Finite Group Databasebeta

← #44560299

Subgroups — #44560299

Order 32128.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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#30632199C211yesmaximal, characteristic
2C2×#22548673C211yesmaximal, characteristic, complemented
3#30632031C212yesmaximal, complemented
4#30632030C212yesmaximal, complemented
5#30632014C212yesmaximal, complemented
6#30632013C212yesmaximal, complemented
7#30631730C211yesmaximal, characteristic, complemented
8#30631689C211yesmaximal, characteristic, complemented
9#30631686C211yesmaximal, characteristic, complemented
10#30631650C211yesmaximal, characteristic, complemented
11#428×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×C4×D251C2211yescharacteristic
14C2×#16860794C2211yescharacteristic
15#22548699—21no—
16#22548695C2211yescharacteristic
17#22548695C2211yescharacteristic
18#22548690C2211yescharacteristic
19#22548690C2211yescharacteristic
20#22548685—21no—
21C4×D1004—21no—
22C4×D1004—21no—
23#22548673—21no—
24#22548673—21no—
25#22548673C2212yes—
26#22548673C2212yes—
27C2×C4⋊Dic251C2211yescharacteristic
28C2×(C2×D251)⋊C4—21no—
29C2×C4×Dic251—21no—
30#22548648C2212yescomplemented
31#22548646C2212yescomplemented
32#22548640C2212yescomplemented
33#22548638C2212yescomplemented
34#22548637C2212yescomplemented
35#22548635C2212yescomplemented
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
44#16860810—22no—
45#16860810—22no—
46C2×Dic502C2311yescharacteristic
47C2×D1004—21no—
48C2×D1004—21no—
49C2×D1004C2311yescharacteristic
50C2×C22⋊D251—21no—
51C22×Dic251—21no—
52C22×Dic251D411yescharacteristic
53C2×C4×D251—21no—
54C2×C4×D251—22no—
55C2×C4×D251—22no—
56C2×C4×D251D411yescharacteristic
57C2×C4×D251D411yescharacteristic
58C2×C4×D251C2311yescharacteristic
59C2×C4×D251C2311yescharacteristic
60#16860794—22no—
61C2×C251⋊C8C2312yes—
62C4⋊Dic251—21no—
63C4⋊Dic251—21no—
64C4⋊Dic251—21no—
65C4⋊Dic251C2312yes—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—21no—
69(C2×D251)⋊C4—21no—
70(C2×D251)⋊C4—21no—
71(C2×D251)⋊C4—21no—
72C4×Dic251—22no—
73C4×Dic251—22no—
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—21no—
86C22×D251—41no—
87C22×D251—41no—
88C22×D251C2×D411yescharacteristic
89C22×D251C2×D411yescharacteristic
90Dic502—21no—
91Dic502—21no—
92D1004—21no—
93D1004—21no—
94D1004—42no—
95D1004—42no—
96C22⋊D251—42no—
97C2×Dic251—22no—
98C2×Dic251—22no—
99C2×Dic251—21no—
100C2×Dic251—42no—
101C2×Dic251C2×D411yescharacteristic
102C2×Dic251C2×D411yescharacteristic
103C4×D251—22no—
104C4×D251—22no—
105C4×D251—21no—
106C4×D251—21no—
107C4×D251—21no—
108C4×D251—21no—
109C4×D251—42no—
110C251⋊C8—22no—
111C23×C251C2×D411yescharacteristic
112C2×C4×C251—21no—
113C2×C4×C251—21no—
114C2×C4×C251—22no—
115C2×C4×C251—42no—
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251#4112yes—
118C2×C4×C251#4112yes—
119C2×C4×C251C2411yescharacteristic
120C8×C251—22no—
121C22×C251—21no—
122C22×C251—21no—
123C22×C251#12312yes—
124C22×C251C22×D411yescharacteristic
125C2×D251—21no—
126C2×D251—21no—
127C2×D251—21no—
128C2×D251—21no—
129C2×D251—21no—
130C2×D251—21no—
131C2×D251—44no—
132C2×D251—41no—
133C2×D251—41no—
134C2×D251—41no—
135C4×C251—42no—
136C4×C251—41no—
137C4×C251—41no—
138C4×C251#12312yes—
139C4×C251C2×#4111yescharacteristic
140C4×C251C2×#4111yescharacteristic
141Dic251—21no—
142Dic251—21no—
143Dic251—41no—
144Dic251—42no—
145C2×C251—22no—
146C2×C251C2×#12311yescharacteristic
147C2×C251#57511yescharacteristic
148C2×C251#57511yescharacteristic
149D251—41no—
150D251—41no—
151D251—81no—
152C251#309311yescharacteristic, complemented, Sylow
153#3093—2511nomaximal, Sylow
154#525—2511no—
155C2×#117—2511no—
156#485—2512no—
157#485—2512no—
158#482—2512no—
159#482—2512no—
160#428—2511no—
161#428D25111yescharacteristic, complemented
162#417—2511no—
163#417—2511no—
164#407—2511no—
165C2×#41—2511no—
166C23×C4—2511no—
167C2×M16—2511no—
168C2×M16C2×D25111yescharacteristic
169#123—5021no—
170C22⋊Q8—2511no—
171C22⋊Q8—2511no—
172#121—2511no—
173#121—2511no—
174#120—5021no—
175C4×D4—5021no—
176C4×D4—5021no—
177#117—5021no—
178#117—5021no—
179#117—2512no—
180#117—2512no—
181#117C2×D25111yescharacteristic
182C2×C4⋊C4—2511no—
183C2×C4⋊C4C2×D25111yescharacteristic
184C2×C22⋊C4—5021no—
185C2×C42—5021no—
186#107—2512no—
187#107C2×D25112yescomplemented
188#106—2512no—
189#106C2×D25112yescomplemented
190#103—2512no—
191#103—2512no—
192#102—2512no—
193#102—2512no—
194C22⋊C8—2512no—
195C22⋊C8—2512no—
196C24—2511no—
197#41—5022no—
198#41—5022no—
199C2×Q8—2511no—
200C2×D4—5021no—
201C2×D4—5021no—
202C2×D4—5021no—
203C2×D4—2511no—
204C22×C4—21no—
205C22×C4—5021no—
206C22×C4—5021no—
207C22×C4—5022no—
208C22×C4—5022no—
209C22×C4—2511no—
210C22×C4—2511no—
211C22×C4—2511no—
212C22×C4—2511no—
213C22×C4—2511no—
214C22×C4C22×D25111yescharacteristic
215M16—22no—
216M16—5022no—
217C2×C8—2512no—
218C2×C8C22×D25112yes—
219C4⋊C4—21no—
220C4⋊C4—5021no—
221C4⋊C4—5021no—
222C4⋊C4—5021no—
223C4⋊C4—2512no—
224C4⋊C4C22×D25112yes—
225C4⋊C4C22×D25111yescharacteristic
226C4⋊C4C22×D25111yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—5022no—
229C22⋊C4—5022no—
230C22⋊C4—5021no—
231C22⋊C4—5021no—
232C22⋊C4—5021no—
233C22⋊C4—5021no—
234C42—21no—
235C42—5022no—
236C42—5022no—
237C23—10041no—
238C23—10041no—
239C23—5021no—
240C23—5022no—
241C23—2511no—
242C23—2511no—
243C23D4×D25111yescharacteristic
244Q8—5021no—
245Q8—5021no—
246D4—10042no—
247D4—10042no—
248D4—10042no—
249D4—5021no—
250D4—5021no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—10042no—
256C2×C4—10042no—
257C2×C4—5022no—
258C2×C4—5021no—
259C2×C4—5022no—
260C2×C4—5021no—
261C2×C4—5021no—
262C2×C4—5021no—
263C2×C4—5021no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—2511no—
267C2×C4—2511no—
268C2×C4D4×D25111yescharacteristic
269C2×C4#1686081012yes—
270C2×C4#1686081112yes—
271C2×C4C23×D25111yescharacteristic
272C8—22no—
273C8—5022no—
274C22—21no—
275C22—21no—
276C22—10044no—
277C22—10041no—
278C22—10041no—
279C22—10041no—
280C22—5021no—
281C22—5021no—
282C22—5021no—
283C22—5021no—
284C22—5021no—
285C22—5021no—
286C22#2254870012yes—
287C22C2×D4×D25111yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—10041no—
292C4—10042no—
293C4—5021no—
294C4—5021no—
295C4#2254870012yes—
296C4C2×#1686081011yescharacteristic
297C4C2×#1686081111yescharacteristic
298C2—22no—
299C2—20081no—
300C2—10041no—
301C2—10041no—
302C2C2×#2254870011yescharacteristic
303C2#3063248311yescharacteristic
304C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
prime radical#30632199C2
hypercenter#428D251
hypocenterC251#3093
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
2. C2×#22548673C2semidirect
3. #30632031C2semidirect
4. #30632030C2semidirect
5. #30632014C2semidirect
6. #30632013C2semidirect
7. #30631730C2semidirect
8. #30631689C2semidirect
9. #30631686C2semidirect
10. #30631650C2semidirect
11. #428×C251C2semidirect
30. #22548648C22semidirect
31. #22548646C22semidirect
32. #22548640C22semidirect
33. #22548638C22semidirect
34. #22548637C22semidirect
35. #22548635C22semidirect
41. #107×C251C22semidirect
42. #106×C251C22semidirect
152. C251#3093semidirect
161. #428D251semidirect
187. #107C2×D251semidirect
189. #106C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306321992
C2×#225486732
#306320312
#306320302
#306320142
#306320132
#306317302
#306316892
#306316862
#306316502
#428×C2512
#3093251