Finite Group Databasebeta

← #44562162

Subgroups — #44562162

Order 32128.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#30632490C212yesmaximal, complemented
2#30632489C211yesmaximal, characteristic, complemented
3#30632062C211yesmaximal, characteristic, complemented
4#30631957C211yesmaximal, characteristic, complemented
5#30631955C212yesmaximal, complemented
6#30631954C211yesmaximal, characteristic, complemented
7#30631855C212yesmaximal, complemented
8#30631853C212yesmaximal, complemented
9#30631804C211yesmaximal, characteristic, complemented
10#30631802C211yesmaximal, characteristic, complemented
11#576×C251C211yesmaximal, characteristic, complemented
12#22548761C2212yescomplemented
13#22548742—22no—
14#22548741—22no—
15#22548741C2212yescomplemented
16#22548740—22no—
17#22548740C2212yescomplemented
18#22548740C2212yescomplemented
19#22548738—21no—
20#22548735—22no—
21#22548735C2211yescharacteristic
22C2×D2008C2211yescharacteristic, complemented
23C2×#16860795—22no—
24C2×#16860795C2212yescomplemented
25D8×D251—21no—
26#22548722C2212yescomplemented
27#22548721C2211yescharacteristic, complemented
28#22548711C2212yescomplemented
29#22548651C2211yescharacteristic, complemented
30#22548650C2211yescharacteristic, complemented
31#22548642C2211yescharacteristic, complemented
32#22548642C2212yescomplemented
33#22548641C2211yescharacteristic, complemented
34#22548632C2212yescomplemented
35#22548631C2212yescomplemented
36C8×Dic251C2211yescharacteristic, complemented
37H(5,𝔽2)×C251C2212yescomplemented
38AGL(1,ℤ/8)×C251—22no—
39AGL(1,ℤ/8)×C251C2212yescomplemented
40#135×C251—21no—
41#135×C251C2211yescharacteristic
42C2×D8×C251—21no—
43C2×D8×C251C2211yescharacteristic, complemented
44#131×C251C2211yescharacteristic, complemented
45#16860811—22no—
46#16860810—22no—
47#16860810C2311yescharacteristic
48C2×D1004—22no—
49C2×D1004C2312yes—
50C2×C22⋊D251—42no—
51D4×D251—22no—
52D4×D251—42no—
53#16860800D411yescharacteristic, complemented
54#16860798—22no—
55#16860798D411yescharacteristic, complemented
56#16860797—22no—
57#16860797—22no—
58#16860797D411yescharacteristic, complemented
59D2008—21no—
60D2008—21no—
61#16860795—22no—
62#16860795—22no—
63#16860795—22no—
64#16860795—22no—
65#16860795—22no—
66#16860795D411yescharacteristic, complemented
67#16860794—22no—
68#16860794C2311yescharacteristic
69#16860794C2312yes—
70#16860793—21no—
71C2×C251⋊C8—22no—
72C2×C251⋊C8C2311yescharacteristic
73C8×D251—21no—
74C4×Dic251C2311yescharacteristic
75#41×C251—22no—
76#41×C251—22no—
77#41×C251C2311yescharacteristic
78#41×C251C2312yes—
79C2×D4×C251—22no—
80C2×D4×C251—22no—
81C2×D4×C251—42no—
82C2×D4×C251C2312yes—
83Q16×C251D411yescharacteristic, complemented
84SD16×C251—22no—
85SD16×C251D412yescomplemented
86D8×C251—22no—
87D8×C251—21no—
88D8×C251—21no—
89D8×C251D411yescharacteristic, complemented
90M16×C251—21no—
91M16×C251C2311yescharacteristic
92C2×C8×C251—21no—
93C2×C8×C251C2311yescharacteristic
94C22×D251—22no—
95C22×D251—42no—
96Dic502C2×D411yescharacteristic
97D1004—22no—
98D1004—22no—
99D1004—42no—
100D1004C2×D411yescharacteristic
101C22⋊D251—21no—
102C22⋊D251—42no—
103C22⋊D251—84no—
104C2×Dic251—21no—
105C4×D251—21no—
106C4×D251—42no—
107C251⋊C8—21no—
108C251⋊C8—22no—
109C251⋊C8C2×D411yescharacteristic
110C251⋊C8C2×D411yescharacteristic
111C23×C251—22no—
112C23×C251—42no—
113Q8×C251C2×D411yescharacteristic
114Q8×C251C2×D412yes—
115D4×C251—21no—
116D4×C251—22no—
117D4×C251—22no—
118D4×C251—22no—
119D4×C251—42no—
120D4×C251—42no—
121D4×C251C2×D411yescharacteristic
122D4×C251C2×D412yes—
123C2×C4×C251—21no—
124C2×C4×C251—22no—
125C2×C4×C251—42no—
126C2×C4×C251C2411yescharacteristic
127C8×C251—21no—
128C8×C251C2×D411yescharacteristic
129C8×C251C2×D411yescharacteristic
130C22×C251—21no—
131C22×C251—22no—
132C22×C251—42no—
133C22×C251—82no—
134C22×C251—162no—
135C22×C251H(5,𝔽2)11yescharacteristic
136C2×D251—21no—
137C2×D251—42no—
138C2×D251—82no—
139C2×D251—162no—
140C4×C251—21no—
141C4×C251—22no—
142C4×C251C22×D411yescharacteristic
143C4×C251C22×D411yescharacteristic
144Dic251—22no—
145Dic251—21no—
146C2×C251—21no—
147C2×C251—42no—
148C2×C251—41no—
149C2×C251—82no—
150C2×C251D4211yescharacteristic
151D251—41no—
152D251—82no—
153C251#329211yescharacteristic, complemented, Sylow
154#3292—2511nomaximal, Sylow
155#576—2512no—
156#576—2511no—
157#576D25111yescharacteristic, complemented
158#493—2511no—
159#472—2512no—
160#472—2511no—
161#472—2511no—
162#453—2512no—
163#453—2512no—
164#443—2511no—
165#443—2511no—
166H(5,𝔽2)—2512no—
167H(5,𝔽2)C2×D25112yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—5022no—
170AGL(1,ℤ/8)—5022no—
171AGL(1,ℤ/8)—5022no—
172AGL(1,ℤ/8)—2512no—
173AGL(1,ℤ/8)—2512no—
174AGL(1,ℤ/8)—2512no—
175AGL(1,ℤ/8)C2×D25112yescomplemented
176#135—21no—
177#135—5022no—
178#135—5021no—
179#135—2511no—
180#135C2×D25111yescharacteristic
181C2×D8—21no—
182C2×D8—5022no—
183C2×D8—5021no—
184C2×D8—2512no—
185C2×D8—2511no—
186C2×D8C2×D25111yescharacteristic, complemented
187#131—2512no—
188#131—2511no—
189#131C2×D25111yescharacteristic, complemented
190#127—2512no—
191#108—2511no—
192#108—2511no—
193#104—2511no—
194#104—2512no—
195#104—2511no—
196#100—2512no—
197#100—2512no—
198C4×C8—2511no—
199#41—22no—
200#41—22no—
201#41—5022no—
202#41—5022no—
203#41—2511no—
204#41C22×D25111yescharacteristic
205#41C22×D25112yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—10042no—
210C2×D4—10042no—
211C2×D4—5022no—
212C2×D4—5022no—
213C2×D4—2512no—
214C2×D4C22×D25112yes—
215Q16—2511no—
216Q16C22⋊D25111yescharacteristic, complemented
217SD16—22no—
218SD16—5022no—
219SD16—5022no—
220SD16—5022no—
221SD16—2511no—
222SD16—2511no—
223SD16C22⋊D25112yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—5022no—
228D8—5022no—
229D8—5022no—
230D8—5022no—
231D8—5022no—
232D8—5021no—
233D8—5021no—
234D8—2511no—
235D8C22⋊D25111yescharacteristic, complemented
236M16—21no—
237M16—5022no—
238M16—5021no—
239M16—2512no—
240M16—2511no—
241M16C22×D25111yescharacteristic
242C2×C8—21no—
243C2×C8—5022no—
244C2×C8—5021no—
245C2×C8—2511no—
246C2×C8C22×D25111yescharacteristic
247C42—2511no—
248C23—22no—
249C23—42no—
250C23—10042no—
251C23—5022no—
252Q8—2511no—
253Q8D4×D25111yescharacteristic
254Q8C2×C22⋊D25112yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—20084no—
262D4—10042no—
263D4—10042no—
264D4—5021no—
265D4—5022no—
266D4—5022no—
267D4—2511no—
268D4D4×D25111yescharacteristic
269D4C2×C22⋊D25112yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—10042no—
274C2×C4—5021no—
275C2×C4—5021no—
276C2×C4C23×D25111yescharacteristic
277C8—21no—
278C8—5022no—
279C8—5021no—
280C8—2511no—
281C8—2511no—
282C8C2×C22⋊D25111yescharacteristic
283C8C2×C22⋊D25111yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—40162no—
290C22—20082no—
291C22—10042no—
292C22—5021no—
293C22#2254876011yescharacteristic
294C4—21no—
295C4—22no—
296C4—5022no—
297C4—5021no—
298C4C2×D4×D25111yescharacteristic
299C4C22×C22⋊D25111yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—20082no—
305C2—10041no—
306C2D4×C22⋊D25111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#576×C251C2
socleC2×C251D42
hypercenter#576D251
hypocenterC251#3292
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #30632490C2semidirect
2. #30632489C2semidirect
3. #30632062C2semidirect
4. #30631957C2semidirect
5. #30631955C2semidirect
6. #30631954C2semidirect
7. #30631855C2semidirect
8. #30631853C2semidirect
9. #30631804C2semidirect
10. #30631802C2semidirect
11. #576×C251C2semidirect
12. #22548761C22semidirect
15. #22548741C22semidirect
17. #22548740C22semidirect
18. #22548740C22semidirect
22. C2×D2008C22semidirect
24. C2×#16860795C22semidirect
26. #22548722C22semidirect
27. #22548721C22semidirect
28. #22548711C22semidirect
29. #22548651C22semidirect
30. #22548650C22semidirect
31. #22548642C22semidirect
32. #22548642C22semidirect
33. #22548641C22semidirect
34. #22548632C22semidirect
35. #22548631C22semidirect
36. C8×Dic251C22semidirect
37. H(5,𝔽2)×C251C22semidirect
39. AGL(1,ℤ/8)×C251C22semidirect
43. C2×D8×C251C22semidirect
44. #131×C251C22semidirect
53. #16860800D4semidirect
55. #16860798D4semidirect
58. #16860797D4semidirect
66. #16860795D4semidirect
83. Q16×C251D4semidirect
85. SD16×C251D4semidirect
89. D8×C251D4semidirect
153. C251#3292semidirect
157. #576D251semidirect
167. H(5,𝔽2)C2×D251semidirect
175. AGL(1,ℤ/8)C2×D251semidirect
186. C2×D8C2×D251semidirect
189. #131C2×D251semidirect
216. Q16C22⋊D251semidirect
223. SD16C22⋊D251semidirect
235. D8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306324902
#306324892
#306320622
#306319572
#306319552
#306319542
#306318552
#306318532
#306318042
#306318022
#576×C2512
#3292251