Finite Group Databasebeta

← #44562154

Subgroups — #44562154

Order 32128.

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

normal subgroups only

This group has 454 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#30632500C211yesmaximal, characteristic, complemented
2#30632496C212yesmaximal, complemented
3#30632492C211yesmaximal, characteristic, complemented
4#30632072C211yesmaximal, characteristic, complemented
5#30631958C211yesmaximal, characteristic, complemented
6#30631953C212yesmaximal, complemented
7#30631863C212yesmaximal, complemented
8#30631862C212yesmaximal, complemented
9#30631803C211yesmaximal, characteristic, complemented
10#30631802C211yesmaximal, characteristic, complemented
11#473×C251C211yesmaximal, characteristic, complemented
12#22548763C2212yescomplemented
13#22548761C2212yes—
14#22548747—22no—
15#22548747—22no—
16#22548746C2212yescomplemented
17#22548743—22no—
18#22548743C2212yes—
19#22548743C2212yescomplemented
20#22548742—22no—
21#22548738—22no—
22#22548738—21no—
23#22548737—21no—
24#22548737C2211yescharacteristic
25#22548735C2211yescharacteristic
26Q16×D251—21no—
27C2×#16860797C2212yescomplemented
28SD16×D251—22no—
29C2×D2008—21no—
30C2×D2008C2211yescharacteristic
31#22548723C2212yescomplemented
32#22548721C2211yescharacteristic, complemented
33#22548704C2212yescomplemented
34#22548650C2211yescharacteristic, complemented
35#22548642C2211yescharacteristic
36#22548641C2211yescharacteristic, complemented
37#22548641C2212yescomplemented
38#22548631C2212yescomplemented
39C8×Dic251C2211yescharacteristic, complemented
40#137×C251C2212yes—
41C2×Q16×C251C2211yescharacteristic, complemented
42#131×C251C2211yescharacteristic
43#108×C251C2211yescharacteristic, complemented
44#101×C251C2212yescomplemented
45#16860811—22no—
46#16860811—22no—
47#16860811—22no—
48#16860810—22no—
49#16860810—42no—
50#16860810C2312yes—
51#16860810C2311yescharacteristic
52Q8×D251—22no—
53Q8×D251—22no—
54C2×D1004—22no—
55C2×D1004C2312yes—
56D4×D251—22no—
57D4×D251—42no—
58Dic1004D411yescharacteristic
59#16860800—22no—
60#16860800D411yescharacteristic
61#16860799—22no—
62#16860799—22no—
63#16860799D412yes—
64#16860798D411yescharacteristic
65#16860797—22no—
66#16860797—22no—
67#16860797—22no—
68#16860797D411yescharacteristic
69D2008—22no—
70D2008—22no—
71D2008—21no—
72D2008—21no—
73D2008D411yescharacteristic
74#16860795D411yescharacteristic
75#16860794C2311yescharacteristic
76#16860793—22no—
77#16860793—21no—
78C2×C251⋊C8C2311yescharacteristic
79C8×D251—22no—
80C8×D251—21no—
81(C2×D251)⋊C4—42no—
82C4×Dic251C2311yescharacteristic
83#41×C251C2311yescharacteristic
84C2×Q8×C251C2312yes—
85Q16×C251—22no—
86Q16×C251—21no—
87Q16×C251—21no—
88SD16×C251—22no—
89M16×C251—21no—
90M16×C251C2311yescharacteristic
91M16×C251C2312yes—
92C2×C8×C251—21no—
93C2×C8×C251C2311yescharacteristic
94C22×D251—22no—
95C22×D251—42no—
96Dic502—42no—
97Dic502C2×D411yescharacteristic
98Dic502C2×D412yes—
99D1004—22no—
100D1004—22no—
101D1004—42no—
102D1004—42no—
103D1004C2×D411yescharacteristic
104D1004C2×D412yes—
105C22⋊D251—22no—
106C22⋊D251—21no—
107C22⋊D251—42no—
108C2×Dic251—21no—
109C4×D251—22no—
110C4×D251—21no—
111C4×D251—42no—
112C4×D251—42no—
113C4×D251—42no—
114C251⋊C8—21no—
115C251⋊C8C2×D411yescharacteristic
116C251⋊C8C2×D411yescharacteristic
117Q8×C251—22no—
118Q8×C251—22no—
119Q8×C251C2×D411yescharacteristic
120D4×C251—21no—
121D4×C251C2×D411yescharacteristic
122C2×C4×C251—21no—
123C2×C4×C251—22no—
124C2×C4×C251C2411yescharacteristic
125C8×C251—21no—
126C8×C251—22no—
127C8×C251C2×D411yescharacteristic
128C8×C251C2×D411yescharacteristic
129C22×C251—21no—
130C22×C251H(5,𝔽2)11yescharacteristic
131C2×D251—21no—
132C2×D251—22no—
133C2×D251—42no—
134C2×D251—82no—
135C2×D251—162no—
136C4×C251—21no—
137C4×C251—42no—
138C4×C251C22×D411yescharacteristic
139C4×C251C22×D411yescharacteristic
140Dic251—22no—
141Dic251—21no—
142Dic251—41no—
143C2×C251—21no—
144C2×C251—41no—
145C2×C251D4211yescharacteristic
146D251—41no—
147D251—42no—
148D251—82no—
149C251#329011yescharacteristic, complemented, Sylow
150#3290—2511nomaximal, Sylow
151#578—2511no—
152#577—2512no—
153#576—2511no—
154#495—2511no—
155#473D25111yescharacteristic, complemented
156#472—2511no—
157#471—2512no—
158#455—2512no—
159#454—2512no—
160#443—2511no—
161#443—2511no—
1622−1+4—2512no—
163H(5,𝔽2)—2512no—
164#137—5022no—
165#137—5022no—
166#137—2512no—
167#137C2×D25112yes—
168AGL(1,ℤ/8)—5022no—
169AGL(1,ℤ/8)—5022no—
170AGL(1,ℤ/8)—2512no—
171AGL(1,ℤ/8)—2512no—
172#135—5021no—
173#135—5022no—
174#135—5021no—
175#135—2511no—
176#135—2511no—
177C2×Q16—5021no—
178C2×Q16C2×D25111yescharacteristic, complemented
179C2×SD16—5022no—
180C2×SD16—2512no—
181C2×D8—5021no—
182C2×D8—2511no—
183#131—2512no—
184#131—2511no—
185#131C2×D25111yescharacteristic
186#124—2512no—
187#108—2511no—
188#108C2×D25111yescharacteristic, complemented
189#104—2511no—
190#104—2511no—
191#104—2512no—
192#101C2×D25112yescomplemented
193#100—2512no—
194C4×C8—2511no—
195#41—10042no—
196#41—5022no—
197#41—5022no—
198#41—5022no—
199#41—5022no—
200#41—2511no—
201#41—2512no—
202#41C22×D25111yescharacteristic
203C2×Q8—5022no—
204C2×Q8—5022no—
205C2×Q8C22×D25112yes—
206C2×D4—10042no—
207C2×D4—5022no—
208C2×D4—5022no—
209C2×D4—2512no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—5022no—
214Q16—2511no—
215Q16—2511no—
216SD16—22no—
217SD16—5022no—
218SD16—5022no—
219SD16—5022no—
220SD16—5022no—
221SD16—5022no—
222SD16—2511no—
223SD16—2511no—
224SD16—2512no—
225D8—5022no—
226D8—5022no—
227D8—5021no—
228D8—5021no—
229D8—2511no—
230D8—2511no—
231M16—21no—
232M16—5022no—
233M16—5021no—
234M16—2511no—
235M16C22×D25112yes—
236M16C22×D25111yescharacteristic
237C2×C8—21no—
238C2×C8—5022no—
239C2×C8—5021no—
240C2×C8—2511no—
241C2×C8C22×D25111yescharacteristic
242C22⋊C4—10042no—
243C42—2511no—
244C23—10042no—
245C23—5022no—
246Q8—22no—
247Q8—22no—
248Q8—10042no—
249Q8—2511no—
250Q8—2512no—
251Q8D4×D25111yescharacteristic
252D4—21no—
253D4—10042no—
254D4—10042no—
255D4—10042no—
256D4—5022no—
257D4—5021no—
258D4—5022no—
259D4—5022no—
260D4—2511no—
261D4—2512no—
262D4D4×D25111yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—10042no—
266C2×C4—10042no—
267C2×C4—10042no—
268C2×C4—5021no—
269C2×C4—5022no—
270C2×C4—5021no—
271C2×C4C23×D25111yescharacteristic
272C8—21no—
273C8—22no—
274C8—5021no—
275C8—2511no—
276C8—2511no—
277C8D4×D25111yescharacteristic
278C8D4×D25111yescharacteristic
279C22—21no—
280C22—40162no—
281C22—20082no—
282C22—10042no—
283C22—5021no—
284C22—5022no—
285C22#2254876111yescharacteristic
286C4—21no—
287C4—42no—
288C4—10041no—
289C4—5022no—
290C4—5021no—
291C4C2×D4×D25111yescharacteristic
292C4C2×D4×D25111yescharacteristic
293C2—21no—
294C2—41no—
295C2—20082no—
296C2—10042no—
297C2—10041no—
298C2#3063232111yescharacteristic
subgroupG/H
centerC2#30632321
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#473×C251C2
socleC2×C251D42
hypercenter#473D251
hypocenterC251#3290
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #30632500C2semidirect
2. #30632496C2semidirect
3. #30632492C2semidirect
4. #30632072C2semidirect
5. #30631958C2semidirect
6. #30631953C2semidirect
7. #30631863C2semidirect
8. #30631862C2semidirect
9. #30631803C2semidirect
10. #30631802C2semidirect
11. #473×C251C2semidirect
12. #22548763C22semidirect
16. #22548746C22semidirect
19. #22548743C22semidirect
27. C2×#16860797C22semidirect
31. #22548723C22semidirect
32. #22548721C22semidirect
33. #22548704C22semidirect
34. #22548650C22semidirect
36. #22548641C22semidirect
37. #22548641C22semidirect
38. #22548631C22semidirect
39. C8×Dic251C22semidirect
41. C2×Q16×C251C22semidirect
43. #108×C251C22semidirect
44. #101×C251C22semidirect
149. C251#3290semidirect
155. #473D251semidirect
178. C2×Q16C2×D251semidirect
188. #108C2×D251semidirect
192. #101C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306325002
#306324962
#306324922
#306320722
#306319582
#306319532
#306318632
#306318622
#306318032
#306318022
#473×C2512
#3290251