Finite Group Databasebeta

← #44560839

Subgroups — #44560839

Order 32128.

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

normal subgroups only

This group has 564 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×#22548748C211yesmaximal, characteristic, complemented
2C2×#22548735C211yesmaximal, characteristic, complemented
3C22×#16860798C211yesmaximal, characteristic, complemented
4#30632258C211yesmaximal, characteristic, complemented
5#30632254C211yesmaximal, characteristic, complemented
6#30632078C212yesmaximal, complemented
7#30632075C212yesmaximal, complemented
8#30632070C212yesmaximal, complemented
9#30632058C212yesmaximal, complemented
10#30631645C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×Dic502C2211yescharacteristic
14#22548748—24no—
15#22548735—24no—
16C2×Dic1004C2212yescomplemented
17C2×#16860800C2211yescharacteristic, complemented
18C2×#16860799C2212yescomplemented
19C2×#16860798—24no—
20C2×#16860798C2212yes—
21C2×#16860798C2212yes—
22C2×#16860798C2211yescharacteristic, complemented
23C2×#16860797C2211yescharacteristic, complemented
24C2×#16860795C2211yescharacteristic, complemented
25C22×C251⋊C8C2211yescharacteristic
26#22548713C2212yescomplemented
27#22548711C2212yes—
28#22548702C2212yes—
29#22548702C2212yescomplemented
30#22548696—22no—
31#22548694—22no—
32#22548686—22no—
33#22548684—22no—
34#22548674C2211yescharacteristic
35#22548625C2212yescomplemented
36C8×Dic251C2212yescomplemented
37C2×#41×C251C2211yescharacteristic
38C22×D4×C251C2211yescharacteristic
39AGL(1,ℤ/8)×C251—24no—
40C2×SD16×C251C2212yes—
41C2×D8×C251C2212yes—
42C2×M16×C251C2211yescharacteristic
43#16860810—22no—
44#16860810—22no—
45C2×Dic502—21no—
46C2×Dic502—22no—
47C2×Dic502—22no—
48C2×Dic502C2312yes—
49C2×Dic502C2311yescharacteristic
50C2×D1004C2311yescharacteristic
51C2×C22⋊D251—24no—
52C2×C22⋊D251—21no—
53C22×Dic251—21no—
54C2×C4×D251—21no—
55Dic1004—24no—
56#16860800—22no—
57#16860799—24no—
58#16860798—24no—
59#16860798—24no—
60#16860798—22no—
61#16860797—22no—
62#16860795—22no—
63C2×C251⋊C8D414yescomplemented
64C2×C251⋊C8C2311yescharacteristic
65C2×C251⋊C8C2311yescharacteristic
66C4⋊Dic251D412yes—
67Dic251⋊C4—24no—
68C22⋊Dic251—24no—
69C22⋊Dic251—24no—
70C22⋊Dic251D412yes—
71C4×Dic251C2312yes—
72C24×C251—21no—
73#41×C251—22no—
74#41×C251—22no—
75C2×Q8×C251C2311yescharacteristic
76C2×D4×C251—21no—
77C2×D4×C251—22no—
78C2×D4×C251—22no—
79C2×D4×C251—21no—
80C2×D4×C251C2311yescharacteristic
81C2×D4×C251C2312yes—
82C22×C4×C251—21no—
83C22×C4×C251C2311yescharacteristic
84SD16×C251—24no—
85D8×C251—24no—
86M16×C251D414yescomplemented
87C2×C8×C251C2312yes—
88C22×D251—21no—
89Dic502—22no—
90Dic502—21no—
91Dic502—22no—
92Dic502—21no—
93Dic502—42no—
94D1004—21no—
95D1004—21no—
96C22⋊D251—42no—
97C22⋊D251—84no—
98C2×Dic251—21no—
99C2×Dic251—22no—
100C2×Dic251—42no—
101C2×Dic251C2×D414yes—
102C4×D251—42no—
103C251⋊C8C2×D412yes—
104C251⋊C8C2×D412yes—
105C23×C251—21no—
106C23×C251—22no—
107C23×C251—42no—
108C23×C251—82no—
109C23×C251C2×D411yescharacteristic
110Q8×C251—21no—
111Q8×C251—21no—
112D4×C251—21no—
113D4×C251—22no—
114D4×C251—22no—
115D4×C251—21no—
116D4×C251—42no—
117D4×C251—42no—
118C2×C4×C251—21no—
119C2×C4×C251—42no—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2×D412yes—
122C2×C4×C251C2×D412yes—
123C2×C4×C251C2411yescharacteristic
124C8×C251C2×D414yes—
125C22×C251—22no—
126C22×C251—41no—
127C22×C251—42no—
128C22×C251—42no—
129C22×C251—44no—
130C22×C251—84no—
131C22×C251#12712yes—
132C22×C251C22×D411yescharacteristic
133C2×D251—41no—
134C2×D251—42no—
135C4×C251—41no—
136C4×C251#12712yes—
137C4×C251C22×D411yescharacteristic
138C4×C251C22×D411yescharacteristic
139Dic251—24no—
140Dic251—41no—
141Dic251—42no—
142C2×C251—22no—
143C2×C251—81no—
144C2×C251—82no—
145C2×C251C2×#12711yescharacteristic
146C2×C251#57712yes—
147D251—81no—
148C251#315411yescharacteristic, complemented, Sylow
149#3154—2511nomaximal, Sylow
150C2×#137—2511no—
151C2×AGL(1,ℤ/8)D25111yescharacteristic, complemented
152C2×#135—2511no—
153C22×SD16—2511no—
154#536—2511no—
155#535—2511no—
156#497—2512no—
157#496—2512no—
158#495—2512no—
159#492—2512no—
160#405—2511no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C22×Q8—2511no—
164C22×D4C2×D25111yescharacteristic
165#137—5024no—
166AGL(1,ℤ/8)—24no—
167#135—5024no—
168C2×Q16—2511no—
169C2×Q16—2512no—
170C2×SD16—5024no—
171C2×SD16—2511no—
172C2×SD16—2512no—
173C2×SD16—2512no—
174C2×SD16—2511no—
175C2×SD16—2512no—
176C2×SD16C2×D25112yes—
177C2×D8—2511no—
178C2×D8C2×D25112yes—
179C2×M16C2×D25111yescharacteristic
180C22×C8—2511no—
181C4⋊Q8—2512no—
182#127—2512no—
183#124—2512no—
184#124—2512no—
185#123—5022no—
186C22⋊Q8—5022no—
187#121—5022no—
188#120—5022no—
189#117—2511no—
190#97—2512no—
191C4×C8—2512no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—5022no—
196#41—5022no—
197C2×Q8—5022no—
198C2×Q8—5022no—
199C2×Q8—5021no—
200C2×Q8—2511no—
201C2×Q8—2512no—
202C2×Q8C22×D25111yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—5021no—
208C2×D4—5024no—
209C2×D4—2511no—
210C2×D4C22×D25111yescharacteristic
211C2×D4C22×D25112yes—
212C22×C4—21no—
213C22×C4—5021no—
214C22×C4—5021no—
215C22×C4C22×D25111yescharacteristic
216Q16—5022no—
217Q16—5024no—
218SD16—24no—
219SD16—5024no—
220SD16—5024no—
221SD16—5022no—
222SD16—5022no—
223SD16—5024no—
224D8—24no—
225D8—5022no—
226M16C22⋊D25114yescomplemented
227C2×C8—2514no—
228C2×C8—2511no—
229C2×C8—2511no—
230C2×C8C22×D25112yes—
231C4⋊C4—5024no—
232C4⋊C4—2512no—
233C22⋊C4—5024no—
234C22⋊C4—5024no—
235C22⋊C4—2512no—
236C42—2512no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—5021no—
242C23D4×D25111yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—10042no—
246Q8—5022no—
247Q8—5021no—
248Q8—5022no—
249Q8—5021no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—20084no—
257D4—10042no—
258D4—5021no—
259D4—5021no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—10042no—
263C2×C4—10042no—
264C2×C4—5021no—
265C2×C4—5022no—
266C2×C4—2514no—
267C2×C4D4×D25112yes—
268C2×C4D4×D25111yescharacteristic
269C2×C4C2×C22⋊D25112yes—
270C2×C4C23×D25111yescharacteristic
271C8—2512no—
272C8—2512no—
273C8C2×C22⋊D25114yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—10041no—
281C22—10042no—
282C22#2254871112yes—
283C22C2×D4×D25111yescharacteristic
284C4—41no—
285C4—10041no—
286C4—10042no—
287C4—5024no—
288C4#2254871112yes—
289C4C2×D4×D25111yescharacteristic
290C4C22×C22⋊D25111yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—20081no—
295C2C2×#2254871111yescharacteristic
296C2#3063249712yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×AGL(1,ℤ/8)×C251C2
socleC22×C251C22×D4
hypercenterC2×AGL(1,ℤ/8)D251
hypocenterC251#3154
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22548748C2semidirect
2. C2×#22548735C2semidirect
3. C22×#16860798C2semidirect
4. #30632258C2semidirect
5. #30632254C2semidirect
6. #30632078C2semidirect
7. #30632075C2semidirect
8. #30632070C2semidirect
9. #30632058C2semidirect
10. #30631645C2semidirect
11. C2×AGL(1,ℤ/8)×C251C2semidirect
16. C2×Dic1004C22semidirect
17. C2×#16860800C22semidirect
18. C2×#16860799C22semidirect
22. C2×#16860798C22semidirect
23. C2×#16860797C22semidirect
24. C2×#16860795C22semidirect
26. #22548713C22semidirect
29. #22548702C22semidirect
35. #22548625C22semidirect
36. C8×Dic251C22semidirect
63. C2×C251⋊C8D4semidirect
86. M16×C251D4semidirect
148. C251#3154semidirect
151. C2×AGL(1,ℤ/8)D251semidirect
226. M16C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487482
C2×#225487352
C22×#168607982
#306322582
#306322542
#306320782
#306320752
#306320702
#306320582
#306316452
C2×AGL(1,ℤ/8)×C2512
#3154251