Finite Group Databasebeta

← #44560461

Subgroups — #44560461

Order 32128.

444 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 119 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#30632239C211yesmaximal, characteristic, complemented
2C2×#22548677C211yesmaximal, characteristic
3#30632022C211yesmaximal, characteristic, complemented
4#30631966C212yesmaximal, complemented
5#30631891C212yesmaximal, complemented
6#30631840C211yesmaximal, characteristic
7#30631837C211yesmaximal, characteristic
8#30631790C212yesmaximal, complemented
9#30631786C212yesmaximal, complemented
10#30631741C211yesmaximal, characteristic, complemented
11#484×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C4×D251—21no—
14C2×#16860799C2212yes—
15C2×#16860797C2212yes—
16#22548714C2211yescharacteristic, complemented
17#22548713C2211yescharacteristic, complemented
18#22548705—21no—
19#22548694C2211yescharacteristic
20#22548692—21no—
21Q8×Dic251C2211yescharacteristic, complemented
22#22548680—21no—
23#22548680C2211yescharacteristic, complemented
24#22548677—21no—
25#22548677—21no—
26#22548677C2212yes—
27#22548677C2212yes—
28#22548671—21no—
29C2×(C2×D251)⋊C4—21no—
30C2×C4×Dic251C2211yescharacteristic
31#22548648C2212yescomplemented
32Dic251⋊C8C2212yescomplemented
33#22548638C2212yescomplemented
34#22548637C2212yes—
35#22548635C2212yes—
36C22⋊(C251⋊C8)C2211yescharacteristic
37#22548625C2212yescomplemented
38C22⋊Q8×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic
40#107×C251C2212yescomplemented
41#103×C251C2212yescomplemented
42C22⋊C8×C251C2211yescharacteristic
43C23×D251—21no—
44C2×Dic502—21no—
45C2×Dic502C2311yescharacteristic
46C2×D1004—24no—
47C2×D1004—21no—
48C2×D1004C2312yes—
49C22×Dic251D411yescharacteristic
50C22×Dic251D411yescharacteristic
51C2×C4×D251—22no—
52C2×C4×D251—22no—
53C2×C4×D251—44no—
54#16860799—24no—
55#16860797—24no—
56C2×C251⋊C8C2312yes—
57C4⋊Dic251—21no—
58C4⋊Dic251C2311yescharacteristic
59Dic251⋊C4—21no—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62Dic251⋊C4—21no—
63Dic251⋊C4—21no—
64C22⋊Dic251—21no—
65(C2×D251)⋊C4—22no—
66(C2×D251)⋊C4—22no—
67C4×Dic251—21no—
68C4×Dic251—21no—
69C4×Dic251D411yescharacteristic
70C4×Dic251D411yescharacteristic
71C4×Dic251C2311yescharacteristic
72C4×Dic251C2311yescharacteristic
73C2×Q8×C251C2311yescharacteristic
74C22×C4×C251—21no—
75C22×C4×C251C2311yescharacteristic
76C2×C8×C251C2312yes—
77C4⋊C4×C251—21no—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2312yes—
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C22×D251—24no—
83C22×D251—22no—
84C22×D251—42no—
85C22×D251—42no—
86Dic502—22no—
87Dic502—42no—
88D1004—24no—
89D1004—22no—
90D1004#4114yes—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251C2×D411yescharacteristic
96C2×Dic251C2×D411yescharacteristic
97C2×Dic251C2×D411yescharacteristic
98C2×Dic251C2×D411yescharacteristic
99C4×D251—44no—
100C4×D251—44no—
101C251⋊C8—22no—
102C23×C251C2×D411yescharacteristic
103Q8×C251—22no—
104C2×C4×C251—22no—
105C2×C4×C251—21no—
106C2×C4×C251—21no—
107C2×C4×C251—22no—
108C2×C4×C251—42no—
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2411yescharacteristic
111C8×C251—22no—
112C22×C251—21no—
113C22×C251—21no—
114C22×C251#13712yes—
115C22×C251C22×D411yescharacteristic
116C2×D251—24no—
117C2×D251—48no—
118C2×D251—42no—
119C2×D251—42no—
120C4×C251—21no—
121C4×C251—41no—
122C4×C251—42no—
123C4×C251—41no—
124C4×C251C2×#4112yes—
125Dic251—21no—
126Dic251—21no—
127Dic251—21no—
128Dic251—41no—
129Dic251—41no—
130Dic251AGL(1,ℤ/8)12yes—
131C2×C251—22no—
132C2×C251#52511yescharacteristic
133C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
134C2×C251C2×#13711yescharacteristic
135D251—44no—
136C251#311011yescharacteristic, complemented, Sylow
137#3110—2511nomaximal, Sylow
138#533—2511no—
139C2×C4×D4—2511no—
140#484—2511no—
141#484D25111yescharacteristic, complemented
142#474—2512no—
143#461—2512no—
144#450—2511no—
145#450—2511no—
146#440—2512no—
147#440—2512no—
148#432—2511no—
149C22×D4—2511no—
150C23×C4—5021no—
151C2×SD16—2512no—
152C2×SD16—2512no—
153C4⋊Q8—2511no—
154C4⋊Q8—2511no—
155#125—5021no—
156C22⋊Q8—5021no—
157C22⋊Q8—2511no—
158C22⋊Q8C2×D25111yescharacteristic
159C4×Q8—5021no—
160C4×Q8—2511no—
161C4×Q8—2511no—
162C4×D4—5021no—
163C4×D4—5021no—
164C4×D4—2512no—
165C4×D4—2512no—
166#117—5021no—
167C2×C4⋊C4C2×D25111yescharacteristic
168C2×C22⋊C4—5021no—
169C2×C42—2511no—
170#107—2512no—
171#107C2×D25112yescomplemented
172C4⋊C8—2512no—
173#103—2512no—
174#103C2×D25112yescomplemented
175#102—2512no—
176#102—2512no—
177C22⋊C8—2511no—
178C22⋊C8C2×D25111yescharacteristic
179#97—2512no—
180C24—5021no—
181C2×Q8—5021no—
182C2×Q8—2511no—
183C2×Q8C22×D25111yescharacteristic
184C2×D4—5024no—
185C2×D4—5021no—
186C2×D4—2512no—
187C22×C4—21no—
188C22×C4—10044no—
189C22×C4—5022no—
190C22×C4—5022no—
191C22×C4—2511no—
192C22×C4—2511no—
193C22×C4C22×D25111yescharacteristic
194SD16—5024no—
195SD16—5024no—
196C2×C8—2512no—
197C2×C8C22×D25112yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—5021no—
201C4⋊C4—5021no—
202C4⋊C4—5021no—
203C4⋊C4—5021no—
204C4⋊C4—5021no—
205C4⋊C4—5021no—
206C4⋊C4—2511no—
207C4⋊C4C22×D25111yescharacteristic
208C4⋊C4C22×D25112yes—
209C22⋊C4—21no—
210C22⋊C4—5022no—
211C22⋊C4—5022no—
212C22⋊C4—5021no—
213C42—5021no—
214C42—5021no—
215C42—2511no—
216C42—2511no—
217C42—2511no—
218C42—2511no—
219C23—10042no—
220C23—10042no—
221C23—5024no—
222C23—5022no—
223C23D4×D25111yescharacteristic
224Q8—22no—
225Q8—10042no—
226Q8—5022no—
227D4—5024no—
228D4—5022no—
229D4—2514no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—10044no—
236C2×C4—10044no—
237C2×C4—5021no—
238C2×C4—5021no—
239C2×C4—5022no—
240C2×C4—5022no—
241C2×C4—2511no—
242C2×C4—2511no—
243C2×C4—2511no—
244C2×C4—2511no—
245C2×C4D4×D25111yescharacteristic
246C2×C4C23×D25111yescharacteristic
247C8—22no—
248C8—5022no—
249C22—21no—
250C22—21no—
251C22—10048no—
252C22—10042no—
253C22—10042no—
254C22—5024no—
255C22#2254874712yes—
256C22C2×D4×D25111yescharacteristic
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—10041no—
262C4—10041no—
263C4—5021no—
264C4—5021no—
265C4—5021no—
266C4—2512no—
267C4#41×D25112yes—
268C2—22no—
269C2—10044no—
270C2#3063219511yescharacteristic
271C2AGL(1,ℤ/8)×D25111yescharacteristic
272C2C2×#2254874711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#484×C251C2
socleC22×C251C22×D4
prime radicalC22×D1004C22
hypercenter#484D251
hypocenterC251#3110
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #30632239C2semidirect
3. #30632022C2semidirect
4. #30631966C2semidirect
5. #30631891C2semidirect
8. #30631790C2semidirect
9. #30631786C2semidirect
10. #30631741C2semidirect
11. #484×C251C2semidirect
16. #22548714C22semidirect
17. #22548713C22semidirect
21. Q8×Dic251C22semidirect
23. #22548680C22semidirect
31. #22548648C22semidirect
32. Dic251⋊C8C22semidirect
33. #22548638C22semidirect
37. #22548625C22semidirect
40. #107×C251C22semidirect
41. #103×C251C22semidirect
136. C251#3110semidirect
141. #484D251semidirect
171. #107C2×D251semidirect
174. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322392
C2×#225486772
#306320222
#306319662
#306318912
#306318402
#306318372
#306317902
#306317862
#306317412
#484×C2512
#3110251