Finite Group Databasebeta

← #44560281

Subgroups — #44560281

Order 32128.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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#30632200C211yesmaximal, characteristic
2C2×#22548673C211yesmaximal, characteristic, complemented
3#30632027C212yesmaximal, complemented
4#30632015C211yesmaximal, characteristic, complemented
5#30632014C212yesmaximal, complemented
6#30632005C211yesmaximal, characteristic, complemented
7#30632001C211yesmaximal, characteristic, complemented
8#30631724C212yesmaximal, complemented
9#30631682C212yesmaximal, complemented
10C22⋊C8×D251C211yesmaximal, characteristic, complemented
11#484×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2211yescharacteristic
13C22×C4×D251C2211yescharacteristic
14C2×C8×D251C2212yes—
15#22548700—21no—
16#22548691C2211yescharacteristic
17#22548690C2211yescharacteristic
18#22548689C2211yescharacteristic
19#22548685—21no—
20#22548677—21no—
21(C22⋊D251)⋊C4—21no—
22#22548673—21no—
23#22548673—21no—
24#22548673C2212yes—
25#22548673C2211yescharacteristic
26#22548673C2212yes—
27C2×C4⋊Dic251C2211yescharacteristic
28C2×(C2×D251)⋊C4—21no—
29C2×C4×Dic251—21no—
30#22548649C2212yescomplemented
31#22548646C2212yescomplemented
32#22548646C2212yescomplemented
33#22548639C2212yescomplemented
34#22548637C2212yescomplemented
35#22548635C2212yescomplemented
36C22⋊(C251⋊C8)C2211yescharacteristic
37(C2×D251)⋊C8C2212yes—
38C22⋊Q8×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic
40#107×C251C2212yescomplemented
41#103×C251C2212yescomplemented
42C22⋊C8×C251C2211yescharacteristic
43C23×D251D411yescharacteristic
44#16860811—24no—
45C2×D1004—21no—
46C2×D1004—21no—
47C2×D1004C2311yescharacteristic
48C2×C22⋊D251—21no—
49C22×Dic251—21no—
50C22×Dic251D411yescharacteristic
51C2×C4×D251—24no—
52C2×C4×D251—21no—
53C2×C4×D251D411yescharacteristic
54C2×C4×D251D411yescharacteristic
55C2×C4×D251C2311yescharacteristic
56C2×C4×D251C2311yescharacteristic
57C2×C251⋊C8C2312yes—
58C8×D251—24no—
59C4⋊Dic251—21no—
60C4⋊Dic251C2312yes—
61C4⋊Dic251C2311yescharacteristic
62Dic251⋊C4—21no—
63(C2×D251)⋊C4—22no—
64(C2×D251)⋊C4—22no—
65(C2×D251)⋊C4—21no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4—21no—
68(C2×D251)⋊C4—21no—
69C4×Dic251—22no—
70C4×Dic251—22no—
71C4×Dic251—21no—
72C2×Q8×C251C2311yescharacteristic
73C22×C4×C251—21no—
74C22×C4×C251C2311yescharacteristic
75C2×C8×C251C2312yes—
76C4⋊C4×C251—21no—
77C4⋊C4×C251—21no—
78C4⋊C4×C251C2312yes—
79C4⋊C4×C251C2311yescharacteristic
80C22⋊C4×C251—21no—
81C22×D251—22no—
82C22×D251—22no—
83C22×D251—21no—
84C22×D251—41no—
85C22×D251C2×D411yescharacteristic
86C22×D251C2×D411yescharacteristic
87D1004—22no—
88D1004—42no—
89D1004—42no—
90C22⋊D251—42no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251—21no—
94C2×Dic251—42no—
95C2×Dic251C2×D411yescharacteristic
96C2×Dic251C2×D411yescharacteristic
97C4×D251—22no—
98C4×D251—22no—
99C4×D251—22no—
100C4×D251—42no—
101C4×D251#4114yes—
102C251⋊C8—22no—
103C23×C251C2×D411yescharacteristic
104Q8×C251—22no—
105C2×C4×C251—22no—
106C2×C4×C251—21no—
107C2×C4×C251—21no—
108C2×C4×C251—22no—
109C2×C4×C251—42no—
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2411yescharacteristic
112C8×C251—22no—
113C22×C251—21no—
114C22×C251—21no—
115C22×C251#13512yes—
116C22×C251C22×D411yescharacteristic
117C2×D251—21no—
118C2×D251—21no—
119C2×D251—21no—
120C2×D251—21no—
121C2×D251—21no—
122C2×D251—44no—
123C2×D251—41no—
124C2×D251—41no—
125C2×D251—41no—
126C2×D251#12312yes—
127C4×C251—21no—
128C4×C251—41no—
129C4×C251—42no—
130C4×C251—41no—
131C4×C251C2×#4112yes—
132Dic251—21no—
133Dic251—42no—
134Dic251—41no—
135Dic251#12312yes—
136C2×C251—22no—
137C2×C251C2×#12311yescharacteristic
138C2×C251C2×#13511yescharacteristic
139C2×C251#57511yescharacteristic
140D251—22no—
141D251—41no—
142D251—81no—
143C251#309111yescharacteristic, complemented, Sylow
144#3091—2511nomaximal, Sylow
145#525—2511no—
146C2×#117—2511no—
147#485—2512no—
148#484D25111yescharacteristic, complemented
149#483—2511no—
150#482—2512no—
151#481—2511no—
152#480—2511no—
153#427—2512no—
154#416—2512no—
155C2×C22⋊C8—2511no—
156C2×#41—2511no—
157C23×C4—2511no—
158C22×C8—2512no—
159#123—5021no—
160C22⋊Q8—2511no—
161C22⋊Q8C2×D25111yescharacteristic
162#121—2511no—
163#121—2511no—
164#120—5021no—
165C4×D4—5021no—
166C4×D4—5021no—
167#117—5021no—
168#117—5021no—
169#117—2512no—
170#117—2512no—
171#117—2511no—
172C2×C4⋊C4—2511no—
173C2×C4⋊C4C2×D25111yescharacteristic
174C2×C22⋊C4—5021no—
175C2×C42—5021no—
176#107—2512no—
177#107C2×D25112yescomplemented
178#106—2512no—
179#106—2512no—
180#103—2512no—
181#103C2×D25112yescomplemented
182#102—2512no—
183#102—2512no—
184C22⋊C8—2511no—
185C22⋊C8—2512no—
186C22⋊C8C2×D25111yescharacteristic
187C24—2511no—
188#41—5024no—
189C2×Q8C22×D25111yescharacteristic
190C2×D4—5021no—
191C2×D4—5021no—
192C2×D4—5021no—
193C2×D4—2511no—
194C22×C4—21no—
195C22×C4—5021no—
196C22×C4—5024no—
197C22×C4—5021no—
198C22×C4—2511no—
199C22×C4—2511no—
200C22×C4—2511no—
201C22×C4—2511no—
202C22×C4—2511no—
203C22×C4C22×D25111yescharacteristic
204C2×C8—5024no—
205C2×C8—2512no—
206C2×C8C22×D25112yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—5021no—
210C4⋊C4—5021no—
211C4⋊C4—2512no—
212C4⋊C4—2511no—
213C4⋊C4C22×D25112yes—
214C4⋊C4C22×D25111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—5022no—
217C22⋊C4—5022no—
218C22⋊C4—5021no—
219C22⋊C4—5021no—
220C22⋊C4—5021no—
221C22⋊C4—5021no—
222C42—5022no—
223C42—5022no—
224C42—5021no—
225C23—10041no—
226C23—5022no—
227C23—5021no—
228C23—5022no—
229C23—2511no—
230C23—2511no—
231C23D4×D25111yescharacteristic
232Q8—22no—
233D4—10042no—
234D4—10042no—
235D4—10042no—
236D4—5022no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—10042no—
243C2×C4—10042no—
244C2×C4—5022no—
245C2×C4—5022no—
246C2×C4—5022no—
247C2×C4—5021no—
248C2×C4—5022no—
249C2×C4—5022no—
250C2×C4—2511no—
251C2×C4—2511no—
252C2×C4—2514no—
253C2×C4D4×D25111yescharacteristic
254C2×C4C23×D25111yescharacteristic
255C8—22no—
256C8—5022no—
257C22—21no—
258C22—21no—
259C22—10044no—
260C22—10041no—
261C22—10041no—
262C22—10041no—
263C22—5021no—
264C22—5021no—
265C22—5021no—
266C22—5021no—
267C22—5021no—
268C22—2512no—
269C22#2254873812yes—
270C22C2×D4×D25111yescharacteristic
271C4—21no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—10042no—
276C4—10041no—
277C4—5021no—
278C4—2512no—
279C4#41×D25112yes—
280C2—22no—
281C2—20081no—
282C2—10041no—
283C2—5022no—
284C2#123×D25111yescharacteristic
285C2C2×#2254873811yescharacteristic
286C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#484×C251C2
socleC22×C251C22×D4
prime radical#30632200C2
hypercenter#484D251
hypocenterC251#3091
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
2. C2×#22548673C2semidirect
3. #30632027C2semidirect
4. #30632015C2semidirect
5. #30632014C2semidirect
6. #30632005C2semidirect
7. #30632001C2semidirect
8. #30631724C2semidirect
9. #30631682C2semidirect
10. C22⋊C8×D251C2semidirect
11. #484×C251C2semidirect
30. #22548649C22semidirect
31. #22548646C22semidirect
32. #22548646C22semidirect
33. #22548639C22semidirect
34. #22548637C22semidirect
35. #22548635C22semidirect
40. #107×C251C22semidirect
41. #103×C251C22semidirect
143. C251#3091semidirect
148. #484D251semidirect
177. #107C2×D251semidirect
181. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322002
C2×#225486732
#306320272
#306320152
#306320142
#306320052
#306320012
#306317242
#306316822
C22⋊C8×D2512
#484×C2512
#3091251