Finite Group Databasebeta

← #44560286

Subgroups — #44560286

Order 32128.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#30632194C211yesmaximal, characteristic
2C2×C4⋊C4×D251C211yesmaximal, characteristic, complemented
3#30632021C212yesmaximal, complemented
4#30632016C212yesmaximal, complemented
5#30632004C212yesmaximal, complemented
6#30631999C212yesmaximal, complemented
7#30631729C211yesmaximal, characteristic, complemented
8#30631727C211yesmaximal, characteristic, complemented
9#30631688C211yesmaximal, characteristic, complemented
10#30631650C211yesmaximal, characteristic, complemented
11#417×C251C211yesmaximal, characteristic, complemented
12C22×C4×D251—21no—
13C22×C4×D251C2211yescharacteristic
14C2×#16860794C2211yescharacteristic
15#22548696—21no—
16#22548691C2211yescharacteristic
17#22548691C2211yescharacteristic
18#22548686C2211yescharacteristic
19#22548686C2211yescharacteristic
20#22548683—21no—
21C4×C22⋊D251—21no—
22C4×C22⋊D251—21no—
23#22548674C2211yescharacteristic
24C2×C4⋊Dic251C2211yescharacteristic
25C2×Dic251⋊C4—21no—
26C4⋊C4×D251—21no—
27C4⋊C4×D251—21no—
28C4⋊C4×D251C2212yes—
29C4⋊C4×D251C2212yes—
30#22548649C2212yescomplemented
31#22548648C2212yescomplemented
32#22548647C2212yescomplemented
33#22548646C2212yescomplemented
34#22548639C2212yescomplemented
35#22548636C2212yescomplemented
36(C2×D251)⋊C8C2212yes—
37(C2×D251)⋊C8C2212yes—
38C2×#41×C251C2211yescharacteristic
39C2×M16×C251C2211yescharacteristic
40C2×C4⋊C4×C251C2211yescharacteristic
41#103×C251C2212yescomplemented
42#102×C251C2212yescomplemented
43C23×D251D411yescharacteristic
44C2×C22⋊D251—21no—
45C2×C22⋊D251—21no—
46C22×Dic251—21no—
47C22×Dic251D411yescharacteristic
48C2×C4×D251—22no—
49C2×C4×D251—22no—
50C2×C4×D251—22no—
51C2×C4×D251—22no—
52C2×C4×D251—44no—
53C2×C4×D251D411yescharacteristic
54C2×C4×D251D411yescharacteristic
55C2×C4×D251C2311yescharacteristic
56C2×C4×D251C2311yescharacteristic
57#16860794—22no—
58C2×C251⋊C8C2312yes—
59C4⋊Dic251—21no—
60C4⋊Dic251C2312yes—
61C4⋊Dic251C2311yescharacteristic
62C4⋊Dic251C2311yescharacteristic
63Dic251⋊C4—22no—
64Dic251⋊C4—22no—
65Dic251⋊C4—21no—
66Dic251⋊C4—21no—
67C22⋊Dic251—21no—
68C22⋊Dic251—21no—
69C22⋊Dic251—21no—
70(C2×D251)⋊C4—21no—
71(C2×D251)⋊C4—21no—
72C4×Dic251—21no—
73#41×C251—22no—
74#41×C251—22no—
75C2×Q8×C251C2311yescharacteristic
76C2×D4×C251—21no—
77C2×D4×C251C2311yescharacteristic
78C22×C4×C251—21no—
79C22×C4×C251—21no—
80C22×C4×C251C2311yescharacteristic
81M16×C251—22no—
82C2×C8×C251C2312yes—
83C4⋊C4×C251—21no—
84C4⋊C4×C251C2312yes—
85C22×D251—22no—
86C22×D251—22no—
87C22×D251—22no—
88C22×D251C2×D411yescharacteristic
89C22×D251C2×D411yescharacteristic
90C22⋊D251—42no—
91C22⋊D251—42no—
92C2×Dic251—22no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—22no—
96C2×Dic251—42no—
97C2×Dic251C2×D411yescharacteristic
98C2×Dic251C2×D411yescharacteristic
99C4×D251—22no—
100C4×D251—22no—
101C4×D251—21no—
102C4×D251—21no—
103C4×D251—21no—
104C4×D251—21no—
105C4×D251—44no—
106C4×D251—44no—
107C251⋊C8—22no—
108C23×C251—21no—
109C23×C251C2×D411yescharacteristic
110Q8×C251—21no—
111Q8×C251—21no—
112D4×C251—21no—
113D4×C251—21no—
114D4×C251—42no—
115C2×C4×C251—22no—
116C2×C4×C251—21no—
117C2×C4×C251—42no—
118C2×C4×C251—42no—
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251#4112yes—
121C2×C4×C251#4112yes—
122C2×C4×C251C2411yescharacteristic
123C8×C251—22no—
124C22×C251—21no—
125C22×C251—21no—
126C22×C251—41no—
127C22×C251—41no—
128C22×C251—41no—
129C22×C251#12312yes—
130C22×C251C22×D411yescharacteristic
131C2×D251—28no—
132C2×D251—21no—
133C2×D251—21no—
134C2×D251—21no—
135C2×D251—21no—
136C2×D251AGL(1,ℤ/8)12yes—
137C2×D251#13712yes—
138C4×C251—41no—
139C4×C251—42no—
140C4×C251#12312yes—
141C4×C251C2×#4111yescharacteristic
142C4×C251C2×#4111yescharacteristic
143Dic251—21no—
144Dic251—21no—
145Dic251—42no—
146Dic251—41no—
147Dic251—41no—
148C2×C251—22no—
149C2×C251—81no—
150C2×C251C2×#12311yescharacteristic
151C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C251C2×#13711yescharacteristic
153D251—22no—
154D251—22no—
155C251#309211yescharacteristic, complemented, Sylow
156#3092—2511nomaximal, Sylow
157#525—2511no—
158C22×C4⋊C4—2511no—
159#484—2512no—
160#483—2512no—
161#481—2512no—
162#480—2512no—
163#428—2511no—
164#428—2511no—
165#417—2511no—
166#417D25111yescharacteristic, complemented
167#407—2511no—
168C2×#41C2×D25111yescharacteristic
169C23×C4—5021no—
170C23×C4—2511no—
171C2×M16—2511no—
172C2×M16C2×D25111yescharacteristic
173#123—5021no—
174C22⋊Q8—2511no—
175C22⋊Q8—2511no—
176#121—2511no—
177#121—2511no—
178#120—5021no—
179C4×D4—5021no—
180C4×D4—5021no—
181#117—2511no—
182C2×C4⋊C4—5021no—
183C2×C4⋊C4—5021no—
184C2×C4⋊C4—5021no—
185C2×C4⋊C4—2511no—
186C2×C4⋊C4—2512no—
187C2×C4⋊C4—2512no—
188C2×C4⋊C4C2×D25111yescharacteristic
189#107—2512no—
190#107—2512no—
191#106—2512no—
192#106—2512no—
193#103—2512no—
194#103C2×D25112yescomplemented
195#102—2512no—
196#102C2×D25112yescomplemented
197C22⋊C8—2512no—
198C22⋊C8—2512no—
199C24—2511no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D25111yescharacteristic
203C2×D4—21no—
204C2×D4—5021no—
205C2×D4—5021no—
206C2×D4C22×D25111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—10044no—
210C22×C4—5022no—
211C22×C4—5022no—
212C22×C4—5021no—
213C22×C4—5022no—
214C22×C4—5022no—
215C22×C4—2511no—
216C22×C4—2511no—
217C22×C4—2511no—
218C22×C4—2511no—
219C22×C4—2511no—
220C22×C4C22×D25111yescharacteristic
221M16—22no—
222M16—5022no—
223C2×C8—2512no—
224C2×C8C22×D25112yes—
225C4⋊C4—21no—
226C4⋊C4—5022no—
227C4⋊C4—5022no—
228C4⋊C4—5021no—
229C4⋊C4—5021no—
230C4⋊C4—5021no—
231C4⋊C4—2512no—
232C4⋊C4—2511no—
233C4⋊C4—2511no—
234C4⋊C4C22×D25112yes—
235C22⋊C4—5021no—
236C22⋊C4—5021no—
237C22⋊C4—5021no—
238C22⋊C4—5021no—
239C22⋊C4—5021no—
240C42—5021no—
241C23—21no—
242C23—5022no—
243C23—5022no—
244C23—5022no—
245C23—2511no—
246C23—2511no—
247C23D4×D25111yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—10042no—
254D4—10042no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—10044no—
260C2×C4—10044no—
261C2×C4—10042no—
262C2×C4—5021no—
263C2×C4—5021no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—5021no—
267C2×C4—5021no—
268C2×C4—5021no—
269C2×C4—5021no—
270C2×C4—5022no—
271C2×C4—5022no—
272C2×C4—2511no—
273C2×C4—2511no—
274C2×C4D4×D25111yescharacteristic
275C2×C4#1686080912yes—
276C2×C4#1686081012yes—
277C2×C4C23×D25111yescharacteristic
278C8—22no—
279C8—5022no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—5028no—
286C22—5021no—
287C22—5021no—
288C22—5021no—
289C22—5021no—
290C22—2512no—
291C22—2512no—
292C22#2254869712yes—
293C22C2×D4×D25111yescharacteristic
294C4—42no—
295C4—41no—
296C4—10042no—
297C4—10041no—
298C4—10041no—
299C4—5021no—
300C4—5021no—
301C4#2254869712yes—
302C4C2×#1686080911yescharacteristic
303C4C2×#1686081011yescharacteristic
304C2—22no—
305C2—81no—
306C2—5022no—
307C2—5022no—
308C2C2×#2254869711yescharacteristic
309C2AGL(1,ℤ/8)×D25111yescharacteristic
310C2#137×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#417×C251C2
socleC22×C251C22×D4
prime radical#30632194C2
hypercenter#417D251
hypocenterC251#3092
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D251C2semidirect
3. #30632021C2semidirect
4. #30632016C2semidirect
5. #30632004C2semidirect
6. #30631999C2semidirect
7. #30631729C2semidirect
8. #30631727C2semidirect
9. #30631688C2semidirect
10. #30631650C2semidirect
11. #417×C251C2semidirect
30. #22548649C22semidirect
31. #22548648C22semidirect
32. #22548647C22semidirect
33. #22548646C22semidirect
34. #22548639C22semidirect
35. #22548636C22semidirect
41. #103×C251C22semidirect
42. #102×C251C22semidirect
155. C251#3092semidirect
166. #417D251semidirect
194. #103C2×D251semidirect
196. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306321942
C2×C4⋊C4×D2512
#306320212
#306320162
#306320042
#306319992
#306317292
#306317272
#306316882
#306316502
#417×C2512
#3092251