Finite Group Databasebeta

← #14752014

Subgroups — #14752014

Order 2816.

906 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622613C211yesmaximal, characteristic
2C2×#10622610C211yesmaximal, characteristic
3#13068491C212yesmaximal, complemented
4#13068485C212yesmaximal, complemented
5#13068400C212yesmaximal
6#13068394C212yesmaximal
7#13068368C211yesmaximal, characteristic, complemented
8#13068365C211yesmaximal, characteristic, complemented
9#13068303C211yesmaximal, characteristic, complemented
10C4×#66867C211yesmaximal, characteristic, complemented
11C11×#2298C211yesmaximal, characteristic
12C2×C11×#117C2211yescharacteristic
13C22×#4318C2211yescharacteristic
14C23×C11⋊C8C2211yescharacteristic
15C11×#388C414yes—
16C11×#387C414yes—
17C4×C11×C4⋊C4C2212yes—
18C4×C11×C22⋊C4C2212yes—
19C11×#376C2211yescharacteristic
20C43×C11C2211yescharacteristic
21C2×C22⋊(C11⋊C8)C2212yes—
22#10622613C418yescomplemented
23#10622610C418yescomplemented
24C2×C4⋊(C11⋊C8)C2212yes—
25C2×#66867C2212yes—
26C2×#66867C2212yes—
27C2×#66867C2212yes—
28C2×#66867C2211yescharacteristic
29C2×#66867C2211yescharacteristic
30C2×#66867C2211yescharacteristic
31C2×C4×C11⋊C8C2212yes—
32C2×C4×C11⋊C8C2211yescharacteristic
33C2×C4×C11⋊C8C2211yescharacteristic
34C2×C4×C11⋊C8C2211yescharacteristic
35#10622162C2212yes—
36#10622162C2212yes—
37#10622162C2212yes—
38#10622162C2212yes—
39C23×C4×C11C2311yescharacteristic
40C11×#117C2×C414yes—
41C11×#117C2×C414yes—
42C2×C11×C4⋊C4C2×C414yes—
43C2×C11×C4⋊C4C2×C412yes—
44C2×C11×C22⋊C4C2×C414yes—
45C2×C11×C22⋊C4C2×C412yes—
46C2×C42×C11C2×C414yes—
47C2×C42×C11C2312yes—
48C2×C42×C11C2312yes—
49C2×C42×C11C2311yescharacteristic
50C2×C42×C11C2311yescharacteristic
51C2×#4318—24no—
52C2×#4318C2×C414yes—
53C22×C11⋊C8—24no—
54C22×C11⋊C8C2×C414yes—
55C22×C11⋊C8C2312yes—
56C22×C11⋊C8C2312yes—
57C22×C11⋊C8C2311yescharacteristic
58C22×C11⋊C8C2311yescharacteristic
59C22×C11⋊C8C2311yescharacteristic
60C22×C11⋊C8C2311yescharacteristic
61C11×#95C2×C414yes—
62C11×#95C2×C414yes—
63C11×#95C2×C414yes—
64C22⋊(C11⋊C8)C2×C418yes—
65C4⋊(C11⋊C8)C2×C418yes—
66#66867—24no—
67#66867—24no—
68#66867—24no—
69#66867C2×C414yes—
70#66867C2×C414yes—
71#66867C2×C414yes—
72C4×C11⋊C8—24no—
73C4×C11⋊C8C2×C414yes—
74C4×C11⋊C8C2×C414yes—
75C4×C11⋊C8C2×C414yes—
76#20475—111nomaximal, Sylow
77C24×C11C22×C411yescharacteristic
78C22×C4×C11—22no—
79C22×C4×C11—22no—
80C22×C4×C11C22×C414yes—
81C22×C4×C11C22×C414yes—
82C22×C4×C11C22×C411yescharacteristic
83C22×C4×C11C22×C412yes—
84C22×C4×C11C22×C412yes—
85C22×C4×C11C22×C412yes—
86C22×C4×C11C22×C412yes—
87C22×C4×C11C2411yescharacteristic
88C11×C4⋊C4—44no—
89C11×C4⋊C4C4218yes—
90C11×C22⋊C4—44no—
91C11×C22⋊C4C4218yes—
92C42×C11—22no—
93C42×C11—22no—
94C42×C11—22no—
95C42×C11—22no—
96C42×C11—44no—
97C42×C11C22×C412yes—
98C42×C11C22×C412yes—
99C42×C11C22×C412yes—
100C42×C11C22×C412yes—
101#4318—28no—
102C2×C11⋊C8—28no—
103C2×C11⋊C8—22no—
104C2×C11⋊C8—22no—
105C2×C11⋊C8—22no—
106C2×C11⋊C8—22no—
107C2×C11⋊C8—22no—
108C2×C11⋊C8—22no—
109C2×C11⋊C8C22×C412yes—
110C2×C11⋊C8C22×C414yes—
111C2×C11⋊C8C22×C412yes—
112C2×C11⋊C8C22×C414yes—
113C2×C11⋊C8C22×C412yes—
114C2×C11⋊C8C22×C412yes—
115C2×C11⋊C8#4114yes—
116C2×C11⋊C8#4114yes—
117C2×#433—111no—
118C2×#405—111no—
119#2298D1111yescharacteristic
120#1771—112no—
121#1770—112no—
122#1753—112no—
123#1752—112no—
124#1746—111no—
125#1745—111no—
126#1726—111no—
127C4×#97—111no—
128C23×C11—22no—
129C23×C11—24no—
130C23×C11C2×C4212yes—
131C23×C11C23×C411yescharacteristic
132C2×C4×C11—22no—
133C2×C4×C11—24no—
134C2×C4×C11—24no—
135C2×C4×C11—22no—
136C2×C4×C11—22no—
137C2×C4×C11—24no—
138C2×C4×C11—24no—
139C2×C4×C11—24no—
140C2×C4×C11—24no—
141C2×C4×C11—24no—
142C2×C4×C11—24no—
143C2×C4×C11—24no—
144C2×C4×C11C2×C4214yes—
145C2×C4×C11C2×C4214yes—
146C2×C4×C11C2×C4212yes—
147C2×C4×C11C23×C411yescharacteristic
148C2×C4×C11C23×C411yescharacteristic
149C2×C4×C11C2×#4112yes—
150C2×C4×C11C2×#4112yes—
151C11⋊C8—22no—
152C11⋊C8—22no—
153C11⋊C8—24no—
154C11⋊C8—24no—
155C11⋊C8#11714yes—
156C11⋊C8#11714yes—
157C22×M16—111no—
158C23×C8—111no—
159C2×#117C2×D1111yescharacteristic
160#433—118no—
161C2×C4⋊C8—112no—
162C2×C22⋊C8—112no—
163#405—118no—
164C2×#97—111no—
165C2×#97—111no—
166C2×#97—111no—
167C2×#97—112no—
168C2×#97—112no—
169C2×#97—112no—
170C2×C4×C8—111no—
171C2×C4×C8—111no—
172C2×C4×C8—111no—
173C2×C4×C8—112no—
174#388Dic1114yes—
175#387Dic1114yes—
176C4×C4⋊C4C2×D1112yes—
177C4×C22⋊C4C2×D1112yes—
178#376C2×D1111yescharacteristic
179C43C2×D1111yescharacteristic
180#336—112no—
181#336—112no—
182#336—112no—
183#336—112no—
184C22×C11—24no—
185C22×C11—22no—
186C22×C11—44no—
187C22×C11C22×C4211yescharacteristic
188C22×C11C4×#4112yes—
189C22×C11#56814yes—
190C4×C11—24no—
191C4×C11—24no—
192C4×C11—22no—
193C4×C11—44no—
194C4×C11—44no—
195C4×C11C2×#11711yescharacteristic
196C4×C11C2×#11711yescharacteristic
197C4×C11C4×#4112yes—
198C23×C4C22×D1111yescharacteristic
199C2×M16—114no—
200C2×M16—224no—
201C22×C8—111no—
202C22×C8—111no—
203C22×C8—114no—
204C22×C8—111no—
205C22×C8—111no—
206C22×C8—112no—
207C22×C8—112no—
208C22×C8—224no—
209#117C4×D1114yes—
210#117C4×D1114yes—
211C2×C4⋊C4C2×Dic1114yes—
212C2×C4⋊C4C2×Dic1112yes—
213C2×C22⋊C4C2×Dic1114yes—
214C2×C22⋊C4C2×Dic1112yes—
215C2×C42C2×Dic1114yes—
216C2×C42C22×D1112yes—
217C2×C42C22×D1112yes—
218C2×C42C22×D1111yescharacteristic
219C2×C42C22×D1111yescharacteristic
220C4⋊C8—118no—
221C22⋊C8—118no—
222#97—114no—
223#97—114no—
224#97—114no—
225#97—224no—
226#97—224no—
227#97—224no—
228C4×C8—114no—
229C4×C8—114no—
230C4×C8—114no—
231C4×C8—224no—
232#95C2×Dic1114yes—
233#95C2×Dic1114yes—
234#95C2×Dic1114yes—
235C2×C11—42no—
236C2×C11C4×#11711yescharacteristic
237C2×C11#287512yes—
238C2×C11#296714yes—
239C24C22×Dic1111yescharacteristic
240C22×C4—22no—
241C22×C4—22no—
242C22×C4C2×C4×D1112yes—
243C22×C4C2×C4×D1112yes—
244C22×C4C22×Dic1114yes—
245C22×C4C22×Dic1114yes—
246C22×C4C22×Dic1111yescharacteristic
247C22×C4C22×Dic1112yes—
248C22×C4C22×Dic1112yes—
249C22×C4C23×D1111yescharacteristic
250M16—228no—
251C2×C8—112no—
252C2×C8—112no—
253C2×C8—114no—
254C2×C8—114no—
255C2×C8—112no—
256C2×C8—114no—
257C2×C8—112no—
258C2×C8—114no—
259C2×C8—228no—
260C2×C8—222no—
261C2×C8—222no—
262C2×C8—222no—
263C2×C8—222no—
264C2×C8—222no—
265C2×C8—222no—
266C4⋊C4—44no—
267C4⋊C4C4×Dic1118yes—
268C22⋊C4—44no—
269C22⋊C4C4×Dic1118yes—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—44no—
275C42C2×C4×D1112yes—
276C42C2×C4×D1112yes—
277C42C2×C4×D1112yes—
278C42C2×C4×D1112yes—
279C11#2047511yescharacteristic, complemented, Sylow
280C23—24no—
281C23—22no—
282C23C2×C4×Dic1112yes—
283C23C23×Dic1111yescharacteristic
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4C2×C4×Dic1114yes—
297C2×C4C2×C4×Dic1114yes—
298C2×C4C2×C4×Dic1112yes—
299C2×C4C22×C4×D1111yescharacteristic
300C2×C4C22×C4×D1111yescharacteristic
301C2×C4#41×D1112yes—
302C2×C4#41×D1112yes—
303C8—114no—
304C8—114no—
305C8—222no—
306C8—222no—
307C8—224no—
308C8—224no—
309C22—24no—
310C22—22no—
311C22—44no—
312C22C22×C4×Dic1111yescharacteristic
313C22#1062335014yes—
314C22#41×Dic1112yes—
315C4—24no—
316C4—24no—
317C4—22no—
318C4—44no—
319C4—44no—
320C4D11×#11711yescharacteristic
321C4D11×#11711yescharacteristic
322C4#1062305312yes—
323C2—42no—
324C2#117×Dic1111yescharacteristic
325C2#1307704412yes—
326C2#1307760314yes—
subgroupG/H
center, FrattiniC22×C4C23×D11
commutatorC22×C11C22×C42
FittingC11×#2298C2
socleC23×C11C23×C4
prime radicalC24×C11C22×C4
cosocleC22×C4×C11C24
hypercenter#2298D11
hypocenterC11#20475
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C22×C4 ◁ #2298

Complements

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

NH ≅ G/Ntype
3. #13068491C2semidirect
4. #13068485C2semidirect
7. #13068368C2semidirect
8. #13068365C2semidirect
9. #13068303C2semidirect
10. C4×#66867C2semidirect
22. #10622613C4semidirect
23. #10622610C4semidirect
279. C11#20475semidirect

Maximal subgroups

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

subgroupindex
C2×#106226132
C2×#106226102
#130684912
#130684852
#130684002
#130683942
#130683682
#130683652
#130683032
C4×#668672
C11×#22982
#2047511