Finite Group Databasebeta

← #14960779

Subgroups — #14960779

Order 2816.

2450 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66936C211yesmaximal, characteristic
2#13076900C212yesmaximal, complemented
3#13075033C218yesmaximal, complemented
4#13074955C218yesmaximal, complemented
5#13074431C212yesmaximal, complemented
6#13073775C212yesmaximal, complemented
7#13073095C212yesmaximal, complemented
8#13072894C212yesmaximal, complemented
9#13071943C212yesmaximal
10#13071214C211yesmaximal, characteristic, complemented
11C2×C11×#543C211yesmaximal, characteristic, complemented
12C23×C22⋊D11C2211yescharacteristic
13C23×C4×D11C2211yescharacteristic
14C11×#543—28no—
15C2×C11×C22⋊Q8C2212yes—
16C2×C11×C22⋊Q8C2212yes—
17C2×C11×C22⋊Q8C2214yescomplemented
18C2×C11×C22⋊Q8C2214yescomplemented
19C22×C11×C22⋊C4C2212yes—
20C22×C11×C22⋊C4C2211yescharacteristic
21C22×C22⋊Dic11C2211yescharacteristic
22C2×#66999C2214yescomplemented
23C2×#66982C2214yes—
24C22×(C2×D11)⋊C4C2211yescharacteristic
25C22×C4⋊Dic11C2211yescharacteristic
26C22×Dic11⋊C4C2211yescharacteristic
27C2×#66936—28no—
28C2×#66936C2218yes—
29#10622860C2212yes—
30#10622844C2214yescomplemented
31#10622843C2214yescomplemented
32#10622842C2214yescomplemented
33#10622826C2214yes—
34#10622825C2212yes—
35#10622824C2214yescomplemented
36#10622823C2214yescomplemented
37C4×C22⋊Dic11C2212yes—
38#10622606C2214yescomplemented
39#10622603C2218yescomplemented
40#10622602C2218yescomplemented
41#10622600C2214yescomplemented
42#10622597C2218yescomplemented
43#10622596C2218yescomplemented
44#10622595C2214yescomplemented
45#10622589C2214yescomplemented
46#10622572C2218yes—
47#10622568C2214yes—
48#10622565C2218yescomplemented
49#10622564C2218yescomplemented
50#10622559C2214yes—
51#10622559C2214yes—
52#10622556C2214yes—
53C25×C11C2311yescharacteristic
54C24×D11C2311yescharacteristic
55C22×Q8×C11C2312yes—
56C22×Q8×C11C2312yes—
57C23×C4×C11C2312yes—
58C23×C4×C11C2312yes—
59C23×C4×C11C2311yescharacteristic
60C23×C4×C11C2311yescharacteristic
61C22×C22⋊D11—24no—
62C22×C22⋊D11—44no—
63C22×C22⋊D11C2314yes—
64C23×Dic11C2311yescharacteristic
65C23×Dic11C2311yescharacteristic
66C23×Dic11C2311yescharacteristic
67C22×C4×D11—24no—
68C22×C4×D11—44no—
69C22×C4×D11C2314yes—
70C11×C22⋊Q8—28no—
71C11×C22⋊Q8—28no—
72C11×C22⋊Q8—216no—
73C11×C22⋊Q8—216no—
74C2×C11×C4⋊C4C2312yes—
75C2×C11×C4⋊C4C2312yes—
76C2×C11×C4⋊C4C2314yes—
77C2×C11×C4⋊C4C2314yes—
78C2×C11×C22⋊C4—28no—
79C2×C11×C22⋊C4D418yes—
80C2×C11×C22⋊C4C2314yes—
81C2×C11×C22⋊C4C2314yes—
82C2×C11×C22⋊C4C2314yes—
83C2×C22⋊Dic11—24no—
84C2×C22⋊Dic11C2314yes—
85C2×C22⋊Dic11C2314yes—
86C2×C22⋊Dic11C2314yes—
87C2×C22⋊Dic11C2314yes—
88#66999—48no—
89#66982—48no—
90C2×(C2×D11)⋊C4—24no—
91C2×(C2×D11)⋊C4C2314yes—
92C2×(C2×D11)⋊C4C2318yes—
93C2×(C2×D11)⋊C4C2318yes—
94C2×C4⋊Dic11—24no—
95C2×C4⋊Dic11C2314yes—
96C2×C4⋊Dic11C2314yes—
97C2×Dic11⋊C4—24no—
98C2×Dic11⋊C4C2314yes—
99C2×Dic11⋊C4C2318yes—
100C2×Dic11⋊C4C2318yes—
101C2×C4×Dic11C2314yes—
102#66936—232no—
103C11⋊#95C2318yes—
104C11⋊#95C2318yes—
105C11⋊#95C2318yes—
106C11⋊#95C2318yes—
107C11⋊#95C2314yes—
108C11⋊#95C2314yes—
109#36720—111nomaximal, Sylow
110C24×C11—24no—
111C24×C11—42no—
112C24×C11—42no—
113C24×C11C2×D414yes—
114C24×C11C2412yes—
115C24×C11C2411yescharacteristic
116C23×D11—22no—
117C23×D11—24no—
118C23×D11—24no—
119C23×D11—42no—
120C23×D11C2412yes—
121C2×Q8×C11—24no—
122C2×Q8×C11—24no—
123C2×Q8×C11—44no—
124C2×Q8×C11—44no—
125C22×C4×C11—24no—
126C22×C4×C11—24no—
127C22×C4×C11—42no—
128C22×C4×C11—42no—
129C22×C4×C11—44no—
130C22×C4×C11—44no—
131C22×C4×C11C2×D414yes—
132C22×C4×C11C2×D414yes—
133C22×C4×C11C2414yes—
134C22×C4×C11C2414yes—
135C22×C4×C11C2412yes—
136C22×C4×C11C2412yes—
137C2×C22⋊D11—28no—
138C2×C22⋊D11—28no—
139C2×C22⋊D11—44no—
140C2×C22⋊D11—416no—
141C22×Dic11—22no—
142C22×Dic11—22no—
143C22×Dic11—22no—
144C22×Dic11—24no—
145C22×Dic11—42no—
146C22×Dic11—42no—
147C22×Dic11C2412yes—
148C22×Dic11C2418yes—
149C22×Dic11C2412yes—
150C22×Dic11C2412yes—
151C2×C4×D11—28no—
152C2×C4×D11—28no—
153C2×C4×D11—44no—
154C2×C4×D11—416no—
155C11×C4⋊C4—24no—
156C11×C4⋊C4—28no—
157C11×C4⋊C4—24no—
158C11×C4⋊C4—28no—
159C11×C22⋊C4—28no—
160C11×C22⋊C4—28no—
161C11×C22⋊C4C2×D4116yes—
162C22⋊Dic11—28no—
163C22⋊Dic11—28no—
164C22⋊Dic11—44no—
165C22⋊Dic11—44no—
166(C2×D11)⋊C4—216no—
167(C2×D11)⋊C4—28no—
168(C2×D11)⋊C4—48no—
169C4⋊Dic11—28no—
170C4⋊Dic11—44no—
171Dic11⋊C4—28no—
172Dic11⋊C4—48no—
173Dic11⋊C4—48no—
174C4×Dic11—44no—
175C2×#543D1111yescharacteristic, complemented
176C22×#123—111no—
177#2848—112no—
178#2687—118no—
179#2680—118no—
180#2627—112no—
181#2561—112no—
182#2502—112no—
183#2483—112no—
184#2405—112no—
185#2312—111no—
186C23×C11—22no—
187C23×C11—22no—
188C23×C11—24no—
189C23×C11—24no—
190C23×C11—24no—
191C23×C11—28no—
192C23×C11—44no—
193C23×C11—44no—
194C23×C11—88no—
195C23×C11C22×D414yes—
196C23×C11C22×D412yes—
197C23×C112−1+414yes—
198C23×C11C2511yescharacteristic
199C22×D11—28no—
200C22×D11—22no—
201C22×D11—22no—
202C22×D11—24no—
203C22×D11—24no—
204C22×D11—416no—
205C22×D11—44no—
206C22×D11—44no—
207C22×D11H(5,𝔽2)14yes—
208Q8×C11—84no—
209Q8×C11—84no—
210C2×C4×C11—24no—
211C2×C4×C11—24no—
212C2×C4×C11—24no—
213C2×C4×C11—24no—
214C2×C4×C11—28no—
215C2×C4×C11—28no—
216C2×C4×C11—44no—
217C2×C4×C11—44no—
218C2×C4×C11—84no—
219C2×C4×C11—84no—
220C2×C4×C11C22×D414yes—
221C2×C4×C11C22×D414yes—
222C22⋊D11—416no—
223C2×Dic11—22no—
224C2×Dic11—22no—
225C2×Dic11—22no—
226C2×Dic11—28no—
227C2×Dic11—24no—
228C2×Dic11—24no—
229C2×Dic11—28no—
230C2×Dic11—24no—
231C2×Dic11—28no—
232C2×Dic11—44no—
233C2×Dic11—44no—
234C2×Dic11—44no—
235C4×D11—416no—
236C23×D4—111no—
237C24×C4—111no—
238#543—28no—
239C2×#123—114no—
240C2×#123—118no—
241C2×#123—228no—
242C2×C22⋊Q8—114no—
243C2×C22⋊Q8C2×D1112yes—
244C2×C22⋊Q8C2×D1112yes—
245C2×C22⋊Q8C2×D1114yescomplemented
246C2×C22⋊Q8C2×D1114yescomplemented
247C22×C4⋊C4—111no—
248C22×C4⋊C4—111no—
249C22×C22⋊C4—111no—
250C22×C22⋊C4—111no—
251C22×C22⋊C4C2×D1112yes—
252C22×C22⋊C4C2×D1111yescharacteristic
253#400—114no—
254#398—118no—
255#398—118no—
256#397—114no—
257#397—118no—
258#395—114no—
259#394—118no—
260#393—114no—
261#391—114no—
262#388—118no—
263#388—118no—
264#386—114no—
265#385—114no—
266#385—118no—
267#385—114no—
268#385—114no—
269#384—114no—
270#381—114no—
271#381—114no—
272#380—112no—
273#380—114no—
274#379—112no—
275#379—114no—
276C4×C22⋊C4—112no—
277C22×C11—22no—
278C22×C11—22no—
279C22×C11—28no—
280C22×C11—24no—
281C22×C11—24no—
282C22×C11—24no—
283C22×C11—48no—
284C22×C11—88no—
285C22×C11#53614yes—
286C22×C11C23×D411yescharacteristic
287C22×C11C2×H(5,𝔽2)12yes—
288C22×C11C2×2−1+412yes—
289C22×C11#58512yes—
290C2×D11—24no—
291C2×D11—28no—
292C2×D11—416no—
293C2×D11—42no—
294C2×D11—42no—
295C2×D11—44no—
296C4×C11—24no—
297C4×C11—24no—
298C4×C11—44no—
299C4×C11—44no—
300Dic11—42no—
301Dic11—42no—
302Dic11—42no—
303Dic11—48no—
304C25—111no—
305C25C22×D1111yescharacteristic
306C22×Q8C22×D1112yes—
307C22×Q8C22×D1112yes—
308C22×D4—114no—
309C22×D4—224no—
310C22×D4—444no—
311C23×C4—111no—
312C23×C4—111no—
313C23×C4—111no—
314C23×C4—114no—
315C23×C4—224no—
316C23×C4—444no—
317C23×C4C22×D1112yes—
318C23×C4C22×D1112yes—
319C23×C4C22×D1111yescharacteristic
320C23×C4C22×D1111yescharacteristic
321#123—2232no—
322#123—448no—
323C22⋊Q8—28no—
324C22⋊Q8—28no—
325C22⋊Q8—216no—
326C22⋊Q8—216no—
327C22⋊Q8—448no—
328C2×C4⋊C4—114no—
329C2×C4⋊C4—114no—
330C2×C4⋊C4—114no—
331C2×C4⋊C4—118no—
332C2×C4⋊C4—118no—
333C2×C4⋊C4—224no—
334C2×C4⋊C4—224no—
335C2×C4⋊C4C22×D1112yes—
336C2×C4⋊C4C22×D1112yes—
337C2×C4⋊C4C22×D1114yes—
338C2×C4⋊C4C22×D1114yes—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—114no—
341C2×C22⋊C4—114no—
342C2×C22⋊C4—114no—
343C2×C22⋊C4—118no—
344C2×C22⋊C4—114no—
345C2×C22⋊C4—114no—
346C2×C22⋊C4—118no—
347C2×C22⋊C4—224no—
348C2×C22⋊C4—224no—
349C2×C22⋊C4C22⋊D1118yes—
350C2×C22⋊C4C22×D1114yes—
351C2×C22⋊C4C22×D1114yes—
352C2×C22⋊C4C22×D1114yes—
353C2×C42—114no—
354#95—114no—
355#95—118no—
356#95—114no—
357#95—118no—
358#95—118no—
359#95—118no—
360C2×C11—24no—
361C2×C11—44no—
362C2×C11C2×#53611yescharacteristic
363C2×C11#348411yescharacteristic
364C2×C11#348511yescharacteristic
365C2×C11#353514yes—
366D11—44no—
367C24—24no—
368C24—42no—
369C24—42no—
370C24—112no—
371C24—224no—
372C24—222no—
373C24—224no—
374C24—442no—
375C24C2×C22⋊D1114yes—
376C24C23×D1112yes—
377C24C23×D1111yescharacteristic
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×D4—228no—
383C2×D4—228no—
384C2×D4—444no—
385C2×D4—4416no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—112no—
393C22×C4—118no—
394C22×C4—112no—
395C22×C4—112no—
396C22×C4—228no—
397C22×C4—224no—
398C22×C4—222no—
399C22×C4—222no—
400C22×C4—222no—
401C22×C4—228no—
402C22×C4—444no—
403C22×C4—4416no—
404C22×C4—442no—
405C22×C4—442no—
406C22×C4C2×C22⋊D1114yes—
407C22×C4C2×C22⋊D1114yes—
408C22×C4C23×D1114yes—
409C22×C4C23×D1114yes—
410C22×C4C23×D1112yes—
411C22×C4C23×D1112yes—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—228no—
417C4⋊C4—228no—
418C4⋊C4—448no—
419C4⋊C4—444no—
420C4⋊C4—448no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—2216no—
424C22⋊C4—228no—
425C22⋊C4—228no—
426C22⋊C4—228no—
427C22⋊C4—448no—
428C22⋊C4—444no—
429C22⋊C4—444no—
430C22⋊C4C2×C22⋊D11116yes—
431C42—444no—
432C11#3672011yescharacteristic, complemented, Sylow
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—28no—
439C23—44no—
440C23—44no—
441C23—88no—
442C23—114no—
443C23—228no—
444C23—224no—
445C23—222no—
446C23—224no—
447C23—222no—
448C23—4416no—
449C23—444no—
450C23—444no—
451C23#6704014yes—
452C23C22×C22⋊D1114yes—
453C23C22×C22⋊D1112yes—
454C23C24×D1111yescharacteristic
455Q8—84no—
456Q8—84no—
457D4—4416no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—228no—
469C2×C4—224no—
470C2×C4—228no—
471C2×C4—224no—
472C2×C4—224no—
473C2×C4—228no—
474C2×C4—222no—
475C2×C4—222no—
476C2×C4—222no—
477C2×C4—4416no—
478C2×C4—444no—
479C2×C4—444no—
480C2×C4—444no—
481C2×C4C22×C22⋊D1114yes—
482C2×C4C22×C22⋊D1114yes—
483C22—22no—
484C22—22no—
485C22—28no—
486C22—24no—
487C22—24no—
488C22—24no—
489C22—48no—
490C22—88no—
491C22—224no—
492C22—228no—
493C22—4416no—
494C22—442no—
495C22—442no—
496C22—444no—
497C22#1062334714yes—
498C22C2×#6704012yes—
499C22D11×H(5,𝔽2)12yes—
500C22#1062345512yes—
501C22C23×C22⋊D1111yescharacteristic
502C4—24no—
503C4—24no—
504C4—44no—
505C4—44no—
506C4—442no—
507C4—442no—
508C4—442no—
509C4—448no—
510C2—24no—
511C2—44no—
512C2—444no—
513C2C2×#1062334711yescharacteristic
514C2#1308224711yescharacteristic
515C2#1308225611yescharacteristic
516C2#1308283014yes—
subgroupG/H
center, FrattiniC23C24×D11
commutator, socle, cosocleC23×C11C25
FittingC2×C11×#543C2
prime radicalC23×C22⋊D11C22
hypercenterC2×#543D11
hypocenterC11#36720
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #13076900C2semidirect
3. #13075033C2semidirect
4. #13074955C2semidirect
5. #13074431C2semidirect
6. #13073775C2semidirect
7. #13073095C2semidirect
8. #13072894C2semidirect
10. #13071214C2semidirect
11. C2×C11×#543C2semidirect
17. C2×C11×C22⋊Q8C22semidirect
18. C2×C11×C22⋊Q8C22semidirect
22. C2×#66999C22semidirect
30. #10622844C22semidirect
31. #10622843C22semidirect
32. #10622842C22semidirect
35. #10622824C22semidirect
36. #10622823C22semidirect
38. #10622606C22semidirect
39. #10622603C22semidirect
40. #10622602C22semidirect
41. #10622600C22semidirect
42. #10622597C22semidirect
43. #10622596C22semidirect
44. #10622595C22semidirect
45. #10622589C22semidirect
48. #10622565C22semidirect
49. #10622564C22semidirect
175. C2×#543D11semidirect
245. C2×C22⋊Q8C2×D11semidirect
246. C2×C22⋊Q8C2×D11semidirect
432. C11#36720semidirect

Maximal subgroups

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

subgroupindex
C22×#669362
#130769002
#130750332
#130749552
#130744312
#130737752
#130730952
#130728942
#130719432
#130712142
C2×C11×#5432
#3672011