Finite Group Databasebeta

← #44560850

Subgroups — #44560850

Order 32128.

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

normal subgroups only

This group has 596 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×#22548743C211yesmaximal, characteristic, complemented
2C2×#22548735C211yesmaximal, characteristic, complemented
3C22×#16860795C211yesmaximal, characteristic, complemented
4#30632254C211yesmaximal, characteristic, complemented
5#30632251C211yesmaximal, characteristic, complemented
6#30632075C212yesmaximal, complemented
7#30632073C212yesmaximal, complemented
8#30632068C212yesmaximal, complemented
9#30632062C212yesmaximal, complemented
10#30631645C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14#22548743—24no—
15#22548735—24no—
16C2×#16860800C2211yescharacteristic, complemented
17C2×#16860799C2212yescomplemented
18C2×#16860798C2211yescharacteristic, complemented
19C2×#16860797C2211yescharacteristic, complemented
20C2×D2008C2212yescomplemented
21C2×#16860795—24no—
22C2×#16860795C2212yes—
23C2×#16860795C2212yes—
24C2×#16860795C2211yescharacteristic, complemented
25C22×C251⋊C8C2211yescharacteristic
26#22548711C2212yescomplemented
27#22548711C2212yescomplemented
28#22548704C2212yescomplemented
29#22548702C2212yescomplemented
30#22548689—22no—
31#22548686—22no—
32#22548684—22no—
33#22548683—22no—
34#22548674C2211yescharacteristic
35#22548625C2212yescomplemented
36C8×Dic251C2212yescomplemented
37C2×#41×C251C2211yescharacteristic
38C22×D4×C251C2211yescharacteristic
39AGL(1,ℤ/8)×C251—24no—
40C2×SD16×C251C2212yescomplemented
41C2×D8×C251C2212yescomplemented
42C2×M16×C251C2211yescharacteristic
43C23×D251—21no—
44#16860810—22no—
45#16860810—22no—
46C2×Dic502C2311yescharacteristic
47C2×D1004—22no—
48C2×D1004—22no—
49C2×D1004—21no—
50C2×D1004C2312yes—
51C2×D1004C2311yescharacteristic
52C2×C22⋊D251—24no—
53C2×C22⋊D251—24no—
54C2×C22⋊D251—21no—
55C2×C4×D251—21no—
56#16860800—22no—
57#16860799—24no—
58#16860798—22no—
59#16860797—22no—
60D2008—24no—
61#16860795—24no—
62#16860795—24no—
63#16860795—22no—
64C2×C251⋊C8D414yescomplemented
65C2×C251⋊C8C2311yescharacteristic
66C2×C251⋊C8C2311yescharacteristic
67C4⋊Dic251D412yes—
68C22⋊Dic251—24no—
69C22⋊Dic251D412yes—
70(C2×D251)⋊C4—24no—
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—22no—
89C22×D251—21no—
90C22×D251—42no—
91C22×D251—82no—
92Dic502—21no—
93Dic502—21no—
94D1004—22no—
95D1004—21no—
96D1004—21no—
97D1004—22no—
98D1004—42no—
99C22⋊D251—42no—
100C22⋊D251—84no—
101C22⋊D251—84no—
102C2×Dic251—21no—
103C2×Dic251C2×D414yes—
104C4×D251—42no—
105C251⋊C8C2×D412yes—
106C251⋊C8C2×D412yes—
107C23×C251—21no—
108C23×C251—22no—
109C23×C251—42no—
110C23×C251—82no—
111C23×C251C2×D411yescharacteristic
112Q8×C251—21no—
113Q8×C251—21no—
114D4×C251—21no—
115D4×C251—22no—
116D4×C251—22no—
117D4×C251—21no—
118D4×C251—42no—
119D4×C251—42no—
120C2×C4×C251—21no—
121C2×C4×C251—42no—
122C2×C4×C251C2×D411yescharacteristic
123C2×C4×C251C2×D412yes—
124C2×C4×C251C2×D412yes—
125C2×C4×C251C2411yescharacteristic
126C8×C251C2×D414yes—
127C22×C251—22no—
128C22×C251—41no—
129C22×C251—42no—
130C22×C251—44no—
131C22×C251—42no—
132C22×C251—84no—
133C22×C251#12712yes—
134C22×C251C22×D411yescharacteristic
135C2×D251—41no—
136C2×D251—42no—
137C2×D251—42no—
138C2×D251—44no—
139C2×D251—84no—
140C4×C251—41no—
141C4×C251#12712yes—
142C4×C251C22×D411yescharacteristic
143C4×C251C22×D411yescharacteristic
144Dic251—24no—
145Dic251—41no—
146C2×C251—22no—
147C2×C251—81no—
148C2×C251—82no—
149C2×C251C2×#12711yescharacteristic
150C2×C251#57612yes—
151D251—81no—
152D251—82no—
153C251#315511yescharacteristic, complemented, Sylow
154#3155—2511nomaximal, Sylow
155C2×AGL(1,ℤ/8)—2511no—
156C2×AGL(1,ℤ/8)D25111yescharacteristic, complemented
157C2×#135—2511no—
158C22×D8—2511no—
159#535—2511no—
160#535—2511no—
161#496—2512no—
162#496—2512no—
163#495—2512no—
164#493—2512no—
165#405—2511no—
166C2×#41—2511no—
167C2×#41C2×D25111yescharacteristic
168C22×D4—2511no—
169C22×D4C2×D25111yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—5024no—
172#135—5024no—
173C2×Q16—2511no—
174C2×SD16—2511no—
175C2×SD16—2511no—
176C2×SD16—2512no—
177C2×SD16C2×D25112yescomplemented
178C2×D8—5024no—
179C2×D8—2512no—
180C2×D8—2512no—
181C2×D8—2511no—
182C2×D8—2512no—
183C2×D8C2×D25112yescomplemented
184C2×M16C2×D25111yescharacteristic
185C22×C8—2511no—
186#127—2512no—
187#127—2512no—
188#124—2512no—
189#124—2512no—
190#121—5022no—
191#121—5022no—
192#120—5022no—
193#120—5022no—
194#117—2511no—
195#97—2512no—
196C4×C8—2512no—
197C24—21no—
198C24—5021no—
199#41—22no—
200#41—22no—
201#41—5022no—
202#41—5022no—
203C2×Q8—2511no—
204C2×Q8C22×D25111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—5021no—
210C2×D4—5024no—
211C2×D4—5024no—
212C2×D4—5021no—
213C2×D4—5022no—
214C2×D4—5022no—
215C2×D4—2512no—
216C2×D4—2511no—
217C2×D4C22×D25112yes—
218C2×D4C22×D25111yescharacteristic
219C22×C4—21no—
220C22×C4—5021no—
221C22×C4C22×D25111yescharacteristic
222Q16—5022no—
223SD16—24no—
224SD16—5022no—
225SD16—5022no—
226SD16—5024no—
227D8—24no—
228D8—5024no—
229D8—5024no—
230D8—5022no—
231D8—5024no—
232M16C22⋊D25114yescomplemented
233C2×C8—2514no—
234C2×C8—2511no—
235C2×C8—2511no—
236C2×C8C22×D25112yes—
237C4⋊C4—2512no—
238C22⋊C4—5024no—
239C22⋊C4—5024no—
240C22⋊C4—2512no—
241C42—2512no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—20082no—
247C23—10042no—
248C23—5022no—
249C23—5021no—
250C23D4×D25111yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—5021no—
254Q8—5021no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—20084no—
262D4—20084no—
263D4—10042no—
264D4—10042no—
265D4—5022no—
266D4—5021no—
267D4—5021no—
268D4—5022no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—10042no—
272C2×C4—5021no—
273C2×C4—2514no—
274C2×C4D4×D25112yes—
275C2×C4D4×D25111yescharacteristic
276C2×C4C2×C22⋊D25112yes—
277C2×C4C23×D25111yescharacteristic
278C8—2512no—
279C8—2512no—
280C8C2×C22⋊D25114yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—20084no—
288C22—10041no—
289C22—10042no—
290C22—10042no—
291C22—10044no—
292C22#2254871112yes—
293C22C2×D4×D25111yescharacteristic
294C4—41no—
295C4—10041no—
296C4—5024no—
297C4#2254871112yes—
298C4C2×D4×D25111yescharacteristic
299C4C22×C22⋊D25111yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—20082no—
304C2—20081no—
305C2C2×#2254871111yescharacteristic
306C2#3063249112yes—
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#3155
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×#22548743C2semidirect
2. C2×#22548735C2semidirect
3. C22×#16860795C2semidirect
4. #30632254C2semidirect
5. #30632251C2semidirect
6. #30632075C2semidirect
7. #30632073C2semidirect
8. #30632068C2semidirect
9. #30632062C2semidirect
10. #30631645C2semidirect
11. C2×AGL(1,ℤ/8)×C251C2semidirect
16. C2×#16860800C22semidirect
17. C2×#16860799C22semidirect
18. C2×#16860798C22semidirect
19. C2×#16860797C22semidirect
20. C2×D2008C22semidirect
24. C2×#16860795C22semidirect
26. #22548711C22semidirect
27. #22548711C22semidirect
28. #22548704C22semidirect
29. #22548702C22semidirect
35. #22548625C22semidirect
36. C8×Dic251C22semidirect
40. C2×SD16×C251C22semidirect
41. C2×D8×C251C22semidirect
64. C2×C251⋊C8D4semidirect
86. M16×C251D4semidirect
153. C251#3155semidirect
156. C2×AGL(1,ℤ/8)D251semidirect
177. C2×SD16C2×D251semidirect
183. C2×D8C2×D251semidirect
232. M16C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×#225487352
C22×#168607952
#306322542
#306322512
#306320752
#306320732
#306320682
#306320622
#306316452
C2×AGL(1,ℤ/8)×C2512
#3155251