Finite Group Databasebeta

← #44560300

Subgroups — #44560300

Order 32128.

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

normal subgroups only

This group has 462 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#121×D251C211yesmaximal, characteristic
2C2×#22548673C211yesmaximal, characteristic, complemented
3#30632014C212yesmaximal, complemented
4#30632010C212yesmaximal, complemented
5#30632006C211yesmaximal, characteristic, complemented
6#30632003C211yesmaximal, characteristic, complemented
7#30632000C211yesmaximal, characteristic, complemented
8#30631728C212yesmaximal, complemented
9#30631690C212yesmaximal, complemented
10#30631649C211yesmaximal, characteristic, complemented
11#480×C251C211yesmaximal, characteristic, complemented
12C2×D4×D251—21no—
13C2×D4×D251C2211yescharacteristic
14C22×C4×D251C2211yescharacteristic
15C2×#16860793C2212yes—
16#22548690C2211yescharacteristic
17#22548689C2211yescharacteristic
18#22548688—21no—
19#22548687—21no—
20#22548686C2211yescharacteristic
21#22548673—21no—
22#22548673—21no—
23#22548673C2212yes—
24#22548673C2212yescomplemented
25C2×C4⋊Dic251C2211yescharacteristic, complemented
26C4⋊C4×D251C2211yescharacteristic
27C2×(C2×D251)⋊C4—21no—
28C22⋊C4×D251—21no—
29C2×C4×Dic251—21no—
30#22548648C2212yescomplemented
31#22548647C2212yescomplemented
32#22548646C2212yescomplemented
33#22548637C2212yescomplemented
34#22548636C2212yescomplemented
35#22548635C2212yescomplemented
36C22⋊(C251⋊C8)C2211yescharacteristic, complemented
37(C2×D251)⋊C8C2212yescomplemented
38#121×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic, complemented
40#107×C251C2212yescomplemented
41#102×C251C2212yescomplemented
42C22⋊C8×C251C2211yescharacteristic, complemented
43C23×D251—21no—
44C23×D251D411yescharacteristic
45C2×D1004—21no—
46C2×D1004C2311yescharacteristic
47C2×C22⋊D251—21no—
48C2×C22⋊D251—21no—
49C2×C22⋊D251—21no—
50D4×D251—24no—
51D4×D251—44no—
52C22×Dic251—21no—
53C22×Dic251D411yescharacteristic
54C2×C4×D251—24no—
55C2×C4×D251—21no—
56C2×C4×D251D411yescharacteristic
57C2×C4×D251D411yescharacteristic
58C2×C4×D251C2311yescharacteristic
59C2×C4×D251C2311yescharacteristic
60#16860793—24no—
61C2×C251⋊C8C2312yes—
62C4⋊Dic251—21no—
63C4⋊Dic251C2312yes—
64C4⋊Dic251C2311yescharacteristic
65Dic251⋊C4—21no—
66C22⋊Dic251—21no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4—21no—
70(C2×D251)⋊C4—21no—
71C4×Dic251—22no—
72C4×Dic251—22no—
73C2×D4×C251—21no—
74C2×D4×C251C2311yescharacteristic
75C22×C4×C251—21no—
76C22×C4×C251C2311yescharacteristic
77C2×C8×C251C2312yes—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2312yes—
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C22×D251—22no—
83C22×D251—22no—
84C22×D251—21no—
85C22×D251—41no—
86C22×D251—42no—
87C22×D251—42no—
88C22×D251—42no—
89C22×D251C2×D411yescharacteristic
90C22×D251C2×D411yescharacteristic
91D1004—22no—
92D1004—42no—
93C22⋊D251—42no—
94C22⋊D251—42no—
95C22⋊D251—42no—
96C2×Dic251—22no—
97C2×Dic251—22no—
98C2×Dic251—21no—
99C2×Dic251—42no—
100C2×Dic251C2×D411yescharacteristic
101C2×Dic251C2×D411yescharacteristic
102C4×D251—22no—
103C4×D251—22no—
104C4×D251—22no—
105C4×D251—42no—
106C4×D251#4114yes—
107C251⋊C8—22no—
108C23×C251—21no—
109C23×C251C2×D411yescharacteristic
110D4×C251—22no—
111D4×C251—42no—
112C2×C4×C251—22no—
113C2×C4×C251—21no—
114C2×C4×C251—22no—
115C2×C4×C251—42no—
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251C2411yescharacteristic
118C8×C251—22no—
119C22×C251—21no—
120C22×C251—21no—
121C22×C251—41no—
122C22×C251—41no—
123C22×C251—41no—
124C22×C251AGL(1,ℤ/8)12yes—
125C22×C251C22×D411yescharacteristic
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—
135C2×D251—84no—
136C2×D251#12312yes—
137C4×C251—21no—
138C4×C251—42no—
139C4×C251—41no—
140C4×C251C2×#4112yes—
141Dic251—21no—
142Dic251—41no—
143Dic251—42no—
144Dic251#12312yes—
145C2×C251—22no—
146C2×C251—81no—
147C2×C251C2×#12311yescharacteristic
148C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C251#57511yescharacteristic
150D251—22no—
151D251—41no—
152D251—81no—
153C251#309411yescharacteristic, complemented, Sylow
154#3094—2511nomaximal, Sylow
155C2×#121—2511no—
156C2×#117—2511no—
157#482—2512no—
158#482—2512no—
159#481—2511no—
160#481—2511no—
161#480—2511no—
162#480D25111yescharacteristic, complemented
163#428—2512no—
164#418—2512no—
165#407—2511no—
166C22×D4—5021no—
167C22×D4—2511no—
168C23×C4—2511no—
169C2×M16—2512no—
170#121—5021no—
171#121—5021no—
172#121—2511no—
173#121—2511no—
174#121—2511no—
175#121C2×D25111yescharacteristic
176#117—5021no—
177#117—5021no—
178#117—2512no—
179#117—2512no—
180C2×C4⋊C4—2511no—
181C2×C4⋊C4—2511no—
182C2×C4⋊C4C2×D25111yescharacteristic, complemented
183C2×C22⋊C4—5021no—
184C2×C22⋊C4—5021no—
185C2×C42—5021no—
186#107—2512no—
187#107C2×D25112yescomplemented
188#106—2512no—
189#106—2512no—
190#102—2512no—
191#102—2512no—
192#102—2512no—
193#102C2×D25112yescomplemented
194C22⋊C8—2511no—
195C22⋊C8—2512no—
196C22⋊C8C2×D25111yescharacteristic, complemented
197C24—5021no—
198C24—2511no—
199C2×D4—21no—
200C2×D4—10044no—
201C2×D4—5021no—
202C2×D4—5021no—
203C2×D4—5021no—
204C2×D4—5024no—
205C2×D4—5021no—
206C2×D4—2511no—
207C2×D4C22×D25111yescharacteristic
208C22×C4—21no—
209C22×C4—5021no—
210C22×C4—5024no—
211C22×C4—5021no—
212C22×C4—2511no—
213C22×C4—2511no—
214C22×C4—2511no—
215C22×C4—2511no—
216C22×C4—2511no—
217C22×C4C22×D25111yescharacteristic
218M16—5024no—
219C2×C8—2512no—
220C2×C8C22×D25112yes—
221C4⋊C4—21no—
222C4⋊C4—5021no—
223C4⋊C4—5021no—
224C4⋊C4—2512no—
225C4⋊C4—2511no—
226C4⋊C4C22×D25112yes—
227C4⋊C4C22×D25111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—5022no—
230C22⋊C4—5022no—
231C22⋊C4—5021no—
232C22⋊C4—5021no—
233C22⋊C4—5021no—
234C42—5022no—
235C42—5022no—
236C23—21no—
237C23—10041no—
238C23—10042no—
239C23—10042no—
240C23—10042no—
241C23—5022no—
242C23—5021no—
243C23—5022no—
244C23—2511no—
245C23—2511no—
246C23D4×D25111yescharacteristic
247D4—22no—
248D4—42no—
249D4—10042no—
250D4—10042no—
251D4—10042no—
252D4—10042no—
253D4—5022no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—10042no—
259C2×C4—10042no—
260C2×C4—5022no—
261C2×C4—5022no—
262C2×C4—5022no—
263C2×C4—5021no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—2511no—
267C2×C4—2511no—
268C2×C4—2514no—
269C2×C4D4×D25111yescharacteristic
270C2×C4C23×D25111yescharacteristic
271C8—22no—
272C8—5022no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—20084no—
279C22—10044no—
280C22—10041no—
281C22—10041no—
282C22—10041no—
283C22—5021no—
284C22—5021no—
285C22—5021no—
286C22—5021no—
287C22—5021no—
288C22—2512no—
289C22#2254873912yes—
290C22C2×D4×D25111yescharacteristic
291C4—21no—
292C4—42no—
293C4—41no—
294C4—10041no—
295C4—10042no—
296C4—5021no—
297C4—2512no—
298C4#41×D25112yes—
299C2—22no—
300C2—81no—
301C2—20081no—
302C2—10041no—
303C2—5022no—
304C2#123×D25111yescharacteristic
305C2C2×#2254873911yescharacteristic
306C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#480×C251C2
socleC22×C251C22×D4
prime radical#121×D251C2
hypercenter#480D251
hypocenterC251#3094
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#22548673C2semidirect
3. #30632014C2semidirect
4. #30632010C2semidirect
5. #30632006C2semidirect
6. #30632003C2semidirect
7. #30632000C2semidirect
8. #30631728C2semidirect
9. #30631690C2semidirect
10. #30631649C2semidirect
11. #480×C251C2semidirect
24. #22548673C22semidirect
25. C2×C4⋊Dic251C22semidirect
30. #22548648C22semidirect
31. #22548647C22semidirect
32. #22548646C22semidirect
33. #22548637C22semidirect
34. #22548636C22semidirect
35. #22548635C22semidirect
36. C22⋊(C251⋊C8)C22semidirect
37. (C2×D251)⋊C8C22semidirect
39. C2×C4⋊C4×C251C22semidirect
40. #107×C251C22semidirect
41. #102×C251C22semidirect
42. C22⋊C8×C251C22semidirect
153. C251#3094semidirect
162. #480D251semidirect
182. C2×C4⋊C4C2×D251semidirect
187. #107C2×D251semidirect
193. #102C2×D251semidirect
196. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#121×D2512
C2×#225486732
#306320142
#306320102
#306320062
#306320032
#306320002
#306317282
#306316902
#306316492
#480×C2512
#3094251