—21no—
111Q8×C251—21no—
112D4×C251—21no—
113D4×C251—21no—
114D4×C251—42no—
115C2×C4×C251—22no—
116C2×C4×C251—21no—
117C2×C4×C251—42no—
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251#4112yes—
121C2×C4×C251C2411yescharacteristic
122C8×C251—22no—
123C22×C251—22no—
124C22×C251—41no—
125C22×C251—42no—
126C22×C251#12112yes—
127C22×C251C22×D411yescharacteristic
128C2×D251—28no—
129C2×D251—22no—
130C2×D251—22no—
131C2×D251—41no—
132C2×D251—42no—
133C2×D251#13514yescomplemented
134C4×C251—41no—
135C4×C251—42no—
136C4×C251#12112yes—
137C4×C251C22×D411yescharacteristic
138C4×C251C2×#4111yescharacteristic
139Dic251—22no—
140Dic251—41no—
141Dic251—42no—
142C2×C251—22no—
143C2×C251—81no—
144C2×C251C2×#12111yescharacteristic
145C2×C251C2×#13512yes—
146D251—24no—
147D251—81no—
148C251#305011yescharacteristic, complemented, Sylow
149#3050—2511nomaximal, Sylow
150C2×#135—2511no—
151C23×C8—2511no—
152#525—2511no—
153#525—2511no—
154#467—2512no—
155#466—2512no—
156#465—2512no—
157#465—2512no—
158#427—2511no—
159#416—2511no—
160#416D25111yescharacteristic, complemented
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C23×C4—2511no—
164#135—5024no—
165C2×Q16—2511no—
166C2×SD16—2511no—
167C2×SD16—2511no—
168C2×D8—2511no—
169C22×C8—5024no—
170C22×C8—2512no—
171C22×C8—2512no—
172C22×C8—2511no—
173C22×C8C2×D25111yescharacteristic
174#123—5021no—
175#123—5021no—
176C22⋊Q8—2512no—
177C22⋊Q8—2512no—
178#121—2512no—
179#121—2512no—
180#120—5021no—
181#120—5021no—
182C4×D4—5022no—
183C4×D4—5022no—
184#117—2511no—
185#117C2×D25111yescharacteristic
186#107—2512no—
187#106—2512no—
188#103—2512no—
189#103C2×D25112yescomplemented
190#102—2512no—
191#102C2×D25112yescomplemented
192C24—2511no—
193#41—22no—
194#41—22no—
195#41—5022no—
196#41—5022no—
197C2×Q8—2511no—
198C2×Q8C22×D25111yescharacteristic
199C2×D4—21no—
200C2×D4—5022no—
201C2×D4—5022no—
202C2×D4—5021no—
203C2×D4—2511no—
204C2×D4C22×D25111yescharacteristic
205C22×C4—21no—
206C22×C4—5021no—
207C22×C4—5022no—
208C22×C4—5022no—
209C22×C4—2511no—
210C22×C4—2512no—
211C22×C4—2512no—
212C22×C4C22×D25111yescharacteristic
213Q16—5022no—
214SD16—5022no—
215SD16—5022no—
216D8—5022no—
217C2×C8—22no—
218C2×C8—5024no—
219C2×C8—5024no—
220C2×C8—2514no—
221C2×C8—2511no—
222C2×C8—2511no—
223C2×C8C22×D25112yes—
224C4⋊C4—5022no—
225C4⋊C4—5022no—
226C4⋊C4—2512no—
227C4⋊C4C22×D25112yes—
228C22⋊C4—21no—
229C22⋊C4—5022no—
230C22⋊C4—5022no—
231C22⋊C4—5022no—
232C22⋊C4—5021no—
233C22⋊C4—5022no—
234C42—21no—
235C42—5021no—
236C23—21no—
237C23—5024no—
238C23—5021no—
239C23—5022no—
240C23—2512no—
241C23D4×D25111yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—5021no—
245Q8—5021no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—10044no—
250D4—10044no—
251D4—10042no—
252D4—5021no—
253D4—5021no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—10042no—
258C2×C4—5022no—
259C2×C4—5021no—
260C2×C4—5022no—
261C2×C4—5022no—
262C2×C4—5022no—
263C2×C4—5024no—
264C2×C4—2512no—
265C2×C4D4×D25112yes—
266C2×C4D4×D25111yescharacteristic
267C2×C4#1686081012yes—
268C2×C4C23×D25111yescharacteristic
269C8—22no—
270C8—2512no—
271C8—2512no—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—10042no—
276C22—10041no—
277C22—5028no—
278C22—5022no—
279C22—5022no—
280C22—2514no—
281C22#2254868812yes—
282C22C2×D4×D25111yescharacteristic
283C4—42no—
284C4—41no—
285C4—10042no—
286C4—10041no—
287C4—5022no—
288C4#2254868812yes—
289C4C2×D4×D25111yescharacteristic
290C4C2×#1686081011yescharacteristic
291C2—22no—
292C2—81no—
293C2—20081no—
294C2—5024no—
295C2C2×#2254868811yescharacteristic
296C2#135×D25112yes—
subgroupG/H
centerC2×C4D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
hypercenter#416D251
hypocenterC251#3050
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#22548735C2semidirect
2. C22×C8×D251C2semidirect
3. #30632199C2semidirect
4. #30632194C2semidirect
5. #30631927C2semidirect
6. #30631922C2semidirect
7. #30631919C2semidirect
8. #30631915C2semidirect
9. #30631723C2semidirect
10. #30631684C2semidirect
11. #416×C251C2semidirect
15. C2×#16860800C22semidirect
16. C2×#16860798C22semidirect
17. C2×#16860797C22semidirect
18. C2×#16860795C22semidirect
34. #22548648C22semidirect
35. #22548646C22semidirect
36. #22548640C22semidirect
37. #22548637C22semidirect
41. #103×C251C22semidirect
42. #102×C251C22semidirect
61. C2×C251⋊C8D4semidirect
133. C2×D251#135semidirect
148. C251#3050semidirect
160. #416D251semidirect
189. #103C2×D251semidirect
191. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487352
C22×C8×D2512
#306321992
#306321942
#306319272
#306319222
#306319192
#306319152
#306317232
#306316842
#416×C2512
#3050251