Finite Group Databasebeta

← #44560307

Subgroups — #44560307

Order 32128.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 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×#22548690C211yesmaximal, characteristic
2#117×D251C211yesmaximal, characteristic, complemented
3#30632014C212yesmaximal, complemented
4#30632013C212yesmaximal, complemented
5#30632007C212yesmaximal, complemented
6#30632002C212yesmaximal, complemented
7#30631727C211yesmaximal, characteristic, complemented
8#30631693C211yesmaximal, characteristic, complemented
9#30631691C211yesmaximal, characteristic, complemented
10#30631650C211yesmaximal, characteristic, complemented
11#428×C251C211yesmaximal, characteristic, complemented
12C22×D1004—21no—
13C22×D1004C2211yescharacteristic
14C22×C4×D251C2211yescharacteristic
15C2×#16860794C2211yescharacteristic
16#22548690—21no—
17#22548690—21no—
18#22548690C2212yes—
19#22548690C2212yes—
20#22548674C2211yescharacteristic
21#22548673C2211yescharacteristic, complemented
22#22548673C2211yescharacteristic, complemented
23#22548672—21no—
24#22548671—21no—
25C4⋊C4×D251C2211yescharacteristic, complemented
26C4⋊C4×D251C2211yescharacteristic, complemented
27C2×(C2×D251)⋊C4—21no—
28C22⋊C4×D251—21no—
29C42×D251—21no—
30#22548648C2212yescomplemented
31#22548646C2212yescomplemented
32#22548637C2212yescomplemented
33#22548637C2212yescomplemented
34#22548635C2212yescomplemented
35#22548635C2212yescomplemented
36(C2×D251)⋊C8C2212yescomplemented
37(C2×D251)⋊C8C2212yescomplemented
38C2×M16×C251C2211yescharacteristic
39#117×C251C2211yescharacteristic
40C2×C4⋊C4×C251C2211yescharacteristic
41#107×C251C2212yescomplemented
42#106×C251C2212yescomplemented
43C23×D251—21no—
44C23×D251D411yescharacteristic
45C2×D1004—22no—
46C2×D1004—22no—
47C2×D1004—21no—
48C2×D1004—22no—
49C2×D1004—22no—
50C2×D1004—44no—
51C2×D1004C2312yes—
52C22×Dic251D411yescharacteristic
53C2×C4×D251—21no—
54C2×C4×D251—21no—
55C2×C4×D251—22no—
56C2×C4×D251—22no—
57C2×C4×D251D411yescharacteristic
58C2×C4×D251D411yescharacteristic
59C2×C4×D251C2311yescharacteristic
60C2×C4×D251C2311yescharacteristic
61#16860794—22no—
62C2×C251⋊C8C2312yes—
63C4⋊Dic251C2311yescharacteristic
64C4⋊Dic251C2311yescharacteristic
65Dic251⋊C4—21no—
66Dic251⋊C4—21no—
67C22⋊Dic251—21no—
68(C2×D251)⋊C4—21no—
69(C2×D251)⋊C4—21no—
70(C2×D251)⋊C4—22no—
71(C2×D251)⋊C4—22no—
72C4×Dic251—21no—
73C4×Dic251—21no—
74C4×Dic251—21no—
75C22×C4×C251—21no—
76C22×C4×C251C2311yescharacteristic
77M16×C251—22no—
78C2×C8×C251C2312yes—
79C4⋊C4×C251—21no—
80C4⋊C4×C251C2311yescharacteristic
81C4⋊C4×C251C2312yes—
82C4⋊C4×C251C2311yescharacteristic
83C22⋊C4×C251—21no—
84C42×C251—21no—
85C22×D251—22no—
86C22×D251—22no—
87C22×D251—22no—
88C22×D251—41no—
89C22×D251—42no—
90C22×D251—42no—
91C22×D251—42no—
92C22×D251C2×D411yescharacteristic
93C22×D251C2×D411yescharacteristic
94D1004—22no—
95D1004—22no—
96D1004—44no—
97D1004—44no—
98D1004—42no—
99C2×Dic251—22no—
100C2×Dic251—21no—
101C2×Dic251—21no—
102C2×Dic251C2×D411yescharacteristic
103C2×Dic251C2×D411yescharacteristic
104C4×D251—22no—
105C4×D251—22no—
106C4×D251—21no—
107C4×D251—21no—
108C4×D251—21no—
109C4×D251—21no—
110C4×D251—42no—
111C4×D251—42no—
112C251⋊C8—22no—
113C23×C251C2×D411yescharacteristic
114C2×C4×C251—21no—
115C2×C4×C251—21no—
116C2×C4×C251—22no—
117C2×C4×C251—42no—
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251#4112yes—
120C2×C4×C251#4112yes—
121C2×C4×C251C2411yescharacteristic
122C8×C251—22no—
123C22×C251—21no—
124C22×C251—21no—
125C22×C251#12312yes—
126C22×C251C22×D411yescharacteristic
127C2×D251—21no—
128C2×D251—21no—
129C2×D251—21no—
130C2×D251—21no—
131C2×D251—21no—
132C2×D251—44no—
133C2×D251—42no—
134C2×D251—42no—
135C2×D251—42no—
136C2×D251—84no—
137C2×D251AGL(1,ℤ/8)12yes—
138C4×C251—42no—
139C4×C251—41no—
140C4×C251—41no—
141C4×C251#12312yes—
142C4×C251C2×#4111yescharacteristic
143C4×C251C2×#4111yescharacteristic
144Dic251—21no—
145Dic251—21no—
146Dic251—41no—
147Dic251—41no—
148C2×C251—22no—
149C2×C251C2×#12311yescharacteristic
150C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C251#57511yescharacteristic
152D251—22no—
153D251—41no—
154D251—82no—
155C251#309411yescharacteristic, complemented, Sylow
156#3094—2511nomaximal, Sylow
157C2×#121—2511no—
158C2×#117—2511no—
159#482—2512no—
160#482—2512no—
161#481—2512no—
162#480—2512no—
163#428—2511no—
164#428D25111yescharacteristic, complemented
165#418—2511no—
166#418—2511no—
167#407—2511no—
168C22×D4—5021no—
169C22×D4—2511no—
170C23×C4—2511no—
171C2×M16—2511no—
172C2×M16C2×D25111yescharacteristic
173#121—5021no—
174#121—5021no—
175#121—2512no—
176#121—2512no—
177#117—5021no—
178#117—5021no—
179#117—2511no—
180#117—2511no—
181#117—2511no—
182#117C2×D25111yescharacteristic
183C2×C4⋊C4—2511no—
184C2×C4⋊C4—2511no—
185C2×C4⋊C4C2×D25111yescharacteristic
186C2×C22⋊C4—5021no—
187C2×C22⋊C4—5021no—
188C2×C42—5021no—
189#107—2512no—
190#107C2×D25112yescomplemented
191#106—2512no—
192#106C2×D25112yescomplemented
193#102—2512no—
194#102—2512no—
195#102—2512no—
196#102—2512no—
197C22⋊C8—2512no—
198C22⋊C8—2512no—
199C24—5021no—
200C24—2511no—
201C2×D4—10044no—
202C2×D4—5022no—
203C2×D4—5022no—
204C2×D4—5021no—
205C2×D4—5022no—
206C2×D4—5022no—
207C2×D4—2512no—
208C22×C4—21no—
209C22×C4—5021no—
210C22×C4—5021no—
211C22×C4—5022no—
212C22×C4—5022no—
213C22×C4—2511no—
214C22×C4—2511no—
215C22×C4—2511no—
216C22×C4—2511no—
217C22×C4—2511no—
218C22×C4C22×D25111yescharacteristic
219M16—22no—
220M16—5022no—
221C2×C8—2512no—
222C2×C8C22×D25112yes—
223C4⋊C4—21no—
224C4⋊C4—5021no—
225C4⋊C4—5021no—
226C4⋊C4—2511no—
227C4⋊C4—2511no—
228C4⋊C4C22×D25111yescharacteristic
229C4⋊C4C22×D25111yescharacteristic
230C4⋊C4C22×D25112yes—
231C22⋊C4—21no—
232C22⋊C4—5021no—
233C22⋊C4—5021no—
234C22⋊C4—5022no—
235C22⋊C4—5022no—
236C22⋊C4—5021no—
237C42—21no—
238C42—5021no—
239C42—5021no—
240C42—5021no—
241C23—10042no—
242C23—10042no—
243C23—10041no—
244C23—10042no—
245C23—5022no—
246C23—5022no—
247C23—5022no—
248C23—2511no—
249C23—2511no—
250C23D4×D25111yescharacteristic
251D4—10044no—
252D4—10044no—
253D4—10042no—
254D4—5022no—
255D4—5022no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—10042no—
261C2×C4—10042no—
262C2×C4—5021no—
263C2×C4—5021no—
264C2×C4—5022no—
265C2×C4—5021no—
266C2×C4—5021no—
267C2×C4—5021no—
268C2×C4—5021no—
269C2×C4—5022no—
270C2×C4—5022no—
271C2×C4—2511no—
272C2×C4—2511no—
273C2×C4D4×D25111yescharacteristic
274C2×C4#1686081012yes—
275C2×C4#1686081112yes—
276C2×C4C23×D25111yescharacteristic
277C8—22no—
278C8—5022no—
279C22—21no—
280C22—21no—
281C22—20084no—
282C22—10044no—
283C22—10042no—
284C22—10042no—
285C22—10042no—
286C22—5021no—
287C22—5021no—
288C22—5021no—
289C22—5021no—
290C22—5021no—
291C22—2512no—
292C22#2254870012yes—
293C22C2×D4×D25111yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—10041no—
298C4—10041no—
299C4—5021no—
300C4—5021no—
301C4#2254870012yes—
302C4C2×#1686081011yescharacteristic
303C4C2×#1686081111yescharacteristic
304C2—22no—
305C2—20082no—
306C2—10041no—
307C2—5022no—
308C2C2×#2254870011yescharacteristic
309C2AGL(1,ℤ/8)×D25111yescharacteristic
310C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548690C2
hypercenter#428D251
hypocenterC251#3094
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. #117×D251C2semidirect
3. #30632014C2semidirect
4. #30632013C2semidirect
5. #30632007C2semidirect
6. #30632002C2semidirect
7. #30631727C2semidirect
8. #30631693C2semidirect
9. #30631691C2semidirect
10. #30631650C2semidirect
11. #428×C251C2semidirect
21. #22548673C22semidirect
22. #22548673C22semidirect
25. C4⋊C4×D251C22semidirect
26. C4⋊C4×D251C22semidirect
30. #22548648C22semidirect
31. #22548646C22semidirect
32. #22548637C22semidirect
33. #22548637C22semidirect
34. #22548635C22semidirect
35. #22548635C22semidirect
36. (C2×D251)⋊C8C22semidirect
37. (C2×D251)⋊C8C22semidirect
41. #107×C251C22semidirect
42. #106×C251C22semidirect
155. C251#3094semidirect
164. #428D251semidirect
190. #107C2×D251semidirect
192. #106C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486902
#117×D2512
#306320142
#306320132
#306320072
#306320022
#306317272
#306316932
#306316912
#306316502
#428×C2512
#3094251