Finite Group Databasebeta

← #481×D251 #44560261

Subgroups — #481×D251 #44560261

Order 32128.

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

normal subgroups only

This group has 512 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
1#121×D251C211yesmaximal, characteristic
2C2×C4⋊C4×D251C211yesmaximal, characteristic, complemented
3#30632007C212yesmaximal, complemented
4#30632006C211yesmaximal, characteristic, complemented
5#30632005C211yesmaximal, characteristic, complemented
6#30632004C212yesmaximal, complemented
7#30632003C211yesmaximal, characteristic, complemented
8#106×D251C212yesmaximal, complemented
9#102×D251C212yesmaximal, complemented
10C22⋊C8×D251C211yesmaximal, characteristic, complemented
11#481×C251C211yesmaximal, characteristic, complemented
12C2×D4×D251—21no—
13C2×D4×D251C2211yescharacteristic
14C22×C4×D251—21no—
15C22×C4×D251C2211yescharacteristic
16C2×C8×D251C2212yes—
17#22548690C2211yescharacteristic
18#22548689C2211yescharacteristic
19#22548688—21no—
20#22548687—21no—
21#22548686C2211yescharacteristic
22C2×C4⋊Dic251C2211yescharacteristic, complemented
23C2×Dic251⋊C4—21no—
24C4⋊C4×D251—21no—
25C4⋊C4×D251—21no—
26C4⋊C4×D251C2211yescharacteristic
27C4⋊C4×D251C2212yes—
28C4⋊C4×D251C2212yescomplemented
29C22⋊C4×D251—21no—
30#22548647C2212yescomplemented
31#22548646C2212yescomplemented
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#106×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—22no—
56C2×C4×D251—22no—
57C2×C4×D251—21no—
58C2×C4×D251—44no—
59C2×C4×D251D411yescharacteristic
60C2×C4×D251D411yescharacteristic
61C2×C4×D251C2311yescharacteristic
62C2×C4×D251C2311yescharacteristic
63C2×C251⋊C8C2312yes—
64C8×D251—24no—
65C4⋊Dic251—21no—
66C4⋊Dic251C2312yes—
67C4⋊Dic251C2311yescharacteristic
68Dic251⋊C4—22no—
69Dic251⋊C4—22no—
70Dic251⋊C4—21no—
71C22⋊Dic251—21no—
72(C2×D251)⋊C4—21no—
73(C2×D251)⋊C4—21no—
74C2×D4×C251—21no—
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C22×C4×C251C2311yescharacteristic
78C2×C8×C251C2312yes—
79C4⋊C4×C251—21no—
80C4⋊C4×C251C2312yes—
81C4⋊C4×C251C2311yescharacteristic
82C22⋊C4×C251—21no—
83C22×D251—22no—
84C22×D251—22no—
85C22×D251—21no—
86C22×D251—42no—
87C22×D251—42no—
88C22×D251—42no—
89C22×D251SD1614yes—
90C22×D251C2×D411yescharacteristic
91C22×D251C2×D411yescharacteristic
92D1004—22no—
93D1004—42no—
94C22⋊D251—42no—
95C22⋊D251—42no—
96C22⋊D251—42no—
97C2×Dic251—22no—
98C2×Dic251—21no—
99C2×Dic251—22no—
100C2×Dic251—42no—
101C2×Dic251C2×D411yescharacteristic
102C2×Dic251C2×D411yescharacteristic
103C4×D251—22no—
104C4×D251—22no—
105C4×D251—22no—
106C4×D251—44no—
107C4×D251—42no—
108C4×D251—44no—
109C4×D251#4114yes—
110C251⋊C8—22no—
111C23×C251—21no—
112C23×C251C2×D411yescharacteristic
113D4×C251—22no—
114D4×C251—42no—
115C2×C4×C251—22no—
116C2×C4×C251—21no—
117C2×C4×C251—22no—
118C2×C4×C251—42no—
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251C2411yescharacteristic
121C8×C251—22no—
122C22×C251—21no—
123C22×C251—21no—
124C22×C251—41no—
125C22×C251—41no—
126C22×C251—41no—
127C22×C251C2×SD1612yes—
128C22×C251C22×D411yescharacteristic
129C2×D251—21no—
130C2×D251—21no—
131C2×D251—28no—
132C2×D251—41no—
133C2×D251—41no—
134C2×D251—41no—
135C2×D251—84no—
136C2×D251#12312yes—
137C2×D251C2×SD1612yes—
138C2×D251C2×SD1612yes—
139C2×D251AGL(1,ℤ/8)12yes—
140C4×C251—21no—
141C4×C251—41no—
142C4×C251—42no—
143C4×C251C2×#4112yes—
144Dic251—21no—
145Dic251—41no—
146Dic251—42no—
147Dic251#12312yes—
148C2×C251—22no—
149C2×C251—81no—
150C2×C251C2×#12311yescharacteristic
151C2×C251C22×SD1611yescharacteristic
152C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
153D251—22no—
154D251—81no—
155D251#48114yesdirect product factor
156C251C2×#48111yescharacteristic, complemented, Sylow
157C2×#481—2511nomaximal, Sylow
158C2×#121—2511no—
159C22×C4⋊C4—2511no—
160#481—2511no—
161#481—2511no—
162#481—2512no—
163#481—2512no—
164#481—2511no—
165#481D25111yescharacteristic, direct product factor
166C2×#106—2512no—
167C2×#102—2512no—
168C2×C22⋊C8—2511no—
169C22×D4—5021no—
170C22×D4—2511no—
171C23×C4—5021no—
172C23×C4—2511no—
173C22×C8—2512no—
174#121—5021no—
175#121—5021no—
176#121—2511no—
177#121—2511no—
178#121—2511no—
179#121C2×D25111yescharacteristic
180C2×C4⋊C4—5021no—
181C2×C4⋊C4—5021no—
182C2×C4⋊C4—5021no—
183C2×C4⋊C4—2511no—
184C2×C4⋊C4—2512no—
185C2×C4⋊C4—2512no—
186C2×C4⋊C4—2511no—
187C2×C4⋊C4C2×D25111yescharacteristic, complemented
188C2×C22⋊C4—5021no—
189#106—2512no—
190#106—2512no—
191#106—2512no—
192#106C2×D25112yescomplemented
193#102—2512no—
194#102—2512no—
195#102—2512no—
196#102C2×D25112yescomplemented
197C22⋊C8—2511no—
198C22⋊C8—2512no—
199C22⋊C8C2×D25111yescharacteristic, complemented
200C24—5021no—
201C24—2511no—
202C2×D4—21no—
203C2×D4—10044no—
204C2×D4—5021no—
205C2×D4—5021no—
206C2×D4—5021no—
207C2×D4—5024no—
208C2×D4—5021no—
209C2×D4—2511no—
210C2×D4C22×D25111yescharacteristic
211C22×C4—21no—
212C22×C4—10044no—
213C22×C4—5021no—
214C22×C4—5022no—
215C22×C4—5022no—
216C22×C4—5024no—
217C22×C4—5021no—
218C22×C4—2511no—
219C22×C4—2511no—
220C22×C4—2511no—
221C22×C4—2511no—
222C22×C4—2511no—
223C22×C4C22×D25111yescharacteristic
224C2×C8—5024no—
225C2×C8—2512no—
226C2×C8C22×D25112yes—
227C4⋊C4—21no—
228C4⋊C4—5022no—
229C4⋊C4—5022no—
230C4⋊C4—5021no—
231C4⋊C4—5021no—
232C4⋊C4—2512no—
233C4⋊C4—2511no—
234C4⋊C4C22×D25112yes—
235C4⋊C4C22×D25111yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—5021no—
238C22⋊C4—5021no—
239C22⋊C4—5021no—
240C23—21no—
241C23—10042no—
242C23—10042no—
243C23—10042no—
244C23—5022no—
245C23—5022no—
246C23—5021no—
247C23—2511no—
248C23—2514no—
249C23—2511no—
250C23D4×D25111yescharacteristic
251D4—22no—
252D4—42no—
253D4—10042no—
254D4—10042no—
255D4—10042no—
256D4—10042no—
257D4—5022no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—10044no—
263C2×C4—10044no—
264C2×C4—10042no—
265C2×C4—10042no—
266C2×C4—5022no—
267C2×C4—5022no—
268C2×C4—5021no—
269C2×C4—5022no—
270C2×C4—5022no—
271C2×C4—5022no—
272C2×C4—2511no—
273C2×C4—2511no—
274C2×C4—2514no—
275C2×C4D4×D25111yescharacteristic
276C2×C4C23×D25111yescharacteristic
277C8—22no—
278C8—5022no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—20084no—
285C22—10041no—
286C22—10041no—
287C22—10041no—
288C22—5028no—
289C22—5021no—
290C22—5021no—
291C22—2512no—
292C22—2512no—
293C22—2512no—
294C22—2512no—
295C22SD16×D25112yes—
296C22C2×D4×D25111yescharacteristic
297C4—21no—
298C4—41no—
299C4—42no—
300C4—10041no—
301C4—10042no—
302C4—5021no—
303C4—2512no—
304C4#41×D25112yes—
305C2—22no—
306C2—81no—
307C2—20081no—
308C2—5022no—
309C2—2514no—
310C2#123×D25111yescharacteristic
311C2C2×SD16×D25111yescharacteristic
312C2AGL(1,ℤ/8)×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#481×C251C2
socleC22×C251C22×D4
prime radical#121×D251C2
hypercenter#481D251
hypocenterC251C2×#481
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D251C2semidirect
3. #30632007C2semidirect
4. #30632006C2semidirect
5. #30632005C2semidirect
6. #30632004C2semidirect
7. #30632003C2semidirect
8. #106×D251C2semidirect
9. #102×D251C2semidirect
10. C22⋊C8×D251C2semidirect
11. #481×C251C2semidirect
22. C2×C4⋊Dic251C22semidirect
28. C4⋊C4×D251C22semidirect
30. #22548647C22semidirect
31. #22548646C22semidirect
32. #22548646C22semidirect
33. #22548637C22semidirect
34. #22548636C22semidirect
35. #22548635C22semidirect
36. C22⋊(C251⋊C8)C22semidirect
37. (C2×D251)⋊C8C22semidirect
39. C2×C4⋊C4×C251C22semidirect
40. #106×C251C22semidirect
41. #102×C251C22semidirect
42. C22⋊C8×C251C22semidirect
155. D251#481direct (N × H)
156. C251C2×#481semidirect
165. #481D251direct (N × H)
187. C2×C4⋊C4C2×D251semidirect
192. #106C2×D251semidirect
196. #102C2×D251semidirect
199. 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×C4⋊C4×D2512
#306320072
#306320062
#306320052
#306320042
#306320032
#106×D2512
#102×D2512
C22⋊C8×D2512
#481×C2512
C2×#481251