Finite Group Databasebeta

← #14759277

Subgroups — #14759277

Order 2816.

1598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622740C212yesmaximal, complemented
2C2×C8×C22⋊D11C211yesmaximal, characteristic, complemented
3C4×#67006C211yesmaximal, characteristic
4#13068913C212yesmaximal, complemented
5#13068908C211yesmaximal, characteristic, complemented
6#13068799C212yesmaximal, complemented
7#13068794C211yesmaximal, characteristic, complemented
8#13068526C211yesmaximal, characteristic, complemented
9#13068397C212yesmaximal, complemented
10#13068392C211yesmaximal, characteristic, complemented
11C23×C11×M16C211yesmaximal, characteristic, complemented
12C24×C4×C11C2211yescharacteristic
13C22×C11×M16—28no—
14C22×C11×M16—24no—
15C22×C11×M16C2212yes—
16C22×C11×M16C2212yes—
17C23×C8×C11C2211yescharacteristic
18C23×C8×C11C2211yescharacteristic
19C2×#67006C411yescharacteristic
20C2×C4×C22⋊D11C2211yescharacteristic
21C2×C4×C22⋊D11C2212yes—
22C22×C8×D11C2211yescharacteristic, complemented
23#10622860C411yescharacteristic
24#10622826C411yescharacteristic
25#10622826C412yes—
26#10622823C412yes—
27#10622823C411yescharacteristic
28C4×C22⋊Dic11C2211yescharacteristic
29C4×C22⋊Dic11C2212yes—
30C2×C22⋊(C11⋊C8)C2211yescharacteristic
31C2×C22⋊(C11⋊C8)C2212yes—
32C2×C22⋊(C11⋊C8)C2211yescharacteristic
33C2×C22⋊(C11⋊C8)C2212yes—
34#10622740—28no—
35C8×C22⋊D11—24no—
36C2×(C2×D11)⋊C8C2211yescharacteristic, complemented
37C2×(C2×D11)⋊C8C2212yescomplemented
38C2×#66879C2212yescomplemented
39C2×Dic11⋊C8C2211yescharacteristic, complemented
40C2×Dic11⋊C8C2212yescomplemented
41C2×C8×Dic11C2211yescharacteristic, complemented
42#10622188C2212yescomplemented
43#10622188C2212yescomplemented
44#10622188C2212yescomplemented
45C25×C11C2×C411yescharacteristic
46C23×C4×C11—24no—
47C23×C4×C11—22no—
48C23×C4×C11—24no—
49C23×C4×C11—22no—
50C23×C4×C11C2×C412yes—
51C23×C4×C11C2×C411yescharacteristic
52C23×C4×C11C2311yescharacteristic
53C23×C4×C11C2312yes—
54C22×C22⋊D11C2×C412yes—
55C22×C22⋊D11C2×C411yescharacteristic
56C22×C4×D11C2311yescharacteristic
57C2×C11×M16—232no—
58C2×C11×M16—24no—
59C2×C11×M16—24no—
60C2×C11×M16—24no—
61C2×C11×M16—24no—
62C2×C11×M16—24no—
63C2×C11×M16D418yescomplemented
64C22×C8×C11—22no—
65C22×C8×C11—24no—
66C22×C8×C11—24no—
67C22×C8×C11D414yescomplemented
68C22×C8×C11C2312yes—
69C22×C8×C11C2311yescharacteristic
70C22×C8×C11C2311yescharacteristic
71#67006—24no—
72C2×C22⋊Dic11C2×C412yes—
73C2×C22⋊Dic11C2×C411yescharacteristic
74C2×C22⋊Dic11C2×C412yes—
75C2×C22⋊Dic11C2×C411yescharacteristic
76C2×C22⋊Dic11C2×C411yescharacteristic
77C2×C22⋊Dic11C2×C412yes—
78C4×C22⋊D11—22no—
79C4×C22⋊D11—24no—
80C4×C22⋊D11—22no—
81C4×C22⋊D11—24no—
82C2×(C2×D11)⋊C4C2×C411yescharacteristic
83C2×(C2×D11)⋊C4C2×C412yes—
84C2×Dic11⋊C4C2×C412yes—
85C2×Dic11⋊C4C2×C411yescharacteristic
86C2×C4×Dic11C2311yescharacteristic
87C2×C4×Dic11C2312yes—
88C22×C11⋊C8C2311yescharacteristic
89C22×C11⋊C8C2312yes—
90C22×C11⋊C8C2311yescharacteristic
91C2×C8×D11—22no—
92C2×C8×D11—24no—
93C11⋊#95C2×C412yes—
94C11⋊#95C2×C412yes—
95C11⋊#95C2×C412yes—
96C22⋊(C11⋊C8)—22no—
97C22⋊(C11⋊C8)—24no—
98C22⋊(C11⋊C8)—22no—
99C22⋊(C11⋊C8)—24no—
100(C2×D11)⋊C8—22no—
101(C2×D11)⋊C8—24no—
102#66879—24no—
103Dic11⋊C8—22no—
104Dic11⋊C8—24no—
105C8×Dic11—22no—
106#21439—111nomaximal, Sylow
107C24×C11—24no—
108C24×C11—22no—
109C24×C11—22no—
110C24×C11—22no—
111C24×C11—42no—
112C24×C11C22×C412yes—
113C24×C11C22×C411yescharacteristic
114C23×D11C22×C411yescharacteristic
115C22×C4×C11—28no—
116C22×C4×C11—28no—
117C22×C4×C11—216no—
118C22×C4×C11—24no—
119C22×C4×C11—22no—
120C22×C4×C11—24no—
121C22×C4×C11—24no—
122C22×C4×C11—24no—
123C22×C4×C11—22no—
124C22×C4×C11—22no—
125C22×C4×C11—22no—
126C22×C4×C11—22no—
127C22×C4×C11—22no—
128C22×C4×C11—22no—
129C22×C4×C11C22×C411yescharacteristic
130C22×C4×C11C22×C412yes—
131C22×C4×C11C2×D412yes—
132C22×C4×C11C2×D414yes—
133C22×C4×C11#4114yes—
134C22×C4×C11#4112yes—
135C22×C4×C11C2411yescharacteristic
136C2×C22⋊D11—22no—
137C2×C22⋊D11—24no—
138C2×C22⋊D11—22no—
139C2×C22⋊D11—24no—
140C2×C22⋊D11—42no—
141C2×C22⋊D11M1614yescomplemented
142C22×Dic11C22×C412yes—
143C22×Dic11C22×C412yes—
144C22×Dic11C22×C411yescharacteristic
145C22×Dic11C22×C411yescharacteristic
146C22×Dic11C22×C411yescharacteristic
147C2×C4×D11—21no—
148C2×C4×D11—22no—
149C2×C4×D11—21no—
150C2×C4×D11—22no—
151C11×M16—216no—
152C11×M16—216no—
153C2×C8×C11—28no—
154C2×C8×C11—28no—
155C2×C8×C11—22no—
156C2×C8×C11—22no—
157C2×C8×C11—22no—
158C2×C8×C11C2×D414yes—
159C2×C8×C11C2×D412yes—
160C2×C8×C11C2×D412yes—
161C2×C8×C11C2×D414yes—
162C22⋊Dic11—24no—
163C22⋊Dic11—24no—
164C22⋊Dic11—22no—
165C22⋊Dic11—22no—
166C22⋊Dic11—22no—
167C22⋊Dic11—42no—
168(C2×D11)⋊C4—22no—
169(C2×D11)⋊C4—24no—
170Dic11⋊C4—22no—
171Dic11⋊C4—24no—
172C4×Dic11—21no—
173C4×Dic11—22no—
174C4×Dic11—21no—
175C4×Dic11—22no—
176C2×C11⋊C8—21no—
177C2×C11⋊C8—21no—
178C2×C11⋊C8—22no—
179C2×C11⋊C8—22no—
180C2×C11⋊C8—22no—
181C2×C11⋊C8—22no—
182C2×C11⋊C8—22no—
183C8×D11—44no—
184C23×M16D1111yescharacteristic, complemented
185C2×#435—112no—
186C2×C8×D4—111no—
187C4×#120—111no—
188#1870—112no—
189#1869—111no—
190#1851—112no—
191#1850—111no—
192#1779—111no—
193#1753—112no—
194#1752—111no—
195C23×C11—22no—
196C23×C11—22no—
197C23×C11—28no—
198C23×C11—22no—
199C23×C11—22no—
200C23×C11—22no—
201C23×C11—22no—
202C23×C11—22no—
203C23×C11—22no—
204C23×C11—48no—
205C23×C11—48no—
206C23×C11—44no—
207C23×C11—44no—
208C23×C11C4×D414yes—
209C23×C11C4×D412yes—
210C23×C11C2×M1612yes—
211C23×C11#13112yes—
212C23×C11C23×C411yescharacteristic
213C22×D11—21no—
214C22×D11—22no—
215C22×D11—21no—
216C22×D11—22no—
217C22×D11C2×M1612yescomplemented
218C2×C4×C11—28no—
219C2×C4×C11—216no—
220C2×C4×C11—22no—
221C2×C4×C11—22no—
222C2×C4×C11—22no—
223C2×C4×C11—22no—
224C2×C4×C11—24no—
225C2×C4×C11—22no—
226C2×C4×C11—24no—
227C2×C4×C11—24no—
228C2×C4×C11—24no—
229C2×C4×C11—24no—
230C2×C4×C11C4×D412yes—
231C2×C4×C11C4×D414yes—
232C2×C4×C11C22×D411yescharacteristic
233C2×C4×C11C22×D412yes—
234C2×C4×C11C2×#4111yescharacteristic
235C2×C4×C11C2×#4112yes—
236C22⋊D11—28no—
237C22⋊D11—48no—
238C2×Dic11—21no—
239C2×Dic11—21no—
240C2×Dic11—22no—
241C2×Dic11—21no—
242C2×Dic11—22no—
243C2×Dic11—21no—
244C2×Dic11—22no—
245C2×Dic11—22no—
246C2×Dic11—22no—
247C2×Dic11—22no—
248C2×Dic11—22no—
249C2×Dic11—22no—
250C2×Dic11C2×M1612yescomplemented
251C4×D11—41no—
252C4×D11—41no—
253C4×D11—42no—
254C8×C11—24no—
255C8×C11—22no—
256C8×C11#12014yescomplemented
257C11⋊C8—41no—
258C11⋊C8—41no—
259C11⋊C8—42no—
260C24×C4C2×D1111yescharacteristic
261C22×M16—28no—
262C22×M16—24no—
263C22×M16C2×D1112yes—
264C22×M16C2×D1112yes—
265C23×C8—111no—
266C23×C8C2×D1111yescharacteristic
267C23×C8C2×D1111yescharacteristic
268C2×#120—111no—
269C2×C4×D4—111no—
270C2×C4×D4—112no—
271#435—228no—
272C8×D4—224no—
273C2×C4⋊C8—112no—
274C2×C4⋊C8—111no—
275C2×C22⋊C8—111no—
276C2×C22⋊C8—112no—
277C2×C22⋊C8—111no—
278C2×C22⋊C8—112no—
279C2×C22⋊C8—112no—
280C2×C22⋊C8—111no—
281C2×#97—112no—
282C2×C4×C8—111no—
283#386—111no—
284#386—112no—
285#385—112no—
286#385—111no—
287#379—111no—
288C4×C22⋊C4—111no—
289C4×C22⋊C4—112no—
290#336—112no—
291#336—112no—
292#336—112no—
293C22×C11—22no—
294C22×C11—24no—
295C22×C11—24no—
296C22×C11—22no—
297C22×C11—24no—
298C22×C11—24no—
299C22×C11—24no—
300C22×C11—416no—
301C22×C11—42no—
302C22×C11—42no—
303C22×C11—42no—
304C22×C11—42no—
305C22×C11#43514yescomplemented
306C22×C11C2×C4×D411yescharacteristic
307C22×C11C2×C4×D412yes—
308C22×C11C22×M1611yescharacteristic
309C22×C11C2×#13111yescharacteristic
310C22×C11#56812yes—
311C2×D11—22no—
312C2×D11—24no—
313C2×D11—41no—
314C2×D11—42no—
315C2×D11—41no—
316C4×C11—22no—
317C4×C11—24no—
318C4×C11C2×#12011yescharacteristic
319C4×C11#52511yescharacteristic
320C4×C11#52512yes—
321Dic11—22no—
322Dic11—41no—
323Dic11—42no—
324Dic11—42no—
325Dic11—41no—
326C25C4×D1111yescharacteristic
327C22×D4—111no—
328C22×D4—112no—
329C23×C4—24no—
330C23×C4—22no—
331C23×C4—24no—
332C23×C4—22no—
333C23×C4—111no—
334C23×C4C4×D1112yes—
335C23×C4C4×D1111yescharacteristic
336C23×C4C22×D1111yescharacteristic
337C23×C4C22×D1112yes—
338C2×M16—232no—
339C2×M16—24no—
340C2×M16—24no—
341C2×M16—24no—
342C2×M16—24no—
343C2×M16—24no—
344C2×M16C22⋊D1118yescomplemented
345C22×C8—22no—
346C22×C8—24no—
347C22×C8—24no—
348C22×C8—111no—
349C22×C8—111no—
350C22×C8—112no—
351C22×C8—222no—
352C22×C8—224no—
353C22×C8C22⋊D1114yescomplemented
354C22×C8C22×D1112yes—
355C22×C8C22×D1111yescharacteristic
356C22×C8C22×D1111yescharacteristic
357#120—224no—
358C4×D4—222no—
359C4×D4—224no—
360C4×D4—222no—
361C4×D4—224no—
362C2×C4⋊C4—111no—
363C2×C4⋊C4—112no—
364C2×C22⋊C4—112no—
365C2×C22⋊C4—111no—
366C2×C22⋊C4—112no—
367C2×C22⋊C4—111no—
368C2×C22⋊C4—111no—
369C2×C22⋊C4—111no—
370C2×C22⋊C4—112no—
371C2×C22⋊C4—112no—
372C2×C42—111no—
373C2×C42—112no—
374C4⋊C8—222no—
375C4⋊C8—224no—
376C22⋊C8—222no—
377C22⋊C8—224no—
378C22⋊C8—222no—
379C22⋊C8—224no—
380C22⋊C8—222no—
381C22⋊C8—224no—
382#97—224no—
383C4×C8—222no—
384#95—112no—
385#95—112no—
386#95—112no—
387C2×C11—24no—
388C2×C11—44no—
389C2×C11C4×#12011yescharacteristic
390C2×C11C2×#43511yescharacteristic
391C2×C11#293011yescharacteristic
392C2×C11#293312yes—
393C2×C11D4×M1612yes—
394D11—42no—
395C24—24no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—42no—
400C24—111no—
401C24C2×C4×D1112yes—
402C24C2×C4×D1111yescharacteristic
403C2×D4—114no—
404C2×D4—222no—
405C2×D4—222no—
406C2×D4—224no—
407C2×D4—224no—
408C2×D4—442no—
409C22×C4—28no—
410C22×C4—216no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—111no—
424C22×C4—112no—
425C22×C4—111no—
426C22×C4—112no—
427C22×C4—111no—
428C22×C4—221no—
429C22×C4—222no—
430C22×C4—221no—
431C22×C4—222no—
432C22×C4C2×C4×D1111yescharacteristic
433C22×C4C2×C4×D1112yes—
434C22×C4#433914yes—
435C22×C4#433912yes—
436C22×C4C2×C22⋊D1112yes—
437C22×C4C2×C22⋊D1114yes—
438C22×C4C23×D1111yescharacteristic
439M16—216no—
440M16—216no—
441C2×C8—28no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—28no—
446C2×C8—222no—
447C2×C8—222no—
448C2×C8—222no—
449C2×C8—221no—
450C2×C8—222no—
451C2×C8—221no—
452C2×C8—222no—
453C2×C8—444no—
454C2×C8C2×C22⋊D1114yes—
455C2×C8C2×C22⋊D1112yes—
456C2×C8C2×C22⋊D1112yes—
457C2×C8C2×C22⋊D1114yes—
458C4⋊C4—222no—
459C4⋊C4—224no—
460C22⋊C4—224no—
461C22⋊C4—222no—
462C22⋊C4—222no—
463C22⋊C4—224no—
464C22⋊C4—222no—
465C22⋊C4—222no—
466C22⋊C4—224no—
467C22⋊C4—442no—
468C42—221no—
469C42—222no—
470C42—221no—
471C42—222no—
472C11#2143911yescharacteristic, complemented, Sylow
473C23—22no—
474C23—22no—
475C23—28no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—48no—
483C23—48no—
484C23—44no—
485C23—44no—
486C23—112no—
487C23—221no—
488C23—222no—
489C23—221no—
490C23—222no—
491C23M16×D1112yes—
492C23#6696012yes—
493C23C4×C22⋊D1114yes—
494C23C4×C22⋊D1112yes—
495C23C22×C4×D1111yescharacteristic
496D4—228no—
497D4—448no—
498C2×C4—28no—
499C2×C4—216no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—112no—
511C2×C4—221no—
512C2×C4—221no—
513C2×C4—222no—
514C2×C4—222no—
515C2×C4—222no—
516C2×C4—222no—
517C2×C4—222no—
518C2×C4—221no—
519C2×C4—222no—
520C2×C4—222no—
521C2×C4—221no—
522C2×C4—222no—
523C2×C4—441no—
524C2×C4—442no—
525C2×C4—441no—
526C2×C4C4×C22⋊D1112yes—
527C2×C4C4×C22⋊D1114yes—
528C2×C4C2×#433911yescharacteristic
529C2×C4C2×#433912yes—
530C2×C4C22×C22⋊D1111yescharacteristic
531C2×C4C22×C22⋊D1112yes—
532C8—24no—
533C8—22no—
534C8—441no—
535C8—442no—
536C8—441no—
537C8#6700614yescomplemented
538C22—22no—
539C22—22no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—416no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—222no—
551C22—224no—
552C22—441no—
553C22—441no—
554C22—442no—
555C22#1062274014yescomplemented
556C22C2×M16×D1111yescharacteristic
557C22C2×#6696011yescharacteristic
558C22#1062327612yes—
559C22C2×C4×C22⋊D1111yescharacteristic
560C22C2×C4×C22⋊D1112yes—
561C4—22no—
562C4—24no—
563C4—222no—
564C4—441no—
565C4—441no—
566C4—442no—
567C4—442no—
568C4#1062332211yescharacteristic
569C4#1062332212yes—
570C4C2×#6700611yescharacteristic
571C2—24no—
572C2—44no—
573C2—442no—
574C2C4×#6700611yescharacteristic
575C2C2×#1062274011yescharacteristic
576C2#1307731911yescharacteristic
577C2#1307734112yes—
578C2M16×C22⋊D1112yes—
subgroupG/H
center, FrattiniC22×C4C23×D11
commutator, socleC23×C11C23×C4
FittingC23×C11×M16C2
prime radicalC2×#67006C4
cosocleC22×C4×C11C24
hypercenterC23×M16D11
hypocenterC11#21439
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C22×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
1. C2×#10622740C2semidirect
2. C2×C8×C22⋊D11C2semidirect
4. #13068913C2semidirect
5. #13068908C2semidirect
6. #13068799C2semidirect
7. #13068794C2semidirect
8. #13068526C2semidirect
9. #13068397C2semidirect
10. #13068392C2semidirect
11. C23×C11×M16C2semidirect
22. C22×C8×D11C22semidirect
36. C2×(C2×D11)⋊C8C22semidirect
37. C2×(C2×D11)⋊C8C22semidirect
38. C2×#66879C22semidirect
39. C2×Dic11⋊C8C22semidirect
40. C2×Dic11⋊C8C22semidirect
41. C2×C8×Dic11C22semidirect
42. #10622188C22semidirect
43. #10622188C22semidirect
44. #10622188C22semidirect
63. C2×C11×M16D4semidirect
67. C22×C8×C11D4semidirect
141. C2×C22⋊D11M16semidirect
184. C23×M16D11semidirect
217. C22×D11C2×M16semidirect
250. C2×Dic11C2×M16semidirect
256. C8×C11#120semidirect
305. C22×C11#435semidirect
344. C2×M16C22⋊D11semidirect
353. C22×C8C22⋊D11semidirect
472. C11#21439semidirect
537. C8#67006semidirect
555. C22#10622740semidirect

Maximal subgroups

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

subgroupindex
C2×#106227402
C2×C8×C22⋊D112
C4×#670062
#130689132
#130689082
#130687992
#130687942
#130685262
#130683972
#130683922
C23×C11×M162
#2143911