Finite Group Databasebeta

← #14780860

Subgroups — #14780860

Order 2816.

2130 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4325C211yesmaximal, characteristic, complemented
2C2×#10623361C211yesmaximal, characteristic
3C2×#10622744C211yesmaximal, characteristic, complemented
4C2×#10622846C211yesmaximal, characteristic, complemented
5#13071202C211yesmaximal, characteristic, complemented
6#13069707C214yesmaximal, complemented
7#13069702C214yesmaximal, complemented
8#13068314C211yesmaximal, characteristic, complemented
9C22×C11×#137C211yesmaximal, characteristic, complemented
10C22×C11×#41C2211yescharacteristic
11C23×Q8×C11C2211yescharacteristic
12C23×D44C2211yescharacteristic
13C2×C11×#137—24no—
14C2×C11×#137—24no—
15C2×C11×#137—24no—
16C22×C11×Q16C2212yescomplemented
17C22×C11×SD16C2212yescomplemented
18C22×C11×M16C2211yescharacteristic
19#10623361—24no—
20#10623361—24no—
21C2×#67000C2212yes—
22C22×#4325—24no—
23C22×#4325—24no—
24C22×#4325—24no—
25C22×#4325C2214yes—
26C2×#66993C2212yes—
27C2×#66990—22no—
28C2×#66983—22no—
29C2×#66979C2211yescharacteristic
30C2×#66979C2211yescharacteristic
31C23×C11⋊C8C2211yescharacteristic
32#10622846—24no—
33#10622843—22no—
34#10622843C2214yescomplemented
35C2×#66899C2212yescomplemented
36C2×#66896C2212yescomplemented
37#10622824—22no—
38#10622744—24no—
39C2×#66885C2214yes—
40#10622188C2214yescomplemented
41C24×D11—21no—
42C2×C11×#41—22no—
43C2×C11×#41—22no—
44C2×C11×#41—22no—
45C2×C11×#41—22no—
46C2×C11×#41—22no—
47C2×C11×#41D414yes—
48C22×Q8×C11—21no—
49C22×Q8×C11—22no—
50C22×Q8×C11—22no—
51C22×Q8×C11—22no—
52C22×Q8×C11—22no—
53C22×Q8×C11—22no—
54C22×Q8×C11D414yescomplemented
55C22×Q8×C11C2312yes—
56C22×Q8×C11C2311yescharacteristic
57C22×D4×C11—21no—
58C22×D4×C11C2311yescharacteristic
59C23×C4×C11—21no—
60C23×C4×C11—21no—
61C23×C4×C11C2311yescharacteristic
62C22×C22⋊D11—24no—
63C22×D44—22no—
64C22×D44—22no—
65C22×D44—22no—
66C22×D44—22no—
67C22×D44—21no—
68C22×D44—22no—
69C22×D44D414yes—
70C22×D44C2312yes—
71C11×#137—232no—
72C2×C11×Q16—24no—
73C2×C11×Q16—24no—
74C2×C11×Q16—24no—
75C2×C11×SD16—24no—
76C2×C11×SD16—24no—
77C2×C11×SD16—24no—
78C2×C11×M16—22no—
79C2×C11×M16—22no—
80C2×C11×M16D414yescomplemented
81C22×C8×C11C2312yes—
82C2×C22⋊Dic11—21no—
83C2×C22⋊Dic11D412yes—
84#67000—24no—
85#67000—24no—
86C2×#4325—232no—
87C2×#4325—28no—
88C2×#4325—28no—
89C2×#4325—28no—
90#66993—24no—
91#66993—24no—
92#66990—216no—
93#66983—216no—
94C2×(C2×D11)⋊C4—24no—
95#66979—22no—
96#66979—22no—
97#66979—22no—
98#66979—22no—
99C2×C4⋊Dic11D412yes—
100C2×C4⋊Dic11C2312yes—
101C2×C4×Dic11—21no—
102C2×C4×Dic11C2312yes—
103C22×C11⋊C8—22no—
104C22×C11⋊C8—22no—
105C22×C11⋊C8D414yescomplemented
106C22×C11⋊C8C2312yes—
107#66899—24no—
108#66896—24no—
109C11⋊#95—28no—
110#66885—28no—
111#23423—111nomaximal, Sylow
112C24×C11—21no—
113C24×C11C2×D411yescharacteristic
114C23×D11—22no—
115C23×D11—24no—
116C23×D11—42no—
117C23×D11—42no—
118C23×D11—84no—
119C11×#41—28no—
120C11×#41—28no—
121C11×#41—44no—
122C11×#41—44no—
123C2×Q8×C11—24no—
124C2×Q8×C11—21no—
125C2×Q8×C11—22no—
126C2×Q8×C11—28no—
127C2×Q8×C11—28no—
128C2×Q8×C11—21no—
129C2×Q8×C11—22no—
130C2×Q8×C11—22no—
131C2×Q8×C11—22no—
132C2×Q8×C11—22no—
133C2×Q8×C11—21no—
134C2×Q8×C11—21no—
135C2×Q8×C11—21no—
136C2×Q8×C11—42no—
137C2×Q8×C11—44no—
138C2×Q8×C11—44no—
139C2×Q8×C11—42no—
140C2×Q8×C11C2×D414yes—
141C2×Q8×C11C2×D412yes—
142C2×D4×C11—24no—
143C2×D4×C11—21no—
144C2×D4×C11—21no—
145C2×D4×C11—21no—
146C2×D4×C11—21no—
147C2×D4×C11—21no—
148C2×D4×C11—42no—
149C2×D4×C11—42no—
150C2×D4×C11C2×D412yes—
151C22×C4×C11—24no—
152C22×C4×C11—22no—
153C22×C4×C11—21no—
154C22×C4×C11—24no—
155C22×C4×C11—42no—
156C22×C4×C11—42no—
157C22×C4×C11—42no—
158C22×C4×C11—42no—
159C22×C4×C11C2×D412yes—
160C22×C4×C11C2×D412yes—
161C22×C4×C11C2×D412yes—
162C22×C4×C11C2×D411yescharacteristic
163C22×C4×C11C2×D412yes—
164C22×C4×C11C2×D412yes—
165C22×C4×C11#4112yes—
166C22×C4×C11C2411yescharacteristic
167C2×C22⋊D11—216no—
168C2×C22⋊D11—88no—
169C22×Dic11—22no—
170C22×Dic11C2×D414yes—
171C2×D44—24no—
172C2×D44—28no—
173C2×D44—28no—
174C2×D44—22no—
175C2×D44—22no—
176C2×D44—22no—
177C2×D44—22no—
178C2×D44—22no—
179C2×D44—44no—
180C2×D44—44no—
181C2×D44—42no—
182C2×D44—42no—
183C2×D44C2×D414yes—
184C11×Q16—216no—
185C11×SD16—216no—
186C11×M16—28no—
187C2×C8×C11—22no—
188C2×C8×C11—22no—
189C2×C8×C11C2×D414yes—
190C22⋊Dic11—24no—
191C22⋊Dic11—42no—
192#4325—232no—
193(C2×D11)⋊C4—216no—
194C4⋊Dic11—24no—
195C4⋊Dic11—22no—
196C4⋊Dic11—22no—
197C4×Dic11—22no—
198C4×Dic11—22no—
199C4×Dic11—42no—
200C2×C11⋊C8—28no—
201C2×C11⋊C8—22no—
202C2×C11⋊C8—22no—
203C2×C11⋊C8C2×D414yes—
204C22×#137D1111yescharacteristic, complemented
205C23×SD16—111no—
206C2×#535—111no—
207C2×#418—111no—
208C2×#416—111no—
209#2310—111no—
210#2015—114no—
211#2014—114no—
212#1729—111no—
213C23×C11—24no—
214C23×C11—22no—
215C23×C11—41no—
216C23×C11—41no—
217C23×C11—44no—
218C23×C11#12012yes—
219C23×C11#12112yes—
220C23×C11#12712yes—
221C23×C11C22×D411yescharacteristic
222C22×D11—24no—
223C22×D11—42no—
224C22×D11—42no—
225C22×D11—48no—
226C22×D11—48no—
227C22×D11—88no—
228C22×D11—88no—
229C22×D11—88no—
230Q8×C11—24no—
231Q8×C11—24no—
232Q8×C11—22no—
233Q8×C11—22no—
234Q8×C11—41no—
235Q8×C11—42no—
236Q8×C11—42no—
237Q8×C11—41no—
238Q8×C11—48no—
239D4×C11—22no—
240D4×C11—22no—
241D4×C11—48no—
242D4×C11—41no—
243D4×C11—41no—
244C2×C4×C11—22no—
245C2×C4×C11—24no—
246C2×C4×C11—42no—
247C2×C4×C11—41no—
248C2×C4×C11—41no—
249C2×C4×C11—42no—
250C2×C4×C11—48no—
251C2×C4×C11—48no—
252C2×C4×C11#12014yes—
253C2×C4×C11#12012yes—
254C2×C4×C11#12014yes—
255C2×C4×C11#12114yes—
256C2×C4×C11#12114yes—
257C2×C4×C11#12112yes—
258C2×C4×C11#12712yes—
259C2×C4×C11C22×D411yescharacteristic
260C2×C4×C11C22×D411yescharacteristic
261C2×C4×C11C22×D411yescharacteristic
262C2×C4×C11C22×D411yescharacteristic
263C2×C4×C11C22×D411yescharacteristic
264C2×C4×C11C2×#4111yescharacteristic
265C22⋊D11—816no—
266C2×Dic11—24no—
267C2×Dic11—28no—
268C2×Dic11—42no—
269C2×Dic11—44no—
270D44—24no—
271D44—24no—
272D44—42no—
273D44—42no—
274D44—48no—
275C8×C11—24no—
276C11⋊C8—24no—
277C22×#41C2×D1111yescharacteristic
278C23×Q8C2×D1111yescharacteristic
279C23×D4—111no—
280C2×#137—24no—
281C2×#137—24no—
282C2×#137—24no—
283C22×Q16C2×D1112yescomplemented
284C22×SD16—114no—
285C22×SD16—224no—
286C22×SD16—224no—
287C22×SD16—224no—
288C22×SD16C2×D1112yescomplemented
289C22×M16C2×D1111yescharacteristic
290C23×C8—111no—
291#535—224no—
292#535—224no—
293C2×#127—112no—
294C2×#124—112no—
295C2×#121—222no—
296C2×#120—222no—
297C2×#117—111no—
298C2×#117—111no—
299#418—224no—
300#416—224no—
301C2×#103—112no—
302C2×#102—112no—
303C2×#102—114no—
304#391—114no—
305#391—222no—
306#381—222no—
307#336—114no—
308C22×C11—24no—
309C22×C11—24no—
310C22×C11—24no—
311C22×C11—42no—
312C22×C11—44no—
313C22×C11—82no—
314C22×C11—82no—
315C22×C11#39214yes—
316C22×C11C2×#12011yescharacteristic
317C22×C11C2×#12111yescharacteristic
318C22×C11C2×#12711yescharacteristic
319C22×C11#57714yes—
320C2×D11—44no—
321C2×D11—48no—
322C2×D11—816no—
323C2×D11—84no—
324C2×D11—84no—
325C4×C11—42no—
326C4×C11—44no—
327C4×C11#39214yes—
328C4×C11C2×#12011yescharacteristic
329C4×C11C2×#12011yescharacteristic
330C4×C11C2×#12111yescharacteristic
331C4×C11C2×#12111yescharacteristic
332Dic11—44no—
333Dic11—82no—
334C25—221no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41C22⋊D1114yes—
341C22×Q8—21no—
342C22×Q8—22no—
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8C22⋊D1114yescomplemented
348C22×Q8C22×D1111yescharacteristic
349C22×Q8C22×D1112yes—
350C22×D4—21no—
351C22×D4—114no—
352C22×D4—112no—
353C22×D4—224no—
354C22×D4—222no—
355C22×D4—221no—
356C22×D4—222no—
357C22×D4—222no—
358C22×D4—222no—
359C22×D4—222no—
360C22×D4C22×D1111yescharacteristic
361C23×C4—21no—
362C23×C4—21no—
363C23×C4C22×D1111yescharacteristic
364#137—232no—
365C2×Q16—24no—
366C2×Q16—24no—
367C2×Q16—24no—
368C2×SD16—24no—
369C2×SD16—24no—
370C2×SD16—24no—
371C2×SD16—228no—
372C2×SD16—228no—
373C2×SD16—2232no—
374C2×SD16—228no—
375C2×M16—22no—
376C2×M16—22no—
377C2×M16C22⋊D1114yescomplemented
378C22×C8—114no—
379C22×C8—112no—
380C22×C8—222no—
381C22×C8—222no—
382C22×C8C22×D1112yes—
383#127—224no—
384#127—224no—
385#124—224no—
386#124—224no—
387#121—2216no—
388#120—2216no—
389#117—222no—
390#117—222no—
391#117—222no—
392#117—222no—
393C2×C4⋊C4—112no—
394C2×C4⋊C4—112no—
395C2×C22⋊C4—112no—
396C2×C22⋊C4—224no—
397C2×C22⋊C4—221no—
398C2×C42—112no—
399C2×C42—221no—
400#103—224no—
401#102—224no—
402#102—228no—
403#95—228no—
404C2×C11—24no—
405C2×C11—82no—
406C2×C11C2×#39211yescharacteristic
407C2×C11#300912yes—
408C2×C11#304312yes—
409C2×C11#315412yes—
410D11—84no—
411C24—21no—
412C24—222no—
413C24—224no—
414C24—442no—
415C24—442no—
416C24—884no—
417C24D4×D1111yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—24no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—28no—
426C2×Q8—28no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8C2×C22⋊D1112yes—
440C2×Q8C2×C22⋊D1114yes—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—114no—
450C2×D4—2216no—
451C2×D4—228no—
452C2×D4—224no—
453C2×D4—228no—
454C2×D4—222no—
455C2×D4—222no—
456C2×D4—222no—
457C2×D4—222no—
458C2×D4—222no—
459C2×D4—444no—
460C2×D4—442no—
461C2×D4—442no—
462C2×D4—444no—
463C2×D4—888no—
464C2×D4C2×C22⋊D1112yes—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—114no—
474C22×C4—222no—
475C22×C4D4×D1112yes—
476C22×C4D4×D1111yescharacteristic
477C22×C4D4×D1112yes—
478C22×C4#434112yes—
479C22×C4C2×C22⋊D1112yes—
480C22×C4C2×C22⋊D1112yes—
481C22×C4C2×C22⋊D1112yes—
482C22×C4C23×D1111yescharacteristic
483Q16—216no—
484SD16—216no—
485SD16—2232no—
486M16—28no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—114no—
490C2×C8—228no—
491C2×C8—222no—
492C2×C8—222no—
493C2×C8C2×C22⋊D1114yes—
494C4⋊C4—224no—
495C4⋊C4—222no—
496C4⋊C4—222no—
497C22⋊C4—2216no—
498C22⋊C4—224no—
499C22⋊C4—442no—
500C42—222no—
501C42—222no—
502C42—442no—
503C11#2342311yescharacteristic, complemented, Sylow
504C23—24no—
505C23—22no—
506C23—44no—
507C23—41no—
508C23—41no—
509C23—224no—
510C23—448no—
511C23—442no—
512C23—442no—
513C23—448no—
514C23—888no—
515C23—888no—
516C23—888no—
517C23#6699012yes—
518C23#6699212yes—
519C23#6699312yes—
520C23C2×D4×D1111yescharacteristic
521Q8—24no—
522Q8—22no—
523Q8—24no—
524Q8—22no—
525Q8—41no—
526Q8—42no—
527Q8—41no—
528Q8—42no—
529Q8—48no—
530D4—22no—
531D4—22no—
532D4—48no—
533D4—41no—
534D4—41no—
535D4—224no—
536D4—224no—
537D4—442no—
538D4—448no—
539D4—442no—
540D4—8816no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—228no—
550C2×C4—224no—
551C2×C4—444no—
552C2×C4—442no—
553C2×C4#6699014yes—
554C2×C4#6699012yes—
555C2×C4#6699114yes—
556C2×C4#6699214yes—
557C2×C4#6699212yes—
558C2×C4#6699312yes—
559C2×C4#6700614yes—
560C2×C4C2×D4×D1111yescharacteristic
561C2×C4C2×D4×D1111yescharacteristic
562C2×C4C2×#434111yescharacteristic
563C2×C4C22×C22⋊D1111yescharacteristic
564C2×C4C22×C22⋊D1111yescharacteristic
565C2×C4C22×C22⋊D1111yescharacteristic
566C8—24no—
567C8—224no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—44no—
572C22—42no—
573C22—82no—
574C22—82no—
575C22—448no—
576C22—444no—
577C22—8816no—
578C22—884no—
579C22—884no—
580C22#1062283614yes—
581C22#1062330914yes—
582C22C2×#6699011yescharacteristic
583C22C2×#6699211yescharacteristic
584C22C2×#6699311yescharacteristic
585C4—42no—
586C4—44no—
587C4—444no—
588C4—882no—
589C4#1062283614yes—
590C4C2×#6699011yescharacteristic
591C4C2×#6699111yescharacteristic
592C4C2×#6699211yescharacteristic
593C4C2×#6700611yescharacteristic
594C2—24no—
595C2—82no—
596C2—884no—
597C2C2×#1062283611yescharacteristic
598C2#1307789212yes—
599C2#1307813912yes—
600C2#1307909512yes—
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC22×C11×#137C2
socleC23×C11C22×D4
prime radicalC2×#10623361C2
hypercenterC22×#137D11
hypocenterC11#23423
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C23×#4325C2semidirect
3. C2×#10622744C2semidirect
4. C2×#10622846C2semidirect
5. #13071202C2semidirect
6. #13069707C2semidirect
7. #13069702C2semidirect
8. #13068314C2semidirect
9. C22×C11×#137C2semidirect
16. C22×C11×Q16C22semidirect
17. C22×C11×SD16C22semidirect
34. #10622843C22semidirect
35. C2×#66899C22semidirect
36. C2×#66896C22semidirect
40. #10622188C22semidirect
54. C22×Q8×C11D4semidirect
80. C2×C11×M16D4semidirect
105. C22×C11⋊C8D4semidirect
204. C22×#137D11semidirect
283. C22×Q16C2×D11semidirect
288. C22×SD16C2×D11semidirect
347. C22×Q8C22⋊D11semidirect
377. C2×M16C22⋊D11semidirect
503. C11#23423semidirect

Maximal subgroups

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

subgroupindex
C23×#43252
C2×#106233612
C2×#106227442
C2×#106228462
#130712022
#130697072
#130697022
#130683142
C22×C11×#1372
#2342311