Finite Group Databasebeta

← #44562161

Subgroups — #44562161

Order 32128.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#30632491C212yesmaximal, complemented
2#30632490C212yesmaximal, complemented
3#30632063C211yesmaximal, characteristic, complemented
4#30631956C212yesmaximal, complemented
5#30631955C212yesmaximal, complemented
6#30631856C212yesmaximal, complemented
7#30631854C211yesmaximal, characteristic, complemented
8#30631801C212yesmaximal, complemented
9#453×C251C211yesmaximal, characteristic, complemented
10#22548761C2212yescomplemented
11#22548760C2211yescharacteristic, complemented
12#22548743C2212yescomplemented
13#22548742—22no—
14#22548741—22no—
15#22548741C2212yescomplemented
16#22548740—22no—
17#22548740C2212yescomplemented
18#22548739—22no—
19#22548736—22no—
20#22548735—22no—
21#22548735C2212yes—
22C2×#16860795—22no—
23C2×#16860795C2212yescomplemented
24C2×#16860795C2212yescomplemented
25D8×D251—22no—
26#22548723C2212yescomplemented
27#22548722C2212yescomplemented
28#22548712C2211yescharacteristic, complemented
29#22548650C2212yescomplemented
30#22548643C2212yescomplemented
31#22548632C2211yescharacteristic, complemented
32#22548631C2212yescomplemented
33C4×C251⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C251C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C251C2212yescomplemented
36#127×C251C2211yescharacteristic, complemented
37#104×C251C2212yescomplemented
38#100×C251C2211yescharacteristic, complemented
39#16860811—22no—
40#16860810—22no—
41#16860810C2312yes—
42#16860809—22no—
43C2×D1004—22no—
44C2×D1004—41no—
45C2×D1004C2312yes—
46C2×C22⋊D251—41no—
47D4×D251—22no—
48D4×D251—22no—
49D4×D251—42no—
50#16860800D412yescomplemented
51#16860799—22no—
52#16860798—22no—
53#16860798D412yescomplemented
54#16860797—22no—
55#16860797D412yescomplemented
56D2008—22no—
57#16860795—22no—
58#16860795—22no—
59#16860795—22no—
60#16860795—22no—
61#16860795—22no—
62#16860795—22no—
63#16860795D412yescomplemented
64#16860794—22no—
65#16860794C2312yes—
66#16860793—22no—
67C2×C251⋊C8—22no—
68C2×C251⋊C8C2312yes—
69C8×D251—22no—
70#41×C251—22no—
71#41×C251C2312yes—
72C2×D4×C251—22no—
73C2×D4×C251—41no—
74C2×D4×C251—41no—
75C2×D4×C251C2311yescharacteristic
76C2×D4×C251C2311yescharacteristic
77SD16×C251—22no—
78D8×C251—22no—
79M16×C251C2312yes—
80C42×C251C2311yescharacteristic
81C22×D251—22no—
82C22×D251—41no—
83C22×D251—42no—
84Dic502C2×D412yes—
85D1004—22no—
86D1004—22no—
87D1004—42no—
88D1004—82no—
89D1004C2×D412yes—
90C22⋊D251—22no—
91C22⋊D251—42no—
92C22⋊D251—42no—
93C2×Dic251—41no—
94C4×D251—22no—
95C4×D251—42no—
96C251⋊C8—22no—
97C251⋊C8C2×D412yes—
98C251⋊C8C2×D412yes—
99C23×C251—21no—
100C23×C251—21no—
101C23×C251—41no—
102Q8×C251C2×D412yes—
103D4×C251—22no—
104D4×C251—22no—
105D4×C251—22no—
106D4×C251—42no—
107D4×C251—82no—
108D4×C251C2×D412yes—
109C2×C4×C251—21no—
110C2×C4×C251—22no—
111C2×C4×C251—41no—
112C2×C4×C251C2411yescharacteristic
113C8×C251—22no—
114C22×C251—22no—
115C22×C251—41no—
116C22×C251—41no—
117C22×C251—81no—
118C22×C251—81no—
119C22×C251—161no—
120C22×C251H(5,𝔽2)11yescharacteristic
121C2×D251—22no—
122C2×D251—42no—
123C2×D251—82no—
124C2×D251—161no—
125C2×D251—162no—
126C4×C251—22no—
127C4×C251—22no—
128C4×C251C22×D412yes—
129Dic251—22no—
130C2×C251—21no—
131C2×C251—42no—
132C2×C251—81no—
133C2×C251—81no—
134C2×C251D4211yescharacteristic
135D251—42no—
136D251—82no—
137C251#329211yescharacteristic, complemented, Sylow
138#3292—2511nomaximal, Sylow
139#576—2512no—
140#576—2512no—
141#493—2511no—
142#472—2512no—
143#472—2512no—
144#453—2511no—
145#453—2512no—
146#453D25111yescharacteristic, complemented
147#443—2512no—
148H(5,𝔽2)—2512no—
149H(5,𝔽2)—2511no—
150H(5,𝔽2)C2×D25111yescharacteristic, complemented
151AGL(1,ℤ/8)—5022no—
152AGL(1,ℤ/8)—5022no—
153AGL(1,ℤ/8)—5022no—
154AGL(1,ℤ/8)—5022no—
155AGL(1,ℤ/8)—2512no—
156AGL(1,ℤ/8)—2512no—
157AGL(1,ℤ/8)—2512no—
158AGL(1,ℤ/8)C2×D25112yescomplemented
159#135—5022no—
160#135—5022no—
161#135—2512no—
162C2×D8—5022no—
163C2×D8—5022no—
164C2×D8—2512no—
165C2×D8—2512no—
166#131—2512no—
167#131—2512no—
168#127—2511no—
169#127C2×D25111yescharacteristic, complemented
170#108—2512no—
171#104—2512no—
172#104C2×D25112yescomplemented
173#100—2511no—
174#100—2512no—
175#100C2×D25111yescharacteristic, complemented
176C4×C8—2511no—
177#41—22no—
178#41—5022no—
179#41—5022no—
180#41—5022no—
181#41—2512no—
182#41C22×D25112yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—10041no—
187C2×D4—10042no—
188C2×D4—10041no—
189C2×D4—5022no—
190C2×D4—5022no—
191C2×D4—5022no—
192C2×D4—2512no—
193C2×D4C22×D25111yescharacteristic
194C2×D4C22×D25111yescharacteristic
195Q16—2512no—
196SD16—22no—
197SD16—5022no—
198SD16—5022no—
199SD16—5022no—
200SD16—2512no—
201SD16—2512no—
202D8—22no—
203D8—5022no—
204D8—5022no—
205D8—5022no—
206D8—5022no—
207D8—5022no—
208D8—5022no—
209D8—5022no—
210D8—2512no—
211M16—5022no—
212M16—5022no—
213M16—2512no—
214M16C22×D25112yes—
215C2×C8—5022no—
216C2×C8—5022no—
217C2×C8—2512no—
218C42C22×D25111yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—10042no—
223C23—10041no—
224C23—5022no—
225Q8—2512no—
226Q8D4×D25112yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—20082no—
233D4—10042no—
234D4—10042no—
235D4—10042no—
236D4—5022no—
237D4—5022no—
238D4—5022no—
239D4—2512no—
240D4D4×D25112yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—10042no—
245C2×C4—10041no—
246C2×C4—5022no—
247C2×C4C23×D25111yescharacteristic
248C8—22no—
249C8—5022no—
250C8—2512no—
251C8—2512no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—40162no—
259C22—40161no—
260C22—20082no—
261C22—10042no—
262C22—5022no—
263C22#2254876011yescharacteristic
264C4—22no—
265C4—22no—
266C4—5022no—
267C4C2×D4×D25112yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—20082no—
273C2—10042no—
274C2#3063231911yescharacteristic
subgroupG/H
centerC2#30632319
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#453×C251C2
socleC2×C251D42
hypercenter#453D251
hypocenterC251#3292
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #30632491C2semidirect
2. #30632490C2semidirect
3. #30632063C2semidirect
4. #30631956C2semidirect
5. #30631955C2semidirect
6. #30631856C2semidirect
7. #30631854C2semidirect
8. #30631801C2semidirect
9. #453×C251C2semidirect
10. #22548761C22semidirect
11. #22548760C22semidirect
12. #22548743C22semidirect
15. #22548741C22semidirect
17. #22548740C22semidirect
23. C2×#16860795C22semidirect
24. C2×#16860795C22semidirect
26. #22548723C22semidirect
27. #22548722C22semidirect
28. #22548712C22semidirect
29. #22548650C22semidirect
30. #22548643C22semidirect
31. #22548632C22semidirect
32. #22548631C22semidirect
33. C4×C251⋊C8C22semidirect
34. H(5,𝔽2)×C251C22semidirect
35. AGL(1,ℤ/8)×C251C22semidirect
36. #127×C251C22semidirect
37. #104×C251C22semidirect
38. #100×C251C22semidirect
50. #16860800D4semidirect
53. #16860798D4semidirect
55. #16860797D4semidirect
63. #16860795D4semidirect
137. C251#3292semidirect
146. #453D251semidirect
150. H(5,𝔽2)C2×D251semidirect
158. AGL(1,ℤ/8)C2×D251semidirect
169. #127C2×D251semidirect
172. #104C2×D251semidirect
175. #100C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324912
#306324902
#306320632
#306319562
#306319552
#306318562
#306318542
#306318012
#453×C2512
#3292251