Finite Group Databasebeta

← #44560311

Subgroups — #44560311

Order 32128.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#22548691C211yesmaximal, characteristic
2#117×D251C211yesmaximal, characteristic
3#30632027C212yesmaximal, complemented
4#30632026C212yesmaximal, complemented
5#30632021C212yesmaximal, complemented
6#30632016C212yesmaximal, complemented
7#30631730C211yesmaximal, characteristic, complemented
8#30631729C211yesmaximal, characteristic, complemented
9#30631696C211yesmaximal, characteristic, complemented
10#30631650C211yesmaximal, characteristic
11#419×C251C211yesmaximal, characteristic, complemented
12C22×C4×D251C2211yescharacteristic
13C2×#16860794C2211yescharacteristic
14#22548691—21no—
15#22548691—21no—
16#22548691C2212yes—
17#22548691C2212yes—
18#22548674C2211yescharacteristic
19#22548673C2211yescharacteristic
20#22548673C2211yescharacteristic
21#22548672—21no—
22#22548671—21no—
23C2×C4⋊Dic251C2211yescharacteristic
24C2×Dic251⋊C4—21no—
25C4⋊C4×D251C2211yescharacteristic
26C4⋊C4×D251C2211yescharacteristic
27C2×(C2×D251)⋊C4—21no—
28C22⋊C4×D251—21no—
29C42×D251—21no—
30#22548649C2212yescomplemented
31#22548648C2212yescomplemented
32#22548647C2212yescomplemented
33#22548646C2212yescomplemented
34#22548639C2212yescomplemented
35#22548639C2212yescomplemented
36(C2×D251)⋊C8C2212yes—
37(C2×D251)⋊C8C2212yes—
38C22×Q8×C251C2211yescharacteristic
39C2×M16×C251C2211yescharacteristic
40#117×C251C2211yescharacteristic
41#103×C251C2212yescomplemented
42#103×C251C2212yescomplemented
43C23×D251D411yescharacteristic
44C22×Dic251—21no—
45C22×Dic251D411yescharacteristic
46C2×C4×D251—21no—
47C2×C4×D251—21no—
48C2×C4×D251—22no—
49C2×C4×D251—22no—
50C2×C4×D251D411yescharacteristic
51C2×C4×D251D411yescharacteristic
52C2×C4×D251C2311yescharacteristic
53C2×C4×D251C2311yescharacteristic
54#16860794—22no—
55C2×C251⋊C8C2312yes—
56C4⋊Dic251—21no—
57C4⋊Dic251C2311yescharacteristic
58C4⋊Dic251C2311yescharacteristic
59C4⋊Dic251C2312yes—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62Dic251⋊C4—22no—
63Dic251⋊C4—22no—
64C22⋊Dic251—21no—
65(C2×D251)⋊C4—21no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—22no—
69C4×Dic251—21no—
70C4×Dic251—21no—
71C4×Dic251—21no—
72C2×Q8×C251—21no—
73C2×Q8×C251—22no—
74C2×Q8×C251—22no—
75C2×Q8×C251C2312yes—
76C22×C4×C251—21no—
77C22×C4×C251C2311yescharacteristic
78M16×C251—22no—
79C2×C8×C251C2312yes—
80C4⋊C4×C251C2311yescharacteristic
81C4⋊C4×C251C2311yescharacteristic
82C22⋊C4×C251—21no—
83C42×C251—21no—
84C22×D251—22no—
85C22×D251—22no—
86C22×D251—41no—
87C22×D251C2×D411yescharacteristic
88C22×D251C2×D411yescharacteristic
89C2×Dic251—22no—
90C2×Dic251—21no—
91C2×Dic251—21no—
92C2×Dic251—22no—
93C2×Dic251—42no—
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C4×D251—22no—
97C4×D251—22no—
98C4×D251—21no—
99C4×D251—21no—
100C4×D251—21no—
101C4×D251—21no—
102C4×D251—42no—
103C4×D251—42no—
104C251⋊C8—22no—
105C23×C251C2×D411yescharacteristic
106Q8×C251—22no—
107Q8×C251—22no—
108Q8×C251—42no—
109C2×C4×C251—21no—
110C2×C4×C251—21no—
111C2×C4×C251—22no—
112C2×C4×C251—42no—
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251#4112yes—
115C2×C4×C251#4112yes—
116C2×C4×C251C2411yescharacteristic
117C8×C251—22no—
118C22×C251—21no—
119C22×C251—21no—
120C22×C251#12312yes—
121C22×C251C22×D411yescharacteristic
122C2×D251—21no—
123C2×D251—21no—
124C2×D251—21no—
125C2×D251—21no—
126C2×D251—21no—
127C2×D251—44no—
128C2×D251#13712yes—
129C4×C251—42no—
130C4×C251—41no—
131C4×C251—41no—
132C4×C251#12312yes—
133C4×C251C2×#4111yescharacteristic
134C4×C251C2×#4111yescharacteristic
135Dic251—21no—
136Dic251—21no—
137Dic251—41no—
138Dic251—41no—
139Dic251—42no—
140C2×C251—22no—
141C2×C251C2×#12311yescharacteristic
142C2×C251C2×#13711yescharacteristic
143C2×C251#57511yescharacteristic
144D251—22no—
145D251—41no—
146C251#309511yescharacteristic, complemented, Sylow
147#3095—2511nomaximal, Sylow
148C2×C22⋊Q8—2511no—
149C2×#117—2511no—
150#485—2512no—
151#485—2512no—
152#484—2512no—
153#483—2512no—
154#428—2511no—
155#428—2511no—
156#419—2511no—
157#419D25111yescharacteristic, complemented
158#407—2511no—
159C22×Q8C2×D25111yescharacteristic
160C23×C4—2511no—
161C2×M16—2511no—
162C2×M16C2×D25111yescharacteristic
163C22⋊Q8—5021no—
164C22⋊Q8—5021no—
165C22⋊Q8—2512no—
166C22⋊Q8—2512no—
167#117—5021no—
168#117—5021no—
169#117—2511no—
170#117—2511no—
171#117—2511no—
172#117C2×D25111yescharacteristic
173C2×C4⋊C4—5021no—
174C2×C4⋊C4—2511no—
175C2×C4⋊C4—2511no—
176C2×C4⋊C4—2511no—
177C2×C22⋊C4—5021no—
178C2×C22⋊C4—5021no—
179C2×C42—5021no—
180#107—2512no—
181#107—2512no—
182#106—2512no—
183#106—2512no—
184#103—2512no—
185#103—2512no—
186#103C2×D25112yescomplemented
187#103C2×D25112yescomplemented
188C22⋊C8—2512no—
189C22⋊C8—2512no—
190C24—2511no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D25112yes—
195C22×C4—21no—
196C22×C4—5021no—
197C22×C4—5021no—
198C22×C4—5021no—
199C22×C4—5022no—
200C22×C4—5022no—
201C22×C4—2511no—
202C22×C4—2511no—
203C22×C4—2511no—
204C22×C4—2511no—
205C22×C4—2511no—
206C22×C4C22×D25111yescharacteristic
207M16—22no—
208M16—5022no—
209C2×C8—2512no—
210C2×C8C22×D25112yes—
211C4⋊C4—5022no—
212C4⋊C4—5022no—
213C4⋊C4—5021no—
214C4⋊C4—5021no—
215C4⋊C4—5021no—
216C4⋊C4—2511no—
217C4⋊C4—2511no—
218C4⋊C4—2512no—
219C4⋊C4C22×D25111yescharacteristic
220C4⋊C4C22×D25111yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—5021no—
223C22⋊C4—5021no—
224C22⋊C4—5022no—
225C22⋊C4—5022no—
226C22⋊C4—5021no—
227C42—21no—
228C42—5021no—
229C42—5021no—
230C42—5021no—
231C23—10041no—
232C23—5022no—
233C23—5022no—
234C23—2511no—
235C23—2511no—
236C23D4×D25111yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—10042no—
245C2×C4—10042no—
246C2×C4—10042no—
247C2×C4—5021no—
248C2×C4—5021no—
249C2×C4—5022no—
250C2×C4—5022no—
251C2×C4—5021no—
252C2×C4—5021no—
253C2×C4—5021no—
254C2×C4—5021no—
255C2×C4—5022no—
256C2×C4—5022no—
257C2×C4—2511no—
258C2×C4—2511no—
259C2×C4D4×D25111yescharacteristic
260C2×C4#1686080912yes—
261C2×C4#1686081012yes—
262C2×C4C23×D25111yescharacteristic
263C8—22no—
264C8—5022no—
265C22—21no—
266C22—21no—
267C22—10044no—
268C22—5021no—
269C22—5021no—
270C22—5021no—
271C22—5021no—
272C22—5021no—
273C22—2512no—
274C22#2254869712yes—
275C22C2×D4×D25111yescharacteristic
276C4—41no—
277C4—41no—
278C4—42no—
279C4—10041no—
280C4—10041no—
281C4—10042no—
282C4—5021no—
283C4—5021no—
284C4#2254869712yes—
285C4C2×#1686080911yescharacteristic
286C4C2×#1686081011yescharacteristic
287C2—22no—
288C2—10041no—
289C2—5022no—
290C2C2×#2254869711yescharacteristic
291C2#137×D25111yescharacteristic
292C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#419×C251C2
socleC22×C251C22×D4
prime radicalC23×D251D4
hypercenter#419D251
hypocenterC251#3095
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #30632027C2semidirect
4. #30632026C2semidirect
5. #30632021C2semidirect
6. #30632016C2semidirect
7. #30631730C2semidirect
8. #30631729C2semidirect
9. #30631696C2semidirect
11. #419×C251C2semidirect
30. #22548649C22semidirect
31. #22548648C22semidirect
32. #22548647C22semidirect
33. #22548646C22semidirect
34. #22548639C22semidirect
35. #22548639C22semidirect
41. #103×C251C22semidirect
42. #103×C251C22semidirect
146. C251#3095semidirect
157. #419D251semidirect
186. #103C2×D251semidirect
187. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486912
#117×D2512
#306320272
#306320262
#306320212
#306320162
#306317302
#306317292
#306316962
#306316502
#419×C2512
#3095251