Finite Group Databasebeta

← #44560750

Subgroups — #44560750

Order 32128.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#30632242C211yesmaximal, characteristic, complemented
2#124×D251C211yesmaximal, characteristic
3#30632049C211yesmaximal, characteristic, complemented
4#30632047C211yesmaximal, characteristic, complemented
5#30632046C211yesmaximal, characteristic, complemented
6#30632043C212yesmaximal, complemented
7#30632040C212yesmaximal, complemented
8#30631694C212yesmaximal, complemented
9#30631690C212yesmaximal
10#30631639C211yesmaximal, characteristic, complemented
11#489×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251C2211yescharacteristic
13C2×D4×D251C2211yescharacteristic
14C2×#16860793C2212yes—
15#22548715C2211yescharacteristic, complemented
16#22548706C2212yescomplemented
17#22548704C2211yescharacteristic
18#22548703C2211yescharacteristic
19#22548702C2211yescharacteristic
20#22548701—21no—
21#22548701—21no—
22#22548695—21no—
23#22548691—21no—
24Q8×Dic251—21no—
25#22548680—21no—
26#22548673C2212yes—
27C22⋊C4×D251—21no—
28C22⋊C4×D251—21no—
29C42×D251C2211yescharacteristic
30#22548640C2212yescomplemented
31#22548639C2212yescomplemented
32#22548638C2212yescomplemented
33#22548637C2212yes—
34#22548636C2212yes—
35#22548635C2212yes—
36#22548626C2211yescharacteristic, complemented
37C8×Dic251C2212yescomplemented
38C4⋊Q8×C251C2211yescharacteristic, complemented
39#124×C251C2211yescharacteristic
40#103×C251C2212yescomplemented
41#102×C251C2212yes—
42#97×C251C2211yescharacteristic, complemented
43C23×D251—21no—
44C2×Dic502—21no—
45C2×Dic502—21no—
46C2×Dic502C2311yescharacteristic
47Q8×D251—24no—
48C2×D1004C2311yescharacteristic
49C2×C22⋊D251—21no—
50D4×D251—24no—
51C2×C4×D251—21no—
52C2×C4×D251D411yescharacteristic
53C2×C4×D251D411yescharacteristic
54C2×C4×D251C2311yescharacteristic
55#16860793—24no—
56C2×C251⋊C8C2312yes—
57C4⋊Dic251—22no—
58C4⋊Dic251—21no—
59C4⋊Dic251C2312yes—
60Dic251⋊C4—22no—
61C22⋊Dic251—21no—
62C22⋊Dic251—21no—
63(C2×D251)⋊C4—22no—
64(C2×D251)⋊C4—21no—
65(C2×D251)⋊C4—21no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4—21no—
68C4×Dic251—22no—
69C4×Dic251D411yescharacteristic
70C4×Dic251D411yescharacteristic
71C4×Dic251C2311yescharacteristic
72C2×Q8×C251—21no—
73C2×Q8×C251C2311yescharacteristic
74C2×D4×C251C2311yescharacteristic
75C2×C8×C251C2312yes—
76C4⋊C4×C251—21no—
77C4⋊C4×C251C2312yes—
78C22⋊C4×C251—21no—
79C22⋊C4×C251—21no—
80C42×C251C2311yescharacteristic
81C22×D251—21no—
82C22×D251—42no—
83C22×D251—42no—
84C22×D251—42no—
85C22×D251C2×D411yescharacteristic
86Dic502—22no—
87Dic502—42no—
88Dic502—42no—
89D1004—22no—
90C22⋊D251—42no—
91C2×Dic251—22no—
92C2×Dic251—21no—
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C4×D251—41no—
97C4×D251—41no—
98C4×D251—42no—
99C4×D251#4114yes—
100C251⋊C8—22no—
101C23×C251—21no—
102Q8×C251—22no—
103Q8×C251—42no—
104D4×C251—22no—
105C2×C4×C251—21no—
106C2×C4×C251—22no—
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2411yescharacteristic
110C8×C251—22no—
111C22×C251—41no—
112C22×C251—41no—
113C22×C251—41no—
114C22×C251C22×D411yescharacteristic
115C2×D251—21no—
116C2×D251—21no—
117C2×D251—41no—
118C2×D251—41no—
119C2×D251—41no—
120C2×D251—84no—
121C2×D251#12412yes—
122C4×C251—21no—
123C4×C251—21no—
124C4×C251—41no—
125C4×C251—42no—
126C4×C251C2×#4112yes—
127Dic251—21no—
128Dic251—21no—
129Dic251—41no—
130Dic251—42no—
131Dic251#12412yes—
132C2×C251—81no—
133C2×C251C2×#12411yescharacteristic
134C2×C251#57511yescharacteristic
135C2×C251#57511yescharacteristic
136D251—22no—
137D251—81no—
138C251#314011yescharacteristic, complemented, Sylow
139#3140—2511nomaximal, Sylow
140#533—2511no—
141C2×#124—2511no—
142#489—2511no—
143#489—2511no—
144#489—2511no—
145#489D25111yescharacteristic, complemented
146#488—2512no—
147#488—2512no—
148#419—2512no—
149#418—2512no—
150C4×M16—2511no—
151C22×Q8—2511no—
152C22×D4—2511no—
153C2×M16—2512no—
154C4⋊Q8—2511no—
155C4⋊Q8C2×D25111yescharacteristic, complemented
156#125—2512no—
157#124—5021no—
158#124—5021no—
159#124—2511no—
160#124—2511no—
161#124—2511no—
162#124C2×D25111yescharacteristic
163C22⋊Q8—5021no—
164C22⋊Q8—5021no—
165C4×Q8—5021no—
166C4×Q8—5021no—
167#117—2512no—
168C2×C22⋊C4—5021no—
169C2×C22⋊C4—5021no—
170C2×C42—2511no—
171#103—2512no—
172#103—2512no—
173#103—2512no—
174#103C2×D25112yescomplemented
175#102—2512no—
176#102—2512no—
177#102—2512no—
178#102C2×D25112yes—
179#97—2511no—
180#97C2×D25111yescharacteristic, complemented
181C4×C8—2512no—
182C24—5021no—
183C2×Q8—21no—
184C2×Q8—5021no—
185C2×Q8—5024no—
186C2×Q8—5021no—
187C2×Q8—2511no—
188C2×Q8C22×D25111yescharacteristic
189C2×D4—5024no—
190C2×D4—5021no—
191C2×D4—2511no—
192C2×D4C22×D25111yescharacteristic
193C22×C4—5021no—
194C22×C4—2511no—
195C22×C4—2511no—
196C22×C4—2511no—
197M16—5024no—
198C2×C8—2512no—
199C2×C8C22×D25112yes—
200C4⋊C4—21no—
201C4⋊C4—5022no—
202C4⋊C4—5022no—
203C4⋊C4—5021no—
204C4⋊C4—2512no—
205C4⋊C4C22×D25112yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—5022no—
209C22⋊C4—5021no—
210C22⋊C4—5021no—
211C22⋊C4—5021no—
212C22⋊C4—5021no—
213C22⋊C4—5021no—
214C22⋊C4—5021no—
215C42—5022no—
216C42—2511no—
217C42—2511no—
218C42—2511no—
219C42C22×D25111yescharacteristic
220C23—21no—
221C23—10042no—
222C23—10042no—
223C23—10042no—
224C23—5021no—
225C23—2511no—
226Q8—22no—
227Q8—42no—
228Q8—10042no—
229Q8—10042no—
230Q8—5022no—
231D4—22no—
232D4—10042no—
233D4—5022no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—10041no—
237C2×C4—10041no—
238C2×C4—10042no—
239C2×C4—5022no—
240C2×C4—5021no—
241C2×C4—2511no—
242C2×C4—2511no—
243C2×C4—2511no—
244C2×C4—2514no—
245C2×C4D4×D25111yescharacteristic
246C2×C4D4×D25111yescharacteristic
247C2×C4C23×D25111yescharacteristic
248C8—22no—
249C8—5022no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—20084no—
254C22—10041no—
255C22—10041no—
256C22—10041no—
257C22—5021no—
258C22—5021no—
259C22—2512no—
260C22C2×D4×D25111yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—10041no—
266C4—10042no—
267C4—5021no—
268C4—5021no—
269C4—2512no—
270C4#41×D25112yes—
271C2—81no—
272C2—20081no—
273C2—5022no—
274C2#124×D25111yescharacteristic
275C2#3063248311yescharacteristic
276C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#489×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
hypercenter#489D251
hypocenterC251#3140
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #30632242C2semidirect
3. #30632049C2semidirect
4. #30632047C2semidirect
5. #30632046C2semidirect
6. #30632043C2semidirect
7. #30632040C2semidirect
8. #30631694C2semidirect
10. #30631639C2semidirect
11. #489×C251C2semidirect
15. #22548715C22semidirect
16. #22548706C22semidirect
30. #22548640C22semidirect
31. #22548639C22semidirect
32. #22548638C22semidirect
36. #22548626C22semidirect
37. C8×Dic251C22semidirect
38. C4⋊Q8×C251C22semidirect
40. #103×C251C22semidirect
42. #97×C251C22semidirect
138. C251#3140semidirect
145. #489D251semidirect
155. C4⋊Q8C2×D251semidirect
174. #103C2×D251semidirect
180. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322422
#124×D2512
#306320492
#306320472
#306320462
#306320432
#306320402
#306316942
#306316902
#306316392
#489×C2512
#3140251