Finite Group Databasebeta

← #44562151

Subgroups — #44562151

Order 32128.

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

normal subgroups only

This group has 454 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#30632495C212yesmaximal, complemented
2#30632489C211yesmaximal, characteristic, complemented
3#30632072C211yesmaximal, characteristic, complemented
4#30631959C211yesmaximal, characteristic, complemented
5#30631957C211yesmaximal, characteristic, complemented
6#30631952C212yesmaximal, complemented
7#30631863C212yesmaximal, complemented
8#30631860C212yesmaximal, complemented
9#30631804C211yesmaximal, characteristic, complemented
10#30631802C211yesmaximal, characteristic, complemented
11#578×C251C211yesmaximal, characteristic, complemented
12#22548761C2212yes—
13#22548746—22no—
14#22548746C2212yes—
15#22548742—22no—
16#22548740—22no—
17#22548740C2212yes—
18#22548740C2212yescomplemented
19#22548738—21no—
20#22548735—22no—
21#22548735C2211yescharacteristic
22C2×#16860797—22no—
23C2×#16860797C2212yescomplemented
24C2×D2008C2211yescharacteristic
25D8×D251—21no—
26#22548722C2212yescomplemented
27#22548721C2211yescharacteristic
28#22548704C2212yescomplemented
29#22548651C2211yescharacteristic, complemented
30#22548650C2211yescharacteristic, complemented
31#22548642C2211yescharacteristic, complemented
32#22548642C2212yescomplemented
33#22548641C2211yescharacteristic
34#22548634C2212yescomplemented
35#22548631C2212yescomplemented
36C8×Dic251C2211yescharacteristic, complemented
372−1+4×C251C2212yescomplemented
38#137×C251—22no—
39#137×C251C2212yescomplemented
40#135×C251—21no—
41#135×C251C2211yescharacteristic
42C2×Q16×C251—21no—
43C2×Q16×C251C2211yescharacteristic, complemented
44#131×C251C2211yescharacteristic, complemented
45#16860811—22no—
46#16860810—22no—
47#16860810C2311yescharacteristic
48C2×D1004—22no—
49C2×D1004C2312yes—
50D4×D251—22no—
51D4×D251—42no—
52#16860800—22no—
53#16860800D411yescharacteristic
54#16860798D411yescharacteristic
55#16860797—22no—
56#16860797—22no—
57#16860797—22no—
58#16860797—22no—
59#16860797—22no—
60#16860797D411yescharacteristic
61D2008—21no—
62D2008—21no—
63#16860795—22no—
64#16860795—22no—
65#16860795D411yescharacteristic
66#16860794—22no—
67#16860794C2311yescharacteristic
68#16860794C2312yes—
69#16860793—21no—
70C2×C251⋊C8—22no—
71C2×C251⋊C8C2311yescharacteristic
72C8×D251—21no—
73(C2×D251)⋊C4—42no—
74C4×Dic251C2311yescharacteristic
75#41×C251—22no—
76#41×C251—22no—
77#41×C251—42no—
78#41×C251C2312yes—
79#41×C251C2311yescharacteristic
80C2×Q8×C251—22no—
81C2×Q8×C251—22no—
82C2×Q8×C251C2312yes—
83Q16×C251—22no—
84Q16×C251—21no—
85Q16×C251—21no—
86Q16×C251D411yescharacteristic
87SD16×C251—22no—
88SD16×C251D412yes—
89D8×C251D411yescharacteristic
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—
103C2×Dic251—21no—
104C4×D251—21no—
105C4×D251—42no—
106C251⋊C8—21no—
107C251⋊C8—22no—
108C251⋊C8C2×D411yescharacteristic
109C251⋊C8C2×D411yescharacteristic
110Q8×C251—22no—
111Q8×C251—22no—
112Q8×C251—42no—
113Q8×C251C2×D411yescharacteristic
114Q8×C251C2×D412yes—
115D4×C251—21no—
116D4×C251—22no—
117D4×C251—42no—
118D4×C251C2×D411yescharacteristic
119D4×C251C2×D412yes—
120C2×C4×C251—21no—
121C2×C4×C251—22no—
122C2×C4×C251—22no—
123C2×C4×C251—42no—
124C2×C4×C251—42no—
125C2×C4×C251C2411yescharacteristic
126C8×C251—21no—
127C8×C251C2×D411yescharacteristic
128C8×C251C2×D411yescharacteristic
129C22×C251—21no—
130C22×C251—22no—
131C22×C251H(5,𝔽2)11yescharacteristic
132C2×D251—21no—
133C2×D251—42no—
134C2×D251—82no—
135C2×D251—162no—
136C4×C251—21no—
137C4×C251—22no—
138C4×C251—42no—
139C4×C251C22×D411yescharacteristic
140C4×C251C22×D411yescharacteristic
141Dic251—21no—
142Dic251—41no—
143C2×C251—21no—
144C2×C251—42no—
145C2×C251—41no—
146C2×C251D4211yescharacteristic
147D251—41no—
148D251—82no—
149C251#329011yescharacteristic, complemented, Sylow
150#3290—2511nomaximal, Sylow
151#578D25111yescharacteristic, complemented
152#577—2512no—
153#576—2511no—
154#495—2511no—
155#473—2511no—
156#472—2511no—
157#471—2512no—
158#455—2512no—
159#454—2512no—
160#443—2511no—
161#443—2511no—
1622−1+4C2×D25112yescomplemented
163H(5,𝔽2)—2512no—
164#137—22no—
165#137—5022no—
166#137—2512no—
167#137C2×D25112yescomplemented
168AGL(1,ℤ/8)—5022no—
169AGL(1,ℤ/8)—5022no—
170AGL(1,ℤ/8)—2512no—
171AGL(1,ℤ/8)—2512no—
172#135—21no—
173#135—5021no—
174#135—5022no—
175#135—2511no—
176#135C2×D25111yescharacteristic
177C2×Q16—21no—
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, complemented
186#124—2512no—
187#108—2511no—
188#108—2511no—
189#104—2511no—
190#104—2512no—
191#104—2511no—
192#101—2512no—
193#100—2512no—
194C4×C8—2511no—
195#41—22no—
196#41—22no—
197#41—42no—
198#41—5022no—
199#41—5022no—
200#41—2511no—
201#41C22×D25111yescharacteristic
202#41C22×D25112yes—
203C2×Q8—22no—
204C2×Q8—22no—
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—
215Q16C22⋊D25111yescharacteristic
216SD16—22no—
217SD16—5022no—
218SD16—5022no—
219SD16—5022no—
220SD16—5022no—
221SD16—5022no—
222SD16—2511no—
223SD16—2511no—
224SD16C22⋊D25112yes—
225D8—5022no—
226D8—5022no—
227D8—5021no—
228D8—5021no—
229D8—2511no—
230D8C22⋊D25111yescharacteristic
231M16—21no—
232M16—5022no—
233M16—5021no—
234M16—2512no—
235M16—2511no—
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—42no—
249Q8—2511no—
250Q8D4×D25111yescharacteristic
251Q8C2×C22⋊D25112yes—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—10042no—
256D4—10042no—
257D4—5021no—
258D4—5022no—
259D4—5022no—
260D4—2511no—
261D4D4×D25111yescharacteristic
262D4C2×C22⋊D25112yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—10042no—
269C2×C4—5021no—
270C2×C4—5021no—
271C2×C4C23×D25111yescharacteristic
272C8—21no—
273C8—5022no—
274C8—5021no—
275C8—2511no—
276C8—2511no—
277C8C2×C22⋊D25111yescharacteristic
278C8C2×C22⋊D25111yescharacteristic
279C22—21no—
280C22—22no—
281C22—40162no—
282C22—20082no—
283C22—10042no—
284C22—5021no—
285C22#2254876011yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—10041no—
290C4—5021no—
291C4C2×D4×D25111yescharacteristic
292C4C22×C22⋊D25111yescharacteristic
293C2—21no—
294C2—42no—
295C2—41no—
296C2—20082no—
297C2—10041no—
298C2D4×C22⋊D25111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#578×C251C2
socleC2×C251D42
hypercenter#578D251
hypocenterC251#3290
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #30632495C2semidirect
2. #30632489C2semidirect
3. #30632072C2semidirect
4. #30631959C2semidirect
5. #30631957C2semidirect
6. #30631952C2semidirect
7. #30631863C2semidirect
8. #30631860C2semidirect
9. #30631804C2semidirect
10. #30631802C2semidirect
11. #578×C251C2semidirect
18. #22548740C22semidirect
23. C2×#16860797C22semidirect
26. #22548722C22semidirect
28. #22548704C22semidirect
29. #22548651C22semidirect
30. #22548650C22semidirect
31. #22548642C22semidirect
32. #22548642C22semidirect
34. #22548634C22semidirect
35. #22548631C22semidirect
36. C8×Dic251C22semidirect
37. 2−1+4×C251C22semidirect
39. #137×C251C22semidirect
43. C2×Q16×C251C22semidirect
44. #131×C251C22semidirect
149. C251#3290semidirect
151. #578D251semidirect
162. 2−1+4C2×D251semidirect
167. #137C2×D251semidirect
178. C2×Q16C2×D251semidirect
185. #131C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324952
#306324892
#306320722
#306319592
#306319572
#306319522
#306318632
#306318602
#306318042
#306318022
#578×C2512
#3290251