Finite Group Databasebeta

← #44560315

Subgroups — #44560315

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×#22548695C211yesmaximal, characteristic
2#117×D251C211yesmaximal, characteristic
3#30632031C212yesmaximal, complemented
4#30632030C212yesmaximal, complemented
5#30632024C212yesmaximal, complemented
6#30632019C212yesmaximal, complemented
7#30631727C211yesmaximal, characteristic, complemented
8#30631697C211yesmaximal, characteristic, complemented
9#30631695C211yesmaximal, characteristic, complemented
10#30631650C211yesmaximal, characteristic
11#428×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2211yescharacteristic
13C22×C4×D251C2211yescharacteristic
14C2×#16860794C2211yescharacteristic
15#22548695—21no—
16#22548695—21no—
17#22548695C2212yes—
18#22548695C2212yes—
19#22548674C2211yescharacteristic
20#22548673C2211yescharacteristic
21#22548673C2211yescharacteristic
22#22548672—21no—
23#22548671—21no—
24C2×C4⋊Dic251—21no—
25C4⋊C4×D251C2211yescharacteristic
26C4⋊C4×D251C2211yescharacteristic
27C2×(C2×D251)⋊C4—21no—
28C22⋊C4×D251—21no—
29C42×D251—21no—
30#22548648C2212yescomplemented
31#22548646C2212yescomplemented
32#22548640C2212yescomplemented
33#22548640C2212yescomplemented
34#22548638C2212yescomplemented
35#22548638C2212yescomplemented
36(C2×D251)⋊C8C2212yes—
37(C2×D251)⋊C8C2212yes—
38C2×M16×C251C2211yescharacteristic
39#117×C251C2211yescharacteristic
40C2×C4⋊C4×C251C2211yescharacteristic
41#107×C251C2212yescomplemented
42#106×C251C2212yescomplemented
43C23×D251D411yescharacteristic
44C2×Dic502—21no—
45C2×Dic502—22no—
46C2×Dic502—22no—
47C2×Dic502C2312yes—
48C22×Dic251—21no—
49C22×Dic251D411yescharacteristic
50C2×C4×D251—21no—
51C2×C4×D251—21no—
52C2×C4×D251—22no—
53C2×C4×D251—22no—
54C2×C4×D251D411yescharacteristic
55C2×C4×D251D411yescharacteristic
56C2×C4×D251C2311yescharacteristic
57C2×C4×D251C2311yescharacteristic
58#16860794—22no—
59C2×C251⋊C8C2312yes—
60C4⋊Dic251—22no—
61C4⋊Dic251—22no—
62C4⋊Dic251C2311yescharacteristic
63C4⋊Dic251C2311yescharacteristic
64Dic251⋊C4—21no—
65Dic251⋊C4—21no—
66C22⋊Dic251—21no—
67(C2×D251)⋊C4—21no—
68(C2×D251)⋊C4—21no—
69(C2×D251)⋊C4—22no—
70(C2×D251)⋊C4—22no—
71C4×Dic251—21no—
72C4×Dic251—21no—
73C4×Dic251—21no—
74C22×C4×C251—21no—
75C22×C4×C251C2311yescharacteristic
76M16×C251—22no—
77C2×C8×C251C2312yes—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2311yescharacteristic
80C4⋊C4×C251C2312yes—
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
89Dic502—22no—
90Dic502—22no—
91Dic502—42no—
92C2×Dic251—22no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—22no—
96C2×Dic251—42no—
97C2×Dic251C2×D411yescharacteristic
98C2×Dic251C2×D411yescharacteristic
99C4×D251—22no—
100C4×D251—22no—
101C4×D251—21no—
102C4×D251—21no—
103C4×D251—21no—
104C4×D251—21no—
105C4×D251—42no—
106C4×D251—42no—
107C251⋊C8—22no—
108C23×C251C2×D411yescharacteristic
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—41no—
130C4×C251—42no—
131C4×C251—41no—
132C4×C251#12312yes—
133C4×C251C2×#4111yescharacteristic
134C4×C251C2×#4111yescharacteristic
135Dic251—21no—
136Dic251—21no—
137Dic251—42no—
138Dic251—41no—
139Dic251—41no—
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#428D25111yescharacteristic, complemented
156#419—2511no—
157#419—2511no—
158#407—2511no—
159C22×Q8—2511no—
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⋊C4C2×D25111yescharacteristic
177C2×C22⋊C4—5021no—
178C2×C22⋊C4—5021no—
179C2×C42—5021no—
180#107—2512no—
181#107C2×D25112yescomplemented
182#106—2512no—
183#106C2×D25112yescomplemented
184#103—2512no—
185#103—2512no—
186#103—2512no—
187#103—2512no—
188C22⋊C8—2512no—
189C22⋊C8—2512no—
190C24—2511no—
191C2×Q8—5022no—
192C2×Q8—5022no—
193C2×Q8—5021no—
194C2×Q8—2512no—
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—21no—
212C4⋊C4—5022no—
213C4⋊C4—5022no—
214C4⋊C4—5021no—
215C4⋊C4—5021no—
216C4⋊C4—2511no—
217C4⋊C4—2511no—
218C4⋊C4C22×D25111yescharacteristic
219C4⋊C4C22×D25111yescharacteristic
220C4⋊C4C22×D25112yes—
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—10042no—
238Q8—5022no—
239Q8—5022no—
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#1686081012yes—
261C2×C4#1686081112yes—
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#2254870012yes—
275C22C2×D4×D25111yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—10042no—
280C4—10041no—
281C4—10041no—
282C4—5021no—
283C4—5021no—
284C4#2254870012yes—
285C4C2×#1686081011yescharacteristic
286C4C2×#1686081111yescharacteristic
287C2—22no—
288C2—10041no—
289C2—5022no—
290C2C2×#2254870011yescharacteristic
291C2#137×D25111yescharacteristic
292C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
prime radicalC23×D251D4
hypercenter#428D251
hypocenterC251#3095
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #30632031C2semidirect
4. #30632030C2semidirect
5. #30632024C2semidirect
6. #30632019C2semidirect
7. #30631727C2semidirect
8. #30631697C2semidirect
9. #30631695C2semidirect
11. #428×C251C2semidirect
30. #22548648C22semidirect
31. #22548646C22semidirect
32. #22548640C22semidirect
33. #22548640C22semidirect
34. #22548638C22semidirect
35. #22548638C22semidirect
41. #107×C251C22semidirect
42. #106×C251C22semidirect
146. C251#3095semidirect
155. #428D251semidirect
181. #107C2×D251semidirect
183. #106C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486952
#117×D2512
#306320312
#306320302
#306320242
#306320192
#306317272
#306316972
#306316952
#306316502
#428×C2512
#3095251