Finite Group Databasebeta

← #14755283

Subgroups — #14755283

Order 2816.

1778 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623353C211yesmaximal, characteristic
2#13081817C211yesmaximal, characteristic, complemented
3C22×#66869C211yesmaximal, characteristic, complemented
4#13077135C211yesmaximal, characteristic, complemented
5#13068995C214yesmaximal, complemented
6#13068988C214yesmaximal, complemented
7#13068355C211yesmaximal, characteristic, complemented
8C2×#10622161C211yesmaximal, characteristic, complemented
9C11×#2297C211yesmaximal, characteristic, complemented
10C22×C11×#41C2211yescharacteristic
11C22×#4339C2211yescharacteristic
12C2×C11×#117C2211yescharacteristic
13C22×C42×C11C2211yescharacteristic
14C2×#67004—21no—
15#10623353—24no—
16#10623353C418yescomplemented
17C2×#67003—22no—
18C2×#67002—21no—
19C2×#66995C2212yes—
20C2×#66985C2212yes—
21C2×#66979C2211yescharacteristic
22C22×#4318C2211yescharacteristic
23C22×#4318C2211yescharacteristic
24#10623054—22no—
25#10623053—22no—
26#10623052—21no—
27D11×#117—22no—
28#10623047—21no—
29#10623035C2214yes—
30#10623017—22no—
31C4×#4339—22no—
32C11×#391C2212yescomplemented
33C11×#390C2212yescomplemented
34C11×#386—22no—
35C4×C11×C22⋊C4—22no—
36#10622846—22no—
37#10622845—22no—
38#10622838C2212yescomplemented
39#10622828C2212yescomplemented
40C2×#66869—24no—
41C2×#66869C2214yes—
42#10622168C2214yescomplemented
43#10622161C2214yescomplemented
44C2×C11×#41—22no—
45C2×C11×#41—21no—
46C2×C11×#41—21no—
47C2×C11×#41—22no—
48C2×C11×#41—21no—
49C2×C11×#41—21no—
50C2×C11×#41—22no—
51C2×C11×#41C2×C414yescomplemented
52C22×Q8×C11C2311yescharacteristic
53C22×D4×C11—21no—
54C22×D4×C11C2311yescharacteristic
55C23×C4×C11—21no—
56C23×C4×C11—21no—
57C23×C4×C11C2311yescharacteristic
58C22×C22⋊D11—21no—
59C2×#4339—21no—
60C2×#4339—21no—
61C2×#4339—22no—
62C2×#4339—22no—
63C2×#4339C2×C414yes—
64C2×#4339D414yes—
65C2×#4339D412yes—
66C2×#4339C2312yes—
67C22×D44D411yescharacteristic
68C22×C4×D11—21no—
69C22×Dic22D411yescharacteristic
70C11×#117—21no—
71C11×#117—21no—
72C11×#117—21no—
73C11×#117—21no—
74C2×C11×C4⋊C4C2312yes—
75C2×C11×C22⋊C4—24no—
76C2×C11×C22⋊C4D412yes—
77C2×C42×C11—22no—
78C2×C42×C11—22no—
79C2×C42×C11—21no—
80C2×C42×C11D412yes—
81C2×C42×C11C2312yes—
82C2×C22⋊Dic11—21no—
83#67004—28no—
84#67003—216no—
85#67002—28no—
86C2×#4326—22no—
87#66995—24no—
88#66995C2×C418yescomplemented
89C2×#4325—22no—
90C2×#4324—22no—
91#66985—24no—
92#66985C2×C418yescomplemented
93C2×#4323—22no—
94C4×C22⋊D11—28no—
95#66979—21no—
96#66979—21no—
97#66979D412yes—
98#66979C2312yes—
99C2×C4⋊Dic11D412yes—
100C2×C4×Dic11—21no—
101C2×#4318—21no—
102C2×#4318—22no—
103C2×#4318—21no—
104C2×#4318—22no—
105C2×#4318—22no—
106C2×#4318C2×C414yescomplemented
107C2×#4318C2312yes—
108C2×#4318C2312yes—
109C22×C11⋊C8—21no—
110C22×C11⋊C8—21no—
111#66946—24no—
112#66945—24no—
113C4⋊C4×D11—24no—
114#66940—24no—
115(C22⋊D11)⋊C4—28no—
116C22⋊C4×D11—24no—
117#66930—24no—
118C4×D44—24no—
119#66925—24no—
120C42×D11—24no—
121C4×Dic22—24no—
122C11×#95—24no—
123#66899—24no—
124#66896—24no—
125#66869—28no—
126#20942—111nomaximal, Sylow
127C24×C11—21no—
128C24×C11C2×D411yescharacteristic
129C23×D11—21no—
130C11×#41—24no—
131C11×#41—24no—
132C11×#41—24no—
133C11×#41—24no—
134C11×#41—42no—
135C11×#41—42no—
136C11×#41—42no—
137C11×#41—42no—
138C2×Q8×C11—21no—
139C2×Q8×C11—21no—
140C2×Q8×C11—21no—
141C2×Q8×C11—21no—
142C2×Q8×C11—21no—
143C2×Q8×C11C22×C412yes—
144C2×D4×C11—24no—
145C2×D4×C11—21no—
146C2×D4×C11—21no—
147C2×D4×C11—21no—
148C2×D4×C11—21no—
149C2×D4×C11—21no—
150C2×D4×C11—41no—
151C2×D4×C11—41no—
152C2×D4×C11—42no—
153C2×D4×C11C22×C412yes—
154C22×C4×C11—21no—
155C22×C4×C11—22no—
156C22×C4×C11—24no—
157C22×C4×C11—44no—
158C22×C4×C11—42no—
159C22×C4×C11—41no—
160C22×C4×C11—41no—
161C22×C4×C11—42no—
162C22×C4×C11C22×C412yes—
163C22×C4×C11C2×D414yes—
164C22×C4×C11C2×D411yescharacteristic
165C22×C4×C11C2×D411yescharacteristic
166C22×C4×C11C2×D412yes—
167C22×C4×C11C2×D411yescharacteristic
168C22×C4×C11C2×D411yescharacteristic
169C22×C4×C11C2×D411yescharacteristic
170C22×C4×C11#4112yes—
171C22×C4×C11#4112yes—
172C22×C4×C11C2411yescharacteristic
173C2×C22⋊D11—24no—
174C2×C22⋊D11—22no—
175C2×C22⋊D11—24no—
176C2×C22⋊D11—22no—
177C22×Dic11—21no—
178C22×Dic11—22no—
179#4339—24no—
180#4339—24no—
181#4339—24no—
182#4339—24no—
183#4339—24no—
184#4339—22no—
185#4339—22no—
186#4339C22⋊C418yescomplemented
187#4339C22×C414yes—
188#4339C2×D414yes—
189C2×D44—21no—
190C2×D44—22no—
191C2×D44C22⋊C414yescomplemented
192C2×D44C2×D412yes—
193C2×C4×D11—28no—
194C2×C4×D11—24no—
195C2×C4×D11—22no—
196C2×C4×D11—24no—
197C2×C4×D11—22no—
198C2×Dic22—21no—
199C2×Dic22—22no—
200C2×Dic22C22⋊C414yescomplemented
201C2×Dic22C2×D412yes—
202C11×C4⋊C4—22no—
203C11×C4⋊C4—22no—
204C11×C22⋊C4—22no—
205C11×C22⋊C4—22no—
206C11×C22⋊C4—48no—
207C42×C11—22no—
208C42×C11—22no—
209C42×C11—22no—
210C42×C11—22no—
211C42×C11—42no—
212C22⋊Dic11—22no—
213C22⋊Dic11—22no—
214#4326—28no—
215#4325—28no—
216#4324—28no—
217#4323—28no—
218(C2×D11)⋊C4—28no—
219C4⋊Dic11—22no—
220C4⋊Dic11C2×D414yes—
221Dic11⋊C4—28no—
222C4×Dic11—28no—
223C4×Dic11—22no—
224C4×Dic11—22no—
225#4318—24no—
226#4318—22no—
227#4318—22no—
228#4318—22no—
229#4318—42no—
230#4318C22×C414yes—
231C2×C11⋊C8—24no—
232C2×C11⋊C8—22no—
233C2×C11⋊C8—22no—
234C2×C11⋊C8—42no—
235C2×#575—111no—
236#3428—111no—
237C22×#104—111no—
238#2896—111no—
239#2297D1111yescharacteristic, complemented
240#1883—114no—
241#1882—114no—
242#1742—111no—
243C2×#339—111no—
244C23×C11—22no—
245C23×C11—42no—
246C23×C11—44no—
247C23×C11—41no—
248C23×C11—41no—
249C23×C11C4×D412yes—
250C23×C11#12011yescharacteristic
251C23×C11#12011yescharacteristic
252C23×C11#12112yes—
253C23×C11C22×D411yescharacteristic
254C22×D11—22no—
255C22×D11—24no—
256C22×D11—82no—
257Q8×C11—22no—
258Q8×C11—22no—
259Q8×C11—41no—
260Q8×C11—41no—
261D4×C11—22no—
262D4×C11—22no—
263D4×C11—44no—
264D4×C11—44no—
265D4×C11—41no—
266D4×C11—41no—
267C2×C4×C11—24no—
268C2×C4×C11—22no—
269C2×C4×C11—24no—
270C2×C4×C11—42no—
271C2×C4×C11—48no—
272C2×C4×C11—41no—
273C2×C4×C11—42no—
274C2×C4×C11—41no—
275C2×C4×C11—44no—
276C2×C4×C11—44no—
277C2×C4×C11—44no—
278C2×C4×C11C2×C22⋊C412yes—
279C2×C4×C11C2×C22⋊C412yes—
280C2×C4×C11C4×D412yes—
281C2×C4×C11C4×D412yes—
282C2×C4×C11C4×D412yes—
283C2×C4×C11#12011yescharacteristic
284C2×C4×C11#12011yescharacteristic
285C2×C4×C11#12112yes—
286C2×C4×C11#12112yes—
287C2×C4×C11#12112yes—
288C2×C4×C11#12312yes—
289C2×C4×C11#12312yes—
290C2×C4×C11C23×C411yescharacteristic
291C2×C4×C11C22×D411yescharacteristic
292C2×C4×C11C22×D411yescharacteristic
293C2×C4×C11C22×D411yescharacteristic
294C2×C4×C11C2×#4111yescharacteristic
295C2×C4×C11C2×#4111yescharacteristic
296C22⋊D11—24no—
297C22⋊D11—24no—
298C22⋊D11—28no—
299C2×Dic11—22no—
300C2×Dic11—24no—
301C2×Dic11—24no—
302C2×Dic11—82no—
303D44—22no—
304D44—24no—
305D44C2×C22⋊C414yes—
306C4×D11—24no—
307C4×D11—24no—
308C4×D11—28no—
309C4×D11—88no—
310Dic22—22no—
311Dic22—24no—
312Dic22C2×C22⋊C414yes—
313C11⋊C8—24no—
314C11⋊C8—42no—
315C22×#41—111no—
316C22×#41C2×D1111yescharacteristic
317#575—118no—
318#575—224no—
319C2×#137—112no—
320C2×#137—221no—
321C2×AGL(1,ℤ/8)—112no—
322C2×AGL(1,ℤ/8)—221no—
323C2×#135—222no—
324C22×M16—111no—
325C22×M16—111no—
326#520—114no—
327#520—222no—
328#519—221no—
329#518—221no—
330#518—222no—
331C4×#41—222no—
332C4×#41—222no—
333C2×#117—111no—
334C2×#117—222no—
335C2×#117C2×D1111yescharacteristic
336C22×C42C2×D1111yescharacteristic
337C2×#104—114no—
338C2×#104—224no—
339#419—112no—
340#418—112no—
341#417—222no—
342#416—222no—
343#391C2×D1112yescomplemented
344#390C2×D1112yescomplemented
345#386—22no—
346C4×C22⋊C4—22no—
347#343—114no—
348#339—114no—
349C22×C11—22no—
350C22×C11—22no—
351C22×C11—22no—
352C22×C11—42no—
353C22×C11—42no—
354C22×C11—44no—
355C22×C11—42no—
356C22×C11—82no—
357C22×C11—82no—
358C22×C11#38612yes—
359C22×C11#38612yes—
360C22×C11C2×C4×D411yescharacteristic
361C22×C11C2×#12011yescharacteristic
362C22×C11C2×#12111yescharacteristic
363C2×D11—24no—
364C2×D11—44no—
365C2×D11—84no—
366C4×C11—44no—
367C4×C11—44no—
368C4×C11—42no—
369C4×C11#38612yes—
370C4×C11#38612yes—
371C4×C11C22×C22⋊C411yescharacteristic
372C4×C11C2×C4×D411yescharacteristic
373C4×C11C2×#12111yescharacteristic
374C4×C11C2×#12311yescharacteristic
375Dic11—24no—
376Dic11—44no—
377C2×#41—22no—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—22no—
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—22no—
384C2×#41—114no—
385C2×#41—112no—
386C2×#41—114no—
387C2×#41—112no—
388C2×#41—222no—
389C2×#41—221no—
390C2×#41—222no—
391C2×#41—221no—
392C2×#41C4×D1114yescomplemented
393C22×Q8—111no—
394C22×Q8C22×D1111yescharacteristic
395C22×D4—21no—
396C22×D4—111no—
397C22×D4—221no—
398C22×D4C22×D1111yescharacteristic
399C23×C4—21no—
400C23×C4—21no—
401C23×C4—221no—
402C23×C4C22×D1111yescharacteristic
403#137—118no—
404#137—224no—
405#137—228no—
406AGL(1,ℤ/8)—118no—
407AGL(1,ℤ/8)—224no—
408AGL(1,ℤ/8)—228no—
409#135—2216no—
410C2×Q16—222no—
411C2×SD16—222no—
412C2×SD16—222no—
413C2×D8—222no—
414C2×M16—114no—
415C2×M16—112no—
416C2×M16—112no—
417C2×M16—221no—
418C2×M16—222no—
419C2×M16—221no—
420C2×M16—222no—
421C2×M16—222no—
422C22×C8—221no—
423C22×C8—221no—
424C4×Q8—224no—
425C4×Q8—224no—
426C4×D4—224no—
427C4×D4—228no—
428C4×D4—224no—
429C4×D4—228no—
430#117—21no—
431#117—21no—
432#117—21no—
433#117—21no—
434#117—112no—
435#117—112no—
436#117—224no—
437#117—224no—
438#117—224no—
439#117—221no—
440#117—221no—
441C2×C4⋊C4—112no—
442C2×C4⋊C4—224no—
443C2×C4⋊C4C22×D1112yes—
444C2×C22⋊C4—24no—
445C2×C22⋊C4—224no—
446C2×C22⋊C4—221no—
447C2×C22⋊C4C22⋊D1112yes—
448C2×C42—22no—
449C2×C42—22no—
450C2×C42—21no—
451C2×C42—224no—
452C2×C42—221no—
453C2×C42C22⋊D1112yes—
454C2×C42C22×D1112yes—
455#104—228no—
456#103—224no—
457#102—224no—
458#95—24no—
459C2×C11—21no—
460C2×C11—22no—
461C2×C11—21no—
462C2×C11—22no—
463C2×C11—82no—
464C2×C11C2×#38611yescharacteristic
465C2×C11#301812yes—
466D11—44no—
467C24—21no—
468C24—221no—
469C24C2×C22⋊D1111yescharacteristic
470#41—24no—
471#41—24no—
472#41—24no—
473#41—24no—
474#41—42no—
475#41—42no—
476#41—42no—
477#41—42no—
478#41—118no—
479#41—114no—
480#41—114no—
481#41—224no—
482#41—224no—
483#41—224no—
484#41—224no—
485#41—224no—
486#41—222no—
487#41—222no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—114no—
494C2×Q8—112no—
495C2×Q8—222no—
496C2×Q8—221no—
497C2×Q8C2×C4×D1112yes—
498C2×D4—24no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—41no—
505C2×D4—41no—
506C2×D4—42no—
507C2×D4—114no—
508C2×D4—112no—
509C2×D4—224no—
510C2×D4—222no—
511C2×D4—224no—
512C2×D4—222no—
513C2×D4—222no—
514C2×D4—221no—
515C2×D4C2×C4×D1112yes—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—24no—
519C22×C4—44no—
520C22×C4—42no—
521C22×C4—41no—
522C22×C4—41no—
523C22×C4—42no—
524C22×C4—228no—
525C22×C4—221no—
526C22×C4—224no—
527C22×C4—224no—
528C22×C4—222no—
529C22×C4—222no—
530C22×C4—222no—
531C22×C4C2×C4×D1112yes—
532C22×C4#433912yes—
533C22×C4D4×D1111yescharacteristic
534C22×C4D4×D1111yescharacteristic
535C22×C4D4×D1112yes—
536C22×C4D4×D1111yescharacteristic
537C22×C4D4×D1111yescharacteristic
538C22×C4#434112yes—
539C22×C4C2×C22⋊D1114yes—
540C22×C4C2×C22⋊D1111yescharacteristic
541C22×C4C23×D1111yescharacteristic
542Q16—228no—
543SD16—228no—
544SD16—228no—
545D8—228no—
546M16—114no—
547M16—224no—
548M16—222no—
549M16—222no—
550M16—222no—
551M16—442no—
552C2×C8—224no—
553C2×C8—222no—
554C2×C8—222no—
555C2×C8—442no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—114no—
559C4⋊C4—228no—
560C4⋊C4—222no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—48no—
564C22⋊C4—228no—
565C22⋊C4—222no—
566C22⋊C4—222no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—42no—
572C42—228no—
573C42—222no—
574C42—222no—
575C11#2094211yescharacteristic, complemented, Sylow
576C23—22no—
577C23—42no—
578C23—44no—
579C23—41no—
580C23—41no—
581C23—222no—
582C23—224no—
583C23—882no—
584C23C4×C22⋊D1112yes—
585C23#6699011yescharacteristic
586C23#6699011yescharacteristic
587C23#6699112yes—
588C23C22×C22⋊D1111yescharacteristic
589Q8—22no—
590Q8—22no—
591Q8—41no—
592Q8—41no—
593Q8—114no—
594Q8—222no—
595Q8—224no—
596D4—22no—
597D4—22no—
598D4—44no—
599D4—44no—
600D4—41no—
601D4—41no—
602D4—114no—
603D4—224no—
604D4—224no—
605D4—228no—
606D4—222no—
607D4—224no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—222no—
620C2×C4—224no—
621C2×C4—224no—
622C2×C4—224no—
623C2×C4—228no—
624C2×C4—224no—
625C2×C4—888no—
626C2×C4—882no—
627C2×C4C22⋊C4×D1112yes—
628C2×C4C22⋊C4×D1112yes—
629C2×C4(C22⋊D11)⋊C412yes—
630C2×C4(C22⋊D11)⋊C412yes—
631C2×C4#6693612yes—
632C2×C4#6693612yes—
633C2×C4#6693712yes—
634C2×C4#6693712yes—
635C2×C4C4×C22⋊D1112yes—
636C2×C4#6699011yescharacteristic
637C2×C4#6699011yescharacteristic
638C2×C4#6699112yes—
639C2×C4C22×C4×D1111yescharacteristic
640C2×C4C2×#433911yescharacteristic
641C2×C4C2×D4×D1111yescharacteristic
642C2×C4C2×D4×D1111yescharacteristic
643C2×C4C2×D4×D1111yescharacteristic
644C2×C4C2×#434111yescharacteristic
645C8—224no—
646C8—442no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—42no—
654C22—82no—
655C22—82no—
656C22—224no—
657C22—444no—
658C22—884no—
659C22#1062256912yes—
660C22#1062256912yes—
661C22C2×C4×C22⋊D1111yescharacteristic
662C22C2×#6699011yescharacteristic
663C22C2×#6699111yescharacteristic
664C4—44no—
665C4—42no—
666C4—44no—
667C4—224no—
668C4—444no—
669C4#1062256912yes—
670C4#1062256912yes—
671C4C2×C22⋊C4×D1111yescharacteristic
672C4C2×(C22⋊D11)⋊C411yescharacteristic
673C4C2×#6693611yescharacteristic
674C4C2×#6693711yescharacteristic
675C2—21no—
676C2—21no—
677C2—22no—
678C2—22no—
679C2—82no—
680C2—444no—
681C2C2×#1062256911yescharacteristic
682C2#1307796412yes—
subgroupG/H
centerC2×C4#66990
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#2297C2
socleC22×C11C2×#120
prime radicalC2×#10623353C2
cosocleC22×C4×C11C24
hypercenter#2297D11
hypocenterC11#20942
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #13081817C2semidirect
3. C22×#66869C2semidirect
4. #13077135C2semidirect
5. #13068995C2semidirect
6. #13068988C2semidirect
7. #13068355C2semidirect
8. C2×#10622161C2semidirect
9. C11×#2297C2semidirect
16. #10623353C4semidirect
32. C11×#391C22semidirect
33. C11×#390C22semidirect
38. #10622838C22semidirect
39. #10622828C22semidirect
42. #10622168C22semidirect
43. #10622161C22semidirect
51. C2×C11×#41C2×C4semidirect
88. #66995C2×C4semidirect
92. #66985C2×C4semidirect
106. C2×#4318C2×C4semidirect
186. #4339C22⋊C4semidirect
191. C2×D44C22⋊C4semidirect
200. C2×Dic22C22⋊C4semidirect
239. #2297D11semidirect
343. #391C2×D11semidirect
344. #390C2×D11semidirect
392. C2×#41C4×D11semidirect
575. C11#20942semidirect

Maximal subgroups

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

subgroupindex
C2×#106233532
#130818172
C22×#668692
#130771352
#130689952
#130689882
#130683552
C2×#106221612
C11×#22972
#2094211