Finite Group Databasebeta

← #45667458

Subgroups — #45667458

Order 32896.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22551560C211yesmaximal, characteristic
2#30656640C211yesmaximal, characteristic, complemented
3#30656637C212yesmaximal, complemented
4#30656631C212yesmaximal, complemented
5#30656629C212yesmaximal, complemented
6#30656626C212yesmaximal, complemented
7#30656616C212yesmaximal, complemented
8#30656592C212yesmaximal, complemented
9#386×C257C211yesmaximal, characteristic, complemented
10C24×D257C2211yescharacteristic
11C22×C22⋊D257C2212yes—
12#22551560C418yescomplemented
13C2×C4⋊Dic257C2211yescharacteristic, complemented
14C2×#16861597C2212yescomplemented
15C2×C22⋊Dic257C2212yes—
16C2×C22⋊Dic257C2211yescharacteristic
17C2×C22⋊Dic257C2212yescomplemented
18C2×#16861595C2212yes—
19C2×#16861595C2212yes—
20C2×#16861595C2212yes—
21C2×#16861595C2212yes—
22C2×#16861595C2212yes—
23C2×C4×Dic257C2211yescharacteristic, complemented
24C2×C4×Dic257C2212yescomplemented
25#22551499C2212yescomplemented
26#22551499C2212yescomplemented
27C22×D4×C257C2211yescharacteristic
28C23×C4×C257C2211yescharacteristic, complemented
29C2×C22⋊C4×C257C2212yes—
30C2×C22⋊C4×C257C2211yescharacteristic, complemented
31#95×C257C2212yescomplemented
32C23×D257—42no—
33C23×D257—41no—
34C23×D257—41no—
35C23×D257—42no—
36C23×D257C2×C414yescomplemented
37C23×D257C2312yes—
38C2×C22⋊D257—42no—
39C2×C22⋊D257—42no—
40C2×C22⋊D257C2×C418yes—
41C22×Dic257C2312yes—
42C22×Dic257C2312yes—
43C22×Dic257C2312yes—
44C4⋊Dic257—41no—
45#16861597—42no—
46C22⋊Dic257—42no—
47C22⋊Dic257—42no—
48C22⋊Dic257C2×C414yescomplemented
49#16861595—42no—
50#16861595—42no—
51#16861595—42no—
52#16861595—42no—
53#16861595C2×C418yescomplemented
54C4×Dic257—41no—
55C4×Dic257—42no—
56C24×C257C2311yescharacteristic
57C24×C257C2311yescharacteristic
58C2×D4×C257—41no—
59C2×D4×C257—41no—
60C2×D4×C257C2×C414yes—
61C22×C4×C257—42no—
62C22×C4×C257—41no—
63C22×C4×C257C2312yes—
64C22×C4×C257C2312yes—
65C22×C4×C257C2311yescharacteristic
66C22⋊C4×C257—42no—
67C22⋊C4×C257—41no—
68C22×D257—22no—
69C22×D257—22no—
70C22×D257—22no—
71C22×D257—22no—
72C22×D257—22no—
73C22×D257—22no—
74C22×D257—48no—
75C22×D257—44no—
76C22×D257—48no—
77C22×D257—44no—
78C22×D257—44no—
79C22×D257C22×C414yes—
80C22⋊D257—48no—
81C2×Dic257—22no—
82C2×Dic257—22no—
83C2×Dic257—22no—
84C2×Dic257—22no—
85C2×Dic257—22no—
86C2×Dic257—22no—
87C2×Dic257—22no—
88C2×Dic257—22no—
89C2×Dic257C22×C414yes—
90C23×C257—21no—
91C23×C257—22no—
92C23×C257—21no—
93C23×C257—22no—
94C23×C257—21no—
95C23×C257—21no—
96C23×C257—22no—
97C23×C257—22no—
98C23×C257C22×C412yes—
99C23×C257C22×C412yes—
100C23×C257C2411yescharacteristic
101D4×C257—44no—
102C2×C4×C257—21no—
103C2×C4×C257—21no—
104C2×C4×C257—22no—
105C2×C4×C257—22no—
106C2×C4×C257—22no—
107C2×C4×C257—22no—
108C2×C4×C257—22no—
109C2×C4×C257—22no—
110C2×C4×C257—44no—
111C2×C4×C257C22×C412yes—
112C22×C257—22no—
113C22×C257—22no—
114C22×C257—24no—
115C22×C257—24no—
116C22×C257—42no—
117C22×C257—41no—
118C22×C257—42no—
119C22×C257—41no—
120C22×C257—41no—
121C22×C257—41no—
122C22×C257C23×C411yescharacteristic
123C22×C257H(5,𝔽2)11yescharacteristic
124C22×C257H(5,𝔽2)12yes—
125C22×C257H(5,𝔽2)11yescharacteristic
126C22×C257H(5,𝔽2)12yes—
127C2×D257—24no—
128C2×D257—24no—
129C2×D257—24no—
130C2×D257—416no—
131C2×D257—42no—
132C2×D257—42no—
133C2×D257—42no—
134C2×D257—42no—
135C4×C257—22no—
136C4×C257—42no—
137C4×C257—42no—
138Dic257—24no—
139Dic257—42no—
140Dic257—42no—
141C2×C257—42no—
142C2×C257—42no—
143C2×C257#51812yes—
144C2×C257#51811yescharacteristic
145C2×C257#56012yes—
146C2×C257#56011yescharacteristic
147C2×C257H(3,𝔽4)11yescharacteristic
148D257—44no—
149C257#237511yescharacteristic, complemented, Sylow
150#2375—2571nomaximal, Sylow
151C2×#120—2571no—
152#390—2571no—
153#390—2572no—
154#388—2572no—
155#388—2572no—
156#388—2572no—
157#386—2572no—
158#386D25711yescharacteristic, complemented
159#379—2572no—
160C25—2571no—
161C22×D4—2572no—
162C22×D4C2×D25711yescharacteristic
163C23×C4C2×D25711yescharacteristic, complemented
164#120—2578no—
165C2×C4⋊C4—2572no—
166C2×C4⋊C4—2571no—
167C2×C22⋊C4—2571no—
168C2×C22⋊C4—2572no—
169C2×C22⋊C4—2572no—
170C2×C22⋊C4—2572no—
171C2×C22⋊C4—2572no—
172C2×C22⋊C4—2572no—
173C2×C22⋊C4—2572no—
174C2×C22⋊C4—2572no—
175C2×C22⋊C4C2×D25712yes—
176C2×C22⋊C4C2×D25711yescharacteristic, complemented
177C2×C42—2572no—
178C2×C42—2571no—
179#95—2572no—
180#95—2572no—
181#95C2×D25712yescomplemented
182C24—2572no—
183C24—2574no—
184C24—10282no—
185C24—10281no—
186C24—10281no—
187C24—10282no—
188C24C22×D25711yescharacteristic
189C24C22×D25711yescharacteristic
190C2×D4—2578no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—10282no—
194C2×D4—10282no—
195C2×D4C4×D25714yes—
196C22×C4—2572no—
197C22×C4—2572no—
198C22×C4—2572no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4C22×D25712yes—
202C22×C4C22×D25712yes—
203C22×C4C22×D25711yescharacteristic
204C4⋊C4—10282no—
205C4⋊C4—10281no—
206C22⋊C4—2574no—
207C22⋊C4—2578no—
208C22⋊C4—42no—
209C22⋊C4—41no—
210C22⋊C4—10282no—
211C22⋊C4—10282no—
212C22⋊C4—10282no—
213C22⋊C4—10282no—
214C22⋊C4—10282no—
215C22⋊C4—10282no—
216C42—10281no—
217C42—10282no—
218C23—2574no—
219C23—21no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—5142no—
228C23—5142no—
229C23—5142no—
230C23—5142no—
231C23—5142no—
232C23—5142no—
233C23—10288no—
234C23—10284no—
235C23—10288no—
236C23—10284no—
237C23—10284no—
238C23C2×C4×D25712yes—
239C23C2×C4×D25712yes—
240C23C23×D25711yescharacteristic
241D4—44no—
242D4—10288no—
243C2×C4—2574no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—5142no—
253C2×C4—5142no—
254C2×C4—5142no—
255C2×C4—5142no—
256C2×C4—5142no—
257C2×C4—5142no—
258C2×C4—5142no—
259C2×C4—5142no—
260C2×C4—44no—
261C2×C4C2×C4×D25712yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—5144no—
267C22—5144no—
268C22—5144no—
269C22—42no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—102816no—
276C22—10282no—
277C22—10282no—
278C22—10282no—
279C22—10282no—
280C22C22×C4×D25711yescharacteristic
281C22#2255163712yes—
282C22#2255163711yescharacteristic
283C22#2255163711yescharacteristic
284C22#2255163812yes—
285C4—22no—
286C4—5144no—
287C4—42no—
288C4—42no—
289C4—10282no—
290C4—10282no—
291C2—42no—
292C2—42no—
293C2—10284no—
294C2#3065720912yes—
295C2#3065721011yescharacteristic
296C2#3065747512yes—
297C2#3065747711yescharacteristic
298C2#3065748011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#386×C257C2
socle, cosocleC23×C257C24
prime radicalC2×#22551560C2
hypercenter#386D257
hypocenterC257#2375
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #30656640C2semidirect
3. #30656637C2semidirect
4. #30656631C2semidirect
5. #30656629C2semidirect
6. #30656626C2semidirect
7. #30656616C2semidirect
8. #30656592C2semidirect
9. #386×C257C2semidirect
12. #22551560C4semidirect
13. C2×C4⋊Dic257C22semidirect
14. C2×#16861597C22semidirect
17. C2×C22⋊Dic257C22semidirect
23. C2×C4×Dic257C22semidirect
24. C2×C4×Dic257C22semidirect
25. #22551499C22semidirect
26. #22551499C22semidirect
28. C23×C4×C257C22semidirect
30. C2×C22⋊C4×C257C22semidirect
31. #95×C257C22semidirect
36. C23×D257C2×C4semidirect
48. C22⋊Dic257C2×C4semidirect
53. #16861595C2×C4semidirect
149. C257#2375semidirect
158. #386D257semidirect
163. C23×C4C2×D257semidirect
176. C2×C22⋊C4C2×D257semidirect
181. #95C2×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515602
#306566402
#306566372
#306566312
#306566292
#306566262
#306566162
#306565922
#386×C2572
#2375257