Finite Group Databasebeta

← #14763569

Subgroups — #14763569

Order 2816.

1022 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623228C211yesmaximal, characteristic
2C2×#10622303C211yesmaximal, characteristic
3#13071241C211yesmaximal, characteristic
4#13069586C211yesmaximal, characteristic, complemented
5#13069435C211yesmaximal, characteristic, complemented
6#13069418C211yesmaximal, characteristic, complemented
7#13068927C212yesmaximal
8#13068922C212yesmaximal
9#13068630C212yesmaximal, complemented
10#13068620C212yesmaximal, complemented
11C11×#1991C211yesmaximal, characteristic, complemented
12C2×C11×#125C2211yescharacteristic
13#10623228—24no—
14#10623228C418yescomplemented
15C2×#66951—22no—
16C2×#66950C2212yes—
17C2×#66945C2211yescharacteristic
18C2×C4⋊C4×D11C2211yescharacteristic
19C2×#66943—22no—
20C2×#66940C2212yes—
21C2×#66925C2211yescharacteristic
22C2×#66922C2211yescharacteristic
23C2×C4×C8×C11C2211yescharacteristic
24C11×#382C2211yescharacteristic
25C2×(C2×D11)⋊C8C2212yes—
26C2×#66880C2212yes—
27C2×#66880C2212yes—
28C2×Dic11⋊C8C2212yes—
29#10622590C2211yescharacteristic
30#10622589C2211yescharacteristic
31C2×#66874C2212yes—
32C2×#66874C2212yes—
33#10622548C2211yescharacteristic
34C2×#66867C2211yescharacteristic
35#10622303—24no—
36#10622281—22no—
37#10622260—22no—
38C11×#340C2212yes—
39C11×#340C2212yes—
40#10622167C2212yes—
41#10622167C2212yes—
42C22×C4×D11—21no—
43C22×C4×D11C2311yescharacteristic
44C22×Dic22C2312yes—
45C22×C8×C11C2312yes—
46C11×#125—22no—
47C11×#125C2×C414yescomplemented
48C2×C11×C4⋊C4—22no—
49C2×C11×C4⋊C4C2311yescharacteristic
50C2×C11×C4⋊C4C2311yescharacteristic
51C2×C11×C4⋊C4C2312yes—
52C2×C42×C11C2311yescharacteristic
53C2×(C2×D11)⋊C4—22no—
54C2×(C2×D11)⋊C4—21no—
55C2×(C2×D11)⋊C4D412yes—
56C2×C4⋊Dic11—22no—
57C2×C4⋊Dic11—22no—
58C2×C4⋊Dic11—22no—
59C2×C4⋊Dic11C2311yescharacteristic
60C2×C4⋊Dic11C2311yescharacteristic
61C2×Dic11⋊C4—22no—
62C2×Dic11⋊C4—21no—
63C2×Dic11⋊C4—22no—
64C2×Dic11⋊C4D412yes—
65C2×C4×Dic11—22no—
66C2×C4×Dic11—21no—
67C2×C4×Dic11C2311yescharacteristic
68C22×C11⋊C8C2312yes—
69#66951—216no—
70#66950—24no—
71#66950C2×C418yes—
72#66945—42no—
73C4⋊C4×D11—42no—
74#66943—216no—
75#66940—24no—
76#66940C2×C418yescomplemented
77#66925C2×C414yes—
78#66925D414yes—
79#66922—22no—
80#66922C2×C414yescomplemented
81C4×C8×C11—22no—
82C11×#95—22no—
83C11⋊#95—22no—
84(C2×D11)⋊C8—24no—
85#66880—42no—
86#66880—42no—
87Dic11⋊C8—24no—
88#66874—42no—
89#66874—42no—
90#66867—22no—
91#21873—111nomaximal, Sylow
92C23×D11C2×D411yescharacteristic
93C22×C4×C11—22no—
94C22×C4×C11—21no—
95C22×C4×C11—21no—
96C22×C4×C11C2×D412yes—
97C22×C4×C11C2411yescharacteristic
98C22×Dic11—22no—
99C22×Dic11—21no—
100C22×Dic11—21no—
101C22×Dic11C2×D412yes—
102C22×Dic11C2×D411yescharacteristic
103C2×C4×D11—22no—
104C2×C4×D11—22no—
105C2×C4×D11—42no—
106C2×C4×D11—82no—
107C2×C4×D11C22×C412yes—
108C2×C4×D11C2×D412yes—
109C2×Dic22—22no—
110C2×Dic22—22no—
111C2×Dic22—22no—
112C2×Dic22—22no—
113C2×Dic22—22no—
114C2×Dic22C22×C414yes—
115C2×C8×C11—22no—
116C2×C8×C11—22no—
117C2×C8×C11—22no—
118C11×C4⋊C4—28no—
119C11×C4⋊C4—22no—
120C11×C4⋊C4—22no—
121C11×C4⋊C4—22no—
122C11×C4⋊C4C22×C414yes—
123C42×C11C22×C412yes—
124C42×C11C2×D412yes—
125(C2×D11)⋊C4—28no—
126(C2×D11)⋊C4—42no—
127(C2×D11)⋊C4C22⋊C418yes—
128C4⋊Dic11—28no—
129C4⋊Dic11—28no—
130C4⋊Dic11—22no—
131C4⋊Dic11—22no—
132C4⋊Dic11—82no—
133Dic11⋊C4—28no—
134Dic11⋊C4—42no—
135Dic11⋊C4—82no—
136Dic11⋊C4C22⋊C418yes—
137C4×Dic11—28no—
138C4×Dic11—42no—
139C4×Dic11C22×C412yes—
140C4×Dic11C2×D412yes—
141C2×C11⋊C8—22no—
142C2×C11⋊C8—22no—
143C2×C11⋊C8—22no—
144C2×#541—111no—
145C2×#433—111no—
146#2317—111no—
147#1992—111no—
148#1991D1111yescharacteristic, complemented
149#1960—111no—
150#1956—111no—
151#1873—112no—
152#1872—112no—
153#1805—112no—
154#1803—112no—
155C23×C11C22×D411yescharacteristic
156C22×D11—21no—
157C22×D11—21no—
158C22×D11—22no—
159C22×D11—22no—
160C22×D11C2×C22⋊C412yes—
161C2×C4×C11—22no—
162C2×C4×C11—22no—
163C2×C4×C11—24no—
164C2×C4×C11—42no—
165C2×C4×C11—42no—
166C2×C4×C11—41no—
167C2×C4×C11—41no—
168C2×C4×C11—42no—
169C2×C4×C11—42no—
170C2×C4×C11C2×C22⋊C414yes—
171C2×C4×C11C23×C411yescharacteristic
172C2×C4×C11C22×D411yescharacteristic
173C2×C4×C11H(5,𝔽2)12yes—
174C2×C4×C112−1+412yes—
175C2×Dic11—21no—
176C2×Dic11—21no—
177C2×Dic11—22no—
178C2×Dic11—22no—
179C2×Dic11—24no—
180C2×Dic11—42no—
181C2×Dic11—42no—
182C2×Dic11—41no—
183C2×Dic11—41no—
184C2×Dic11—42no—
185C2×Dic11—42no—
186C2×Dic11C2×C22⋊C414yes—
187C2×Dic11C2×C22⋊C412yes—
188C4×D11—24no—
189C4×D11—24no—
190C4×D11—84no—
191Dic22—24no—
192Dic22—24no—
193Dic22—42no—
194Dic22—42no—
195C8×C11—42no—
196C11⋊C8—42no—
197#541—118no—
198#541—224no—
199C2×C4⋊Q8—111no—
200C2×#126—222no—
201C2×#125—222no—
202C2×#125C2×D1111yescharacteristic
203C2×C22⋊Q8—112no—
204C2×C4×Q8—112no—
205C2×#117—111no—
206C2×#117—111no—
207C22×C4⋊C4—111no—
208#433—224no—
209C2×C4⋊C8—112no—
210C2×#103—112no—
211C2×#103—112no—
212C2×#103—112no—
213C2×#103—112no—
214C2×C22⋊C8—112no—
215C2×#97—111no—
216C2×C4×C8C2×D1111yescharacteristic
217#389—222no—
218#385—222no—
219#384—111no—
220#384—111no—
221#382—111no—
222#382C2×D1111yescharacteristic
223#340—112no—
224#340—112no—
225#340C2×D1112yes—
226#340C2×D1112yes—
227C22×C11C22×C22⋊C411yescharacteristic
228C22×C11#53511yescharacteristic
229C22×C11#53611yescharacteristic
230C22×C11#57712yes—
231C22×C11#57812yes—
232C2×D11—22no—
233C2×D11—24no—
234C2×D11—41no—
235C2×D11—41no—
236C2×D11—42no—
237C4×C11—24no—
238C4×C11—44no—
239C4×C11—81no—
240C4×C11—81no—
241C4×C11#52012yes—
242C4×C11#53712yes—
243Dic11—22no—
244Dic11—24no—
245Dic11—44no—
246Dic11—81no—
247Dic11—81no—
248C22×Q8—112no—
249C23×C4—111no—
250C23×C4—221no—
251C22×C8—112no—
252C22×C8C22×D1112yes—
253C4⋊Q8—114no—
254C4⋊Q8—222no—
255#126—2216no—
256#125—22no—
257#125—2216no—
258#125C4×D1114yescomplemented
259C22⋊Q8—118no—
260C22⋊Q8—224no—
261C4×Q8—118no—
262C4×Q8—224no—
263#117—114no—
264#117—114no—
265#117—442no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—112no—
268C2×C4⋊C4—111no—
269C2×C4⋊C4—111no—
270C2×C4⋊C4—222no—
271C2×C4⋊C4—222no—
272C2×C4⋊C4—222no—
273C2×C4⋊C4—221no—
274C2×C4⋊C4—222no—
275C2×C4⋊C4—222no—
276C2×C4⋊C4—442no—
277C2×C4⋊C4C22×D1111yescharacteristic
278C2×C4⋊C4C22×D1111yescharacteristic
279C2×C4⋊C4C22×D1112yes—
280C2×C22⋊C4—112no—
281C2×C22⋊C4—222no—
282C2×C22⋊C4—221no—
283C2×C42—111no—
284C2×C42—222no—
285C2×C42—221no—
286C2×C42C22×D1111yescharacteristic
287C4⋊C8—224no—
288#103—442no—
289#103—442no—
290#103—442no—
291#103—442no—
292C22⋊C8—224no—
293#97—222no—
294C4×C8—22no—
295#95—22no—
296#95—222no—
297C2×C11#231111yescharacteristic
298C2×C11#289812yes—
299C2×C11#322211yescharacteristic
300C2×C11#322311yescharacteristic
301C2×C11#324212yes—
302D11—42no—
303C24—111no—
304C2×Q8—114no—
305C2×Q8—222no—
306C2×Q8—222no—
307C2×Q8—222no—
308C2×Q8—222no—
309C2×Q8—222no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—112no—
314C22×C4—111no—
315C22×C4—112no—
316C22×C4—112no—
317C22×C4—222no—
318C22×C4—222no—
319C22×C4—222no—
320C22×C4—221no—
321C22×C4—221no—
322C22×C4—442no—
323C22×C4—882no—
324C22×C4D4×D1112yes—
325C22×C4C23×D1111yescharacteristic
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—222no—
330C2×C8—222no—
331C2×C8—222no—
332C4⋊C4—28no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—118no—
337C4⋊C4—228no—
338C4⋊C4—228no—
339C4⋊C4—228no—
340C4⋊C4—222no—
341C4⋊C4—222no—
342C4⋊C4—442no—
343C4⋊C4—882no—
344C4⋊C4—882no—
345C4⋊C4C2×C4×D1114yes—
346C22⋊C4—118no—
347C22⋊C4—228no—
348C22⋊C4—442no—
349C42—112no—
350C42—112no—
351C42—228no—
352C42—442no—
353C42C2×C4×D1112yes—
354C42D4×D1112yes—
355C11#2187311yescharacteristic, complemented, Sylow
356C23—112no—
357C23—221no—
358C23—222no—
359C23—222no—
360C23—221no—
361C23C2×D4×D1111yescharacteristic
362Q8—224no—
363Q8—224no—
364Q8—442no—
365Q8—442no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—114no—
376C2×C4—112no—
377C2×C4—221no—
378C2×C4—222no—
379C2×C4—222no—
380C2×C4—221no—
381C2×C4—224no—
382C2×C4—224no—
383C2×C4—224no—
384C2×C4—442no—
385C2×C4—442no—
386C2×C4—442no—
387C2×C4—442no—
388C2×C4—441no—
389C2×C4—441no—
390C2×C4—884no—
391C2×C4C22⋊C4×D1114yes—
392C2×C4C22×C4×D1111yescharacteristic
393C2×C4C2×D4×D1111yescharacteristic
394C2×C4#6704012yes—
395C2×C4#6704212yes—
396C8—42no—
397C8—442no—
398C22—222no—
399C22—224no—
400C22—442no—
401C22—441no—
402C22—441no—
403C22C2×C22⋊C4×D1111yescharacteristic
404C22#1062320511yescharacteristic
405C22#1062325511yescharacteristic
406C22#1062330112yes—
407C22#1062330212yes—
408C4—24no—
409C4—44no—
410C4—81no—
411C4—81no—
412C4—222no—
413C4—224no—
414C4—444no—
415C4—881no—
416C4—881no—
417C4#1062310512yes—
418C4#1062316012yes—
419C2—442no—
420C2#1307120711yescharacteristic
421C2#1307714512yes—
422C2#1307969111yescharacteristic
423C2#1307970211yescharacteristic
424C2#1307988712yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1991C2
socleC23×C11C22×D4
prime radicalC23×D11C2×D4
cosocleC22×C4×C11C24
hypercenter#1991D11
hypocenterC11#21873
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
4. #13069586C2semidirect
5. #13069435C2semidirect
6. #13069418C2semidirect
9. #13068630C2semidirect
10. #13068620C2semidirect
11. C11×#1991C2semidirect
14. #10623228C4semidirect
47. C11×#125C2×C4semidirect
76. #66940C2×C4semidirect
80. #66922C2×C4semidirect
148. #1991D11semidirect
258. #125C4×D11semidirect
355. C11#21873semidirect

Maximal subgroups

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

subgroupindex
C2×#106232282
C2×#106223032
#130712412
#130695862
#130694352
#130694182
#130689272
#130689222
#130686302
#130686202
C11×#19912
#2187311