Finite Group Databasebeta

← #483×D251 #44560239

Subgroups — #483×D251 #44560239

Order 32128.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22⋊Q8×D251C211yesmaximal, characteristic
2C2×C4⋊C4×D251C211yesmaximal, characteristic
3#30632019C212yesmaximal, complemented
4#30632018C211yesmaximal, characteristic, complemented
5#30632017C211yesmaximal, characteristic, complemented
6#30632016C212yesmaximal, complemented
7#30632015C211yesmaximal, characteristic, complemented
8#106×D251C212yesmaximal, complemented
9#103×D251C212yesmaximal, complemented
10C22⋊C8×D251C211yesmaximal, characteristic
11#483×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251C2211yescharacteristic
13C22×C4×D251—21no—
14C22×C4×D251C2211yescharacteristic
15C2×C8×D251C2212yes—
16#22548695C2211yescharacteristic
17#22548694C2211yescharacteristic
18#22548693—21no—
19#22548692—21no—
20#22548691C2211yescharacteristic
21C2×C4⋊Dic251C2211yescharacteristic
22C2×Dic251⋊C4—21no—
23C4⋊C4×D251—21no—
24C4⋊C4×D251—21no—
25C4⋊C4×D251—21no—
26C4⋊C4×D251C2211yescharacteristic
27C4⋊C4×D251C2212yes—
28C4⋊C4×D251C2212yes—
29C22⋊C4×D251—21no—
30#22548647C2212yescomplemented
31#22548646C2212yescomplemented
32#22548646C2212yescomplemented
33#22548640C2212yescomplemented
34#22548639C2212yescomplemented
35#22548638C2212yescomplemented
36C22⋊(C251⋊C8)C2211yescharacteristic
37(C2×D251)⋊C8C2212yes—
38C22⋊Q8×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic
40#106×C251C2212yescomplemented
41#103×C251C2212yescomplemented
42C22⋊C8×C251C2211yescharacteristic
43C23×D251D411yescharacteristic
44C2×Dic502—21no—
45C2×Dic502C2311yescharacteristic
46Q8×D251—24no—
47C22×Dic251—21no—
48C22×Dic251D411yescharacteristic
49C2×C4×D251—24no—
50C2×C4×D251—22no—
51C2×C4×D251—22no—
52C2×C4×D251—21no—
53C2×C4×D251—21no—
54C2×C4×D251—44no—
55C2×C4×D251D411yescharacteristic
56C2×C4×D251D411yescharacteristic
57C2×C4×D251C2311yescharacteristic
58C2×C4×D251C2311yescharacteristic
59C2×C251⋊C8C2312yes—
60C8×D251—24no—
61C4⋊Dic251—21no—
62C4⋊Dic251—21no—
63C4⋊Dic251C2312yes—
64C4⋊Dic251C2311yescharacteristic
65Dic251⋊C4—22no—
66Dic251⋊C4—22no—
67Dic251⋊C4—21no—
68Dic251⋊C4—21no—
69Dic251⋊C4—21no—
70C22⋊Dic251—21no—
71(C2×D251)⋊C4—21no—
72(C2×D251)⋊C4—21no—
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—22no—
83C22×D251—22no—
84C22×D251SD1614yes—
85C22×D251C2×D411yescharacteristic
86C22×D251C2×D411yescharacteristic
87Dic502—22no—
88Dic502—42no—
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—22no—
99C4×D251—42no—
100C4×D251—44no—
101C4×D251—44no—
102C4×D251—42no—
103C4×D251#4114yes—
104C251⋊C8—22no—
105C23×C251C2×D411yescharacteristic
106Q8×C251—22no—
107C2×C4×C251—22no—
108C2×C4×C251—21no—
109C2×C4×C251—21no—
110C2×C4×C251—22no—
111C2×C4×C251—42no—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2411yescharacteristic
114C8×C251—22no—
115C22×C251—21no—
116C22×C251—21no—
117C22×C251C2×SD1612yes—
118C22×C251C22×D411yescharacteristic
119C2×D251—21no—
120C2×D251—21no—
121C2×D251—28no—
122C2×D251#12312yes—
123C2×D251C2×SD1612yes—
124C2×D251C2×SD1612yes—
125C2×D251#13712yes—
126C4×C251—21no—
127C4×C251—41no—
128C4×C251—42no—
129C4×C251—41no—
130C4×C251C2×#4112yes—
131Dic251—21no—
132Dic251—41no—
133Dic251—42no—
134Dic251—41no—
135Dic251#12312yes—
136C2×C251—22no—
137C2×C251C2×#12311yescharacteristic
138C2×C251C22×SD1611yescharacteristic
139C2×C251C2×#13711yescharacteristic
140D251—22no—
141D251#48314yesdirect product factor
142C251C2×#48311yescharacteristic, complemented, Sylow
143C2×#483—2511nomaximal, Sylow
144C2×C22⋊Q8—2511no—
145C22×C4⋊C4—2511no—
146#483—2511no—
147#483—2511no—
148#483—2512no—
149#483—2512no—
150#483—2511no—
151#483D25111yescharacteristic, direct product factor
152C2×#106—2512no—
153C2×#103—2512no—
154C2×C22⋊C8—2511no—
155C22×Q8—2511no—
156C23×C4—5021no—
157C23×C4—2511no—
158C22×C8—2512no—
159C22⋊Q8—5021no—
160C22⋊Q8—5021no—
161C22⋊Q8—2511no—
162C22⋊Q8—2511no—
163C22⋊Q8—2511no—
164C22⋊Q8C2×D25111yescharacteristic
165C2×C4⋊C4—5021no—
166C2×C4⋊C4—5021no—
167C2×C4⋊C4—5021no—
168C2×C4⋊C4—5021no—
169C2×C4⋊C4—2511no—
170C2×C4⋊C4—2512no—
171C2×C4⋊C4—2512no—
172C2×C4⋊C4—2511no—
173C2×C4⋊C4C2×D25111yescharacteristic
174C2×C22⋊C4—5021no—
175#106—2512no—
176#106—2512no—
177#106—2512no—
178#106C2×D25112yescomplemented
179#103—2512no—
180#103—2512no—
181#103—2512no—
182#103C2×D25112yescomplemented
183C22⋊C8—2511no—
184C22⋊C8—2512no—
185C22⋊C8C2×D25111yescharacteristic
186C24—2511no—
187C2×Q8—5021no—
188C2×Q8—5024no—
189C2×Q8—2511no—
190C2×Q8C22×D25111yescharacteristic
191C22×C4—21no—
192C22×C4—10044no—
193C22×C4—5021no—
194C22×C4—5021no—
195C22×C4—5022no—
196C22×C4—5022no—
197C22×C4—5024no—
198C22×C4—5021no—
199C22×C4—2511no—
200C22×C4—2511no—
201C22×C4—2511no—
202C22×C4—2511no—
203C22×C4—2511no—
204C22×C4C22×D25111yescharacteristic
205C2×C8—5024no—
206C2×C8—2512no—
207C2×C8C22×D25112yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—5022no—
211C4⋊C4—5022no—
212C4⋊C4—5021no—
213C4⋊C4—5021no—
214C4⋊C4—5021no—
215C4⋊C4—5021no—
216C4⋊C4—5021no—
217C4⋊C4—2512no—
218C4⋊C4—2511no—
219C4⋊C4C22×D25112yes—
220C4⋊C4C22×D25111yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—5021no—
223C22⋊C4—5021no—
224C22⋊C4—5021no—
225C23—5022no—
226C23—5022no—
227C23—2511no—
228C23—2514no—
229C23—2511no—
230C23D4×D25111yescharacteristic
231Q8—22no—
232Q8—10042no—
233Q8—5022no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—10044no—
240C2×C4—10044no—
241C2×C4—10042no—
242C2×C4—10042no—
243C2×C4—10042no—
244C2×C4—5022no—
245C2×C4—5022no—
246C2×C4—5021no—
247C2×C4—5021no—
248C2×C4—5022no—
249C2×C4—5022no—
250C2×C4—5022no—
251C2×C4—2511no—
252C2×C4—2511no—
253C2×C4—2514no—
254C2×C4D4×D25111yescharacteristic
255C2×C4C23×D25111yescharacteristic
256C8—22no—
257C8—5022no—
258C22—21no—
259C22—21no—
260C22—5028no—
261C22—5021no—
262C22—5021no—
263C22—2512no—
264C22—2512no—
265C22—2512no—
266C22—2512no—
267C22SD16×D25112yes—
268C22C2×D4×D25111yescharacteristic
269C4—21no—
270C4—42no—
271C4—41no—
272C4—41no—
273C4—10042no—
274C4—10041no—
275C4—10041no—
276C4—5021no—
277C4—2512no—
278C4#41×D25112yes—
279C2—22no—
280C2—5022no—
281C2—2514no—
282C2#123×D25111yescharacteristic
283C2C2×SD16×D25111yescharacteristic
284C2#137×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#483×C251C2
socleC22×C251C22×D4
prime radicalC23×D251D4
hypercenter#483D251
hypocenterC251C2×#483
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #30632019C2semidirect
4. #30632018C2semidirect
5. #30632017C2semidirect
6. #30632016C2semidirect
7. #30632015C2semidirect
8. #106×D251C2semidirect
9. #103×D251C2semidirect
11. #483×C251C2semidirect
30. #22548647C22semidirect
31. #22548646C22semidirect
32. #22548646C22semidirect
33. #22548640C22semidirect
34. #22548639C22semidirect
35. #22548638C22semidirect
40. #106×C251C22semidirect
41. #103×C251C22semidirect
141. D251#483direct (N × H)
142. C251C2×#483semidirect
151. #483D251direct (N × H)
178. #106C2×D251semidirect
182. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2512
C2×C4⋊C4×D2512
#306320192
#306320182
#306320172
#306320162
#306320152
#106×D2512
#103×D2512
C22⋊C8×D2512
#483×C2512
C2×#483251