Finite Group Databasebeta

← #44562189

Subgroups — #44562189

Order 32128.

462 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 141 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#30632394C211yesmaximal, characteristic, complemented
2#30632033C211yesmaximal, characteristic, complemented
3#30632003C212yesmaximal, complemented
4#30631977C211yesmaximal, characteristic, complemented
5#30631914C212yesmaximal, complemented
6#30631884C211yesmaximal, characteristic, complemented
7#30631838C212yesmaximal, complemented
8C4×#16860798C211yesmaximal, characteristic, complemented
9D4×C251⋊C8C211yesmaximal, characteristic, complemented
10C2×#22548636C212yesmaximal, complemented
11D42×C251C211yesmaximal, characteristic
12C2×#16860798C2211yescharacteristic
13C2×#16860798C2212yescomplemented
14C22×C251⋊C8C2212yescomplemented
15#22548715C2211yescharacteristic, complemented
16#22548706—21no—
17#22548694C2212yescomplemented
18#22548692—22no—
19C4×Dic502C2211yescharacteristic, complemented
20D4×Dic251—21no—
21C2×C4⋊Dic251C2212yescomplemented
22#22548646C2212yescomplemented
23#22548646C2211yescharacteristic, complemented
24C4⋊(C251⋊C8)C2211yescharacteristic, complemented
25#22548638C2211yescharacteristic, complemented
26#22548636C2212yes—
27#22548636C2211yescharacteristic
28#22548636C2212yescomplemented
29#22548636C2212yescomplemented
30#22548636C2212yescomplemented
31C22⋊(C251⋊C8)C2212yescomplemented
32C4×C251⋊C8C2211yescharacteristic, complemented
33C22×D4×C251—21no—
34C22×D4×C251C2212yes—
35#127×C251C2211yescharacteristic
36#121×C251—21no—
37#121×C251C2212yes—
38#120×C251—22no—
39C4×D4×C251C2211yescharacteristic
40C4×D4×C251C2211yescharacteristic
41C2×Dic502—21no—
42C2×Dic502C2311yescharacteristic
43C22×Dic251—22no—
44#16860798—22no—
45#16860798—42no—
46C2×C251⋊C8—42no—
47C2×C251⋊C8C2311yescharacteristic
48C2×C251⋊C8C2311yescharacteristic
49C2×C251⋊C8C2312yes—
50C4⋊Dic251—22no—
51C4⋊Dic251—21no—
52C4⋊Dic251C2311yescharacteristic
53C4⋊Dic251C2312yes—
54Dic251⋊C4—21no—
55Dic251⋊C4—22no—
56C22⋊Dic251—22no—
57C4×Dic251—21no—
58C24×C251—22no—
59C2×D4×C251—24no—
60C2×D4×C251—22no—
61C2×D4×C251—22no—
62C2×D4×C251—21no—
63C2×D4×C251—21no—
64C2×D4×C251—42no—
65C2×D4×C251—44no—
66C2×D4×C251D411yescharacteristic
67C2×D4×C251C2312yes—
68C2×D4×C251C2311yescharacteristic
69C22×C4×C251—21no—
70C22×C4×C251C2312yes—
71C4⋊C4×C251D411yescharacteristic
72C4⋊C4×C251C2311yescharacteristic
73C22⋊C4×C251—21no—
74C22⋊C4×C251—22no—
75C22⋊C4×C251D412yes—
76C42×C251C2311yescharacteristic
77Dic502—21no—
78Dic502—21no—
79Dic502—81no—
80C2×Dic251—21no—
81C2×Dic251—21no—
82C2×Dic251—22no—
83C2×Dic251—82no—
84C251⋊C8—21no—
85C251⋊C8—21no—
86C251⋊C8—22no—
87C23×C251—21no—
88C23×C251—22no—
89C23×C251—44no—
90C23×C251—44no—
91C23×C251—82no—
92C23×C251C2×D412yes—
93D4×C251—22no—
94D4×C251—21no—
95D4×C251—22no—
96D4×C251—44no—
97D4×C251—44no—
98D4×C251—42no—
99D4×C251—81no—
100D4×C251SD1614yescomplemented
101D4×C251#4112yes—
102C2×C4×C251—22no—
103C2×C4×C251—22no—
104C2×C4×C251—21no—
105C2×C4×C251—42no—
106C2×C4×C251C2×D411yescharacteristic
107C2×C4×C251C2×D412yes—
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2411yescharacteristic
110C22×C251—22no—
111C22×C251—22no—
112C22×C251—22no—
113C22×C251—41no—
114C22×C251—41no—
115C22×C251—42no—
116C22×C251—42no—
117C22×C251—42no—
118C22×C251—88no—
119C22×C251C2×SD1614yes—
120C22×C251C22×D411yescharacteristic
121C4×C251—22no—
122C4×C251—21no—
123C4×C251—41no—
124C4×C251C2×SD1612yes—
125C4×C251C2×#4111yescharacteristic
126C4×C251H(5,𝔽2)11yescharacteristic
127Dic251—41no—
128Dic251—41no—
129Dic251—42no—
130C2×C251—24no—
131C2×C251—42no—
132C2×C251—82no—
133C2×C251#54611yescharacteristic
134C2×C251C22×SD1611yescharacteristic
135C2×C251#57611yescharacteristic
136C251#329511yescharacteristic, complemented, Sylow
137#3295—2511nomaximal, Sylow
138#554—2511no—
139D42D25111yescharacteristic
140#486—2511no—
141#481—2512no—
142#476—2511no—
143#465—2512no—
144#460—2511no—
145#450—2512no—
146C4×SD16—2511no—
147C8×D4—2511no—
148C2×#102—2512no—
149C22×D4—21no—
150C22×D4C2×D25112yes—
151C2×SD16—2512no—
152C2×SD16—2511no—
153C22×C8—2512no—
154C4⋊Q8—2511no—
155#127C2×D25111yescharacteristic
156#125—5021no—
157C22⋊Q8—5022no—
158C22⋊Q8—2512no—
159#121—21no—
160#121C2×D25112yes—
161#120—22no—
162C4×Q8—2511no—
163C4×D4—5021no—
164C4×D4C2×D25111yescharacteristic
165C4×D4C2×D25111yescharacteristic
166C2×C4⋊C4—2512no—
167#106—2511no—
168#106—2512no—
169C4⋊C8—2511no—
170#103—2511no—
171#102—2511no—
172#102—2512no—
173#102—2512no—
174#102—2512no—
175#102—2512no—
176C22⋊C8—2512no—
177C4×C8—2511no—
178C24—22no—
179C2×Q8—5021no—
180C2×Q8—2511no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4C22⋊D25111yescharacteristic
189C2×D4C22×D25112yes—
190C2×D4C22×D25111yescharacteristic
191C22×C4—21no—
192C22×C4—5022no—
193C22×C4C22×D25112yes—
194SD16—10042no—
195SD16—5022no—
196C2×C8—10042no—
197C2×C8—2511no—
198C2×C8—2511no—
199C2×C8—2512no—
200C4⋊C4—5022no—
201C4⋊C4—5021no—
202C4⋊C4—5022no—
203C4⋊C4—5021no—
204C4⋊C4—2511no—
205C4⋊C4—2512no—
206C4⋊C4C22⋊D25111yescharacteristic
207C4⋊C4C22×D25111yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—5022no—
211C22⋊C4C22⋊D25112yes—
212C42—5021no—
213C42C22×D25111yescharacteristic
214C23—21no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—82no—
219C23C2×C22⋊D25112yes—
220Q8—20081no—
221Q8—5021no—
222Q8—5021no—
223D4—22no—
224D4—21no—
225D4—22no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—81no—
230D4#1686079814yescomplemented
231D4#1686080912yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—20082no—
237C2×C4—5021no—
238C2×C4—5021no—
239C2×C4—5022no—
240C2×C4C2×C22⋊D25111yescharacteristic
241C2×C4C2×C22⋊D25112yes—
242C2×C4C2×C22⋊D25111yescharacteristic
243C2×C4C23×D25111yescharacteristic
244C8—5021no—
245C8—5021no—
246C8—5022no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—88no—
256C22C2×#1686079814yes—
257C22C22×C22⋊D25111yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—10041no—
262C4—10042no—
263C4—10041no—
264C4C2×#1686079812yes—
265C4C2×#1686080911yescharacteristic
266C4#2254876011yescharacteristic
267C2—24no—
268C2—42no—
269C2—82no—
270C2#3063233111yescharacteristic
271C2C22×#1686079811yescharacteristic
272C2#3063249011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting, prime radicalD42×C251C2
socleC22×C251C22×D4
hypercenterD42D251
hypocenterC251#3295
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #30632394C2semidirect
2. #30632033C2semidirect
3. #30632003C2semidirect
4. #30631977C2semidirect
5. #30631914C2semidirect
6. #30631884C2semidirect
7. #30631838C2semidirect
8. C4×#16860798C2semidirect
9. D4×C251⋊C8C2semidirect
10. C2×#22548636C2semidirect
13. C2×#16860798C22semidirect
14. C22×C251⋊C8C22semidirect
15. #22548715C22semidirect
17. #22548694C22semidirect
19. C4×Dic502C22semidirect
21. C2×C4⋊Dic251C22semidirect
22. #22548646C22semidirect
23. #22548646C22semidirect
24. C4⋊(C251⋊C8)C22semidirect
25. #22548638C22semidirect
28. #22548636C22semidirect
29. #22548636C22semidirect
30. #22548636C22semidirect
31. C22⋊(C251⋊C8)C22semidirect
32. C4×C251⋊C8C22semidirect
100. D4×C251SD16semidirect
136. C251#3295semidirect
230. D4#16860798semidirect

Maximal subgroups

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

subgroupindex
#306323942
#306320332
#306320032
#306319772
#306319142
#306318842
#306318382
C4×#168607982
D4×C251⋊C82
C2×#225486362
D42×C2512
#3295251