Finite Group Databasebeta

← #44560280

Subgroups — #44560280

Order 32128.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#30632198C211yesmaximal, characteristic
2C2×#22548674C211yesmaximal, characteristic, complemented
3#30632029C212yesmaximal, complemented
4#30632023C211yesmaximal, characteristic, complemented
5#30632018C211yesmaximal, characteristic, complemented
6#30632012C212yesmaximal, complemented
7#30632006C211yesmaximal, characteristic, complemented
8#30632001C211yesmaximal, characteristic, complemented
9#30631725C212yesmaximal, complemented
10#30631684C212yesmaximal, complemented
11C2×C22⋊C8×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13#22548698—21no—
14#22548694C2211yescharacteristic
15#22548694C2211yescharacteristic
16#22548689C2211yescharacteristic
17#22548689C2211yescharacteristic
18#22548684—21no—
19C4×C22⋊D251—21no—
20C4×C22⋊D251—21no—
21#22548674—21no—
22#22548674—21no—
23#22548674C2212yes—
24#22548674C2211yescharacteristic
25#22548674C2212yes—
26C2×C4⋊Dic251C2211yescharacteristic
27C2×C4⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251—21no—
29C2×C4×Dic251—21no—
30#22548649C2212yescomplemented
31#22548649C2212yescomplemented
32#22548647C2212yescomplemented
33#22548647C2212yescomplemented
34#22548640C2212yescomplemented
35#22548640C2212yescomplemented
36#22548637C2212yescomplemented
37#22548637C2212yescomplemented
38C23×C4×C251C2211yescharacteristic
39C22×C8×C251C2212yes—
40C22⋊C8×C251C2211yescharacteristic
41C22⋊C8×C251C2211yescharacteristic
42C22⋊C8×C251C2212yes—
43#16860810—24no—
44C2×Dic502C2311yescharacteristic
45C2×D1004C2311yescharacteristic
46C2×C22⋊D251—21no—
47C2×C22⋊D251—21no—
48C2×C22⋊D251—21no—
49C22×Dic251—21no—
50C22×Dic251—21no—
51C2×C4×D251—21no—
52C4⋊Dic251—21no—
53C4⋊Dic251—21no—
54C4⋊Dic251C2312yes—
55C4⋊Dic251C2312yes—
56C4⋊Dic251C2311yescharacteristic
57C4⋊Dic251C2311yescharacteristic
58Dic251⋊C4—21no—
59Dic251⋊C4—21no—
60C22⋊Dic251—22no—
61C22⋊Dic251—22no—
62C22⋊Dic251—21no—
63C22⋊Dic251—21no—
64C22⋊Dic251—21no—
65(C2×D251)⋊C4—21no—
66(C2×D251)⋊C4—21no—
67C4×Dic251—22no—
68C4×Dic251—22no—
69C4×Dic251—21no—
70C24×C251D411yescharacteristic
71C22×C4×C251—24no—
72C22×C4×C251D411yescharacteristic
73C22×C4×C251D411yescharacteristic
74C22×C4×C251D411yescharacteristic
75C22×C4×C251C2311yescharacteristic
76C22×C4×C251C2311yescharacteristic
77C22×C4×C251C2311yescharacteristic
78C2×C8×C251—24no—
79C2×C8×C251C2312yes—
80C2×C8×C251C2312yes—
81C22×D251—21no—
82Dic502—22no—
83D1004—22no—
84C22⋊D251—42no—
85C22⋊D251—42no—
86C22⋊D251—42no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251—21no—
90C2×Dic251—21no—
91C2×Dic251—21no—
92C2×Dic251—42no—
93C2×Dic251—42no—
94C4×D251—42no—
95C23×C251—22no—
96C23×C251—22no—
97C23×C251—41no—
98C23×C251C2×D411yescharacteristic
99C23×C251C2×D411yescharacteristic
100C23×C251C2×D411yescharacteristic
101C2×C4×C251—22no—
102C2×C4×C251—22no—
103C2×C4×C251—22no—
104C2×C4×C251—22no—
105C2×C4×C251—22no—
106C2×C4×C251C2×D411yescharacteristic
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251#4114yes—
110C2×C4×C251C2411yescharacteristic
111C8×C251—22no—
112C8×C251—22no—
113C22×C251—21no—
114C22×C251—21no—
115C22×C251—21no—
116C22×C251—21no—
117C22×C251—21no—
118C22×C251—21no—
119C22×C251—21no—
120C22×C251—44no—
121C22×C251#12312yes—
122C22×C251#13512yes—
123C22×C251C22×D411yescharacteristic
124C2×D251—41no—
125C2×D251—41no—
126C2×D251—41no—
127C4×C251—21no—
128C4×C251—21no—
129C4×C251#12312yes—
130C4×C251C2×#4112yes—
131Dic251—41no—
132Dic251—42no—
133Dic251—41no—
134Dic251—42no—
135Dic251—41no—
136C2×C251—22no—
137C2×C251—22no—
138C2×C251—41no—
139C2×C251C2×#12311yescharacteristic
140C2×C251C2×#13511yescharacteristic
141C2×C251#57511yescharacteristic
142D251—81no—
143C251#309111yescharacteristic, complemented, Sylow
144#3091—2511nomaximal, Sylow
145#525—2511no—
146C2×#117—2511no—
147#485—2512no—
148#484—2511no—
149#483—2511no—
150#482—2512no—
151#481—2511no—
152#480—2511no—
153#427—2512no—
154#416—2512no—
155C2×C22⋊C8D25111yescharacteristic, complemented
156C2×#41—2511no—
157C23×C4C2×D25111yescharacteristic
158C22×C8C2×D25112yes—
159#123—5021no—
160C22⋊Q8—2511no—
161C22⋊Q8—2511no—
162#121—2511no—
163#121—2511no—
164#120—5021no—
165C4×D4—5021no—
166C4×D4—5021no—
167#117—5021no—
168#117—5021no—
169#117—2512no—
170#117—2512no—
171#117—2511no—
172C2×C4⋊C4—2511no—
173C2×C4⋊C4—2511no—
174C2×C22⋊C4—5021no—
175C2×C42—5021no—
176#107—2512no—
177#107—2512no—
178#106—2512no—
179#106—2512no—
180#103—2512no—
181#103—2512no—
182#102—2512no—
183#102—2512no—
184C22⋊C8C2×D25111yescharacteristic
185C22⋊C8C2×D25111yescharacteristic
186C22⋊C8C2×D25112yes—
187C24D100411yescharacteristic
188#41—5024no—
189C2×Q8—2511no—
190C2×D4—5021no—
191C2×D4—5021no—
192C2×D4—5021no—
193C2×D4—2511no—
194C22×C4—24no—
195C22×C4—5021no—
196C22×C4—5021no—
197C22×C4—5021no—
198C22×C4D100411yescharacteristic
199C22×C4D100411yescharacteristic
200C22×C4D100411yescharacteristic
201C22×C4C22×D25111yescharacteristic
202C22×C4C22×D25111yescharacteristic
203C22×C4C22×D25111yescharacteristic
204C2×C8—24no—
205C2×C8C22×D25112yes—
206C2×C8C22×D25112yes—
207C4⋊C4—5021no—
208C4⋊C4—5021no—
209C4⋊C4—5021no—
210C4⋊C4—5021no—
211C4⋊C4—2512no—
212C4⋊C4—2512no—
213C4⋊C4—2511no—
214C4⋊C4—2511no—
215C22⋊C4—5022no—
216C22⋊C4—5022no—
217C22⋊C4—5021no—
218C22⋊C4—5021no—
219C22⋊C4—5021no—
220C22⋊C4—5021no—
221C22⋊C4—5021no—
222C42—5022no—
223C42—5022no—
224C42—5021no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—5021no—
229C23C2×D100411yescharacteristic
230C23C2×D100411yescharacteristic
231C23C2×D100411yescharacteristic
232Q8—5022no—
233D4—10042no—
234D4—10042no—
235D4—10042no—
236D4—5022no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—10042no—
243C2×C4—10042no—
244C2×C4—10042no—
245C2×C4—5022no—
246C2×C4—5022no—
247C2×C4—5021no—
248C2×C4—5021no—
249C2×C4—5021no—
250C2×C4C2×D100411yescharacteristic
251C2×C4C2×D100411yescharacteristic
252C2×C4C2×D100411yescharacteristic
253C2×C4#1686080914yes—
254C2×C4C23×D25111yescharacteristic
255C8—22no—
256C8—22no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—10041no—
266C22—10041no—
267C22—10041no—
268C22#2254869912yes—
269C22#2254873712yes—
270C22C22×D100411yescharacteristic
271C4—21no—
272C4—21no—
273C4—10042no—
274C4—10041no—
275C4—10042no—
276C4—10041no—
277C4—10041no—
278C4#2254869912yes—
279C4C2×#1686080912yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—20081no—
284C2C2×#2254869911yescharacteristic
285C2C2×#2254873711yescharacteristic
286C2#3063248711yescharacteristic
subgroupG/H
centerC2×C4C2×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C22⋊C8×C251C2
socleC22×C251C22×D4
prime radical#30632198C2
hypercenterC2×C22⋊C8D251
hypocenterC251#3091
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#22548674C2semidirect
3. #30632029C2semidirect
4. #30632023C2semidirect
5. #30632018C2semidirect
6. #30632012C2semidirect
7. #30632006C2semidirect
8. #30632001C2semidirect
9. #30631725C2semidirect
10. #30631684C2semidirect
11. C2×C22⋊C8×C251C2semidirect
30. #22548649C22semidirect
31. #22548649C22semidirect
32. #22548647C22semidirect
33. #22548647C22semidirect
34. #22548640C22semidirect
35. #22548640C22semidirect
36. #22548637C22semidirect
37. #22548637C22semidirect
143. C251#3091semidirect
155. C2×C22⋊C8D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
C2×#225486742
#306320292
#306320232
#306320182
#306320122
#306320062
#306320012
#306317252
#306316842
C2×C22⋊C8×C2512
#3091251