Finite Group Databasebeta

← #14765609

Subgroups — #14765609

Order 2816.

780 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622428C211yesmaximal, characteristic
2#13071292C211yesmaximal, characteristic
3#13071238C211yesmaximal, characteristic
4#13069357C211yesmaximal, characteristic, complemented
5#13069348C211yesmaximal, characteristic, complemented
6#13069308C211yesmaximal, characteristic, complemented
7#13069289C211yesmaximal, characteristic, complemented
8#13069195C211yesmaximal, characteristic, complemented
9#13069184C211yesmaximal, characteristic, complemented
10#13068920C211yesmaximal, characteristic
11#13068870C211yesmaximal, characteristic, complemented
12#13068802C211yesmaximal, characteristic
13#13068710C211yesmaximal, characteristic
14#13068381C211yesmaximal, characteristic
15C11×#1863C211yesmaximal, characteristic, complemented
16C2×C4×Q8×C11C2211yescharacteristic
17C2×#66999C2211yescharacteristic
18C2×#66945C2211yescharacteristic
19C2×C4⋊C4×D11C2211yescharacteristic
20C2×#66925C2211yescharacteristic
21C2×C11×#107C2211yescharacteristic
22C2×C11×#103C2211yescharacteristic
23C2×C11×#103C2211yescharacteristic
24C11×#389C2211yescharacteristic
25#10622843C2211yescharacteristic
26C2×#66899C2211yescharacteristic
27C2×#66899C2211yescharacteristic
28C2×(C2×D11)⋊C8C2211yescharacteristic
29C2×(C2×D11)⋊C8C2211yescharacteristic
30C2×#66881C2211yescharacteristic
31C2×#66879C2211yescharacteristic
32C2×C8×Dic11C2211yescharacteristic
33#10622603—21no—
34#10622602—21no—
35#10622600C2211yescharacteristic
36#10622598C2211yescharacteristic
37#10622589—21no—
38#10622586C2211yescharacteristic
39#10622586C2211yescharacteristic
40#10622585—21no—
41C2×#66871C2211yescharacteristic
42C2×#66871C2211yescharacteristic
43#10622553—21no—
44C4×C4⋊Dic11C2211yescharacteristic
45#10622546—21no—
46#10622541C2211yescharacteristic
47#10622428C418yescomplemented
48#10622281—21no—
49#10622263—21no—
50C11×#340C2211yescharacteristic
51C11×#336C2211yescharacteristic
52#10622189C2211yescharacteristic
53#10622189C2211yescharacteristic
54#10622189C2211yescharacteristic
55#10622167C2211yescharacteristic
56#10622167C2211yescharacteristic
57#10622167C2211yescharacteristic
58#10622167C2211yescharacteristic
59#10622162C2211yescharacteristic
60C22×Q8×C11C2311yescharacteristic
61C22×C4×D11—21no—
62C22×C4×D11C2311yescharacteristic
63C22×C8×C11C2311yescharacteristic
64C22×C8×C11C2311yescharacteristic
65C4×Q8×C11—41no—
66C4×Q8×C11—41no—
67C2×C11×C4⋊C4—21no—
68C2×C11×C4⋊C4—21no—
69C2×C11×C4⋊C4C2311yescharacteristic
70C2×C11×C4⋊C4C2311yescharacteristic
71C2×C11×C4⋊C4C2311yescharacteristic
72C2×C42×C11—21no—
73C2×C42×C11C2311yescharacteristic
74#66999—22no—
75#66999C2×C414yes—
76C2×(C2×D11)⋊C4—21no—
77C2×(C2×D11)⋊C4—21no—
78C2×(C2×D11)⋊C4—21no—
79C2×(C2×D11)⋊C4D411yescharacteristic
80C2×(C2×D11)⋊C4D411yescharacteristic
81C2×C4⋊Dic11—21no—
82C2×C4⋊Dic11C2311yescharacteristic
83C2×C4⋊Dic11C2311yescharacteristic
84C2×C4⋊Dic11C2311yescharacteristic
85C2×C4⋊Dic11C2311yescharacteristic
86C2×Dic11⋊C4—21no—
87C2×Dic11⋊C4—21no—
88C2×Dic11⋊C4—21no—
89C2×Dic11⋊C4—21no—
90C2×Dic11⋊C4—21no—
91C2×Dic11⋊C4D411yescharacteristic
92C2×Dic11⋊C4D411yescharacteristic
93C2×C4×Dic11—21no—
94C2×C4×Dic11—21no—
95C2×C4×Dic11C2311yescharacteristic
96C22×C11⋊C8C2311yescharacteristic
97C22×C11⋊C8C2311yescharacteristic
98#66945—22no—
99#66945C2×C414yes—
100C4⋊C4×D11—41no—
101C4⋊C4×D11—41no—
102#66925—41no—
103#66925—41no—
104C11×#107—41no—
105C11×#103—41no—
106C11×#103C2×C414yescomplemented
107C11×#95—21no—
108#66899—41no—
109#66899C2×C414yescomplemented
110C11⋊#95—21no—
111C11⋊#95—21no—
112C11⋊#95—21no—
113C11⋊#95—21no—
114C11⋊#95—21no—
115C11⋊#95—21no—
116C11⋊#95—21no—
117C11⋊#95—21no—
118(C2×D11)⋊C8—22no—
119(C2×D11)⋊C8C2×C414yes—
120#66881C2×C414yescomplemented
121#66879—41no—
122C8×Dic11—41no—
123#66871—41no—
124#66871C2×C414yescomplemented
125#22094—111nomaximal, Sylow
126C23×D11C2×D411yescharacteristic
127C2×Q8×C11—21no—
128C2×Q8×C11—21no—
129C2×Q8×C11—21no—
130C2×Q8×C11—21no—
131C2×Q8×C11—21no—
132C2×Q8×C11C22×C412yes—
133C22×C4×C11—21no—
134C22×C4×C11—21no—
135C22×C4×C11—21no—
136C22×C4×C11—21no—
137C22×C4×C11C2×D411yescharacteristic
138C22×C4×C11C2×D411yescharacteristic
139C22×C4×C11C2411yescharacteristic
140C22×Dic11—21no—
141C22×Dic11—21no—
142C22×Dic11—21no—
143C22×Dic11—21no—
144C22×Dic11C2×D411yescharacteristic
145C22×Dic11C2×D411yescharacteristic
146C22×Dic11C2×D411yescharacteristic
147C2×C4×D11—21no—
148C2×C4×D11—21no—
149C2×C4×D11—21no—
150C2×C4×D11—21no—
151C2×C4×D11—42no—
152C2×C4×D11—81no—
153C2×C4×D11—81no—
154C2×C4×D11C22×C412yes—
155C2×C4×D11#4112yes—
156C2×C8×C11—21no—
157C2×C8×C11—21no—
158C2×C8×C11—21no—
159C2×C8×C11—21no—
160C2×C8×C11—21no—
161C2×C8×C11C22×C412yes—
162C11×C4⋊C4—21no—
163C11×C4⋊C4—21no—
164C11×C4⋊C4—21no—
165C11×C4⋊C4—21no—
166C11×C4⋊C4—21no—
167C11×C4⋊C4—42no—
168C11×C4⋊C4—81no—
169C11×C4⋊C4C22×C412yes—
170C42×C11—21no—
171C42×C11—21no—
172C42×C11—42no—
173(C2×D11)⋊C4—24no—
174(C2×D11)⋊C4—24no—
175(C2×D11)⋊C4—41no—
176(C2×D11)⋊C4—41no—
177(C2×D11)⋊C4—81no—
178C4⋊Dic11—21no—
179C4⋊Dic11—21no—
180C4⋊Dic11—21no—
181C4⋊Dic11—21no—
182C4⋊Dic11—21no—
183C4⋊Dic11—21no—
184C4⋊Dic11—81no—
185C4⋊Dic11C22×C412yes—
186C4⋊Dic11C22×C412yes—
187Dic11⋊C4—24no—
188Dic11⋊C4—41no—
189Dic11⋊C4—41no—
190Dic11⋊C4—42no—
191Dic11⋊C4—81no—
192Dic11⋊C4—81no—
193Dic11⋊C4—81no—
194C4×Dic11—24no—
195C4×Dic11—21no—
196C4×Dic11—21no—
197C4×Dic11—81no—
198C2×C11⋊C8—21no—
199C2×C11⋊C8—21no—
200C2×C11⋊C8—21no—
201C2×C11⋊C8—21no—
202C2×C11⋊C8—21no—
203C2×C11⋊C8C22×C412yes—
204C2×#485—111no—
205#2323—111no—
206#2317—111no—
207#1944—111no—
208#1942—111no—
209#1935—111no—
210#1933—111no—
211#1919—111no—
212#1918—111no—
213#1872—111no—
214#1863—111no—
215#1863D1111yescharacteristic, complemented
216#1852—111no—
217#1825—111no—
218#1750—111no—
219C23×C11C22×D411yescharacteristic
220C22×D11—21no—
221C22×D11—21no—
222C22×D11—21no—
223C22×D11—21no—
224C22×D11—21no—
225C22×D11—21no—
226C22×D11C4×D412yes—
227Q8×C11—22no—
228Q8×C11—22no—
229Q8×C11—41no—
230Q8×C11—41no—
231C2×C4×C11—22no—
232C2×C4×C11—22no—
233C2×C4×C11—21no—
234C2×C4×C11—21no—
235C2×C4×C11—21no—
236C2×C4×C11—21no—
237C2×C4×C11—21no—
238C2×C4×C11—21no—
239C2×C4×C11—41no—
240C2×C4×C11—41no—
241C2×C4×C11—41no—
242C2×C4×C11—41no—
243C2×C4×C11—41no—
244C2×C4×C11—41no—
245C2×C4×C11—41no—
246C2×C4×C11—41no—
247C2×C4×C11—41no—
248C2×C4×C11—41no—
249C2×C4×C11C23×C411yescharacteristic
250C2×C4×C11C2×#4111yescharacteristic
251C2×C4×C11H(5,𝔽2)11yescharacteristic
252C2×C4×C11H(5,𝔽2)11yescharacteristic
253C2×C4×C112−1+411yescharacteristic
254C2×C4×C112−1+411yescharacteristic
255C2×Dic11—21no—
256C2×Dic11—21no—
257C2×Dic11—22no—
258C2×Dic11—22no—
259C2×Dic11—21no—
260C2×Dic11—21no—
261C2×Dic11—21no—
262C2×Dic11—21no—
263C2×Dic11—21no—
264C2×Dic11—21no—
265C2×Dic11—41no—
266C2×Dic11—41no—
267C2×Dic11—41no—
268C2×Dic11—41no—
269C2×Dic11—41no—
270C2×Dic11—41no—
271C2×Dic11—41no—
272C2×Dic11—41no—
273C2×Dic11—41no—
274C2×Dic11—41no—
275C2×Dic11C4×D412yes—
276C4×D11—22no—
277C4×D11—22no—
278C4×D11—22no—
279C4×D11—22no—
280C4×D11—84no—
281C8×C11—22no—
282C8×C11—41no—
283C11⋊C8—22no—
284C11⋊C8—41no—
285C2×C22⋊Q8—111no—
286C2×C4×Q8C2×D1111yescharacteristic
287C2×#117—111no—
288C2×#117—111no—
289C22×C4⋊C4—111no—
290#485—118no—
291C2×#107—111no—
292C2×#107—111no—
293C2×#107C2×D1111yescharacteristic
294C2×#106—111no—
295C2×#103—111no—
296C2×#103—111no—
297C2×#103C2×D1111yescharacteristic
298C2×#103C2×D1111yescharacteristic
299C2×C22⋊C8—111no—
300C2×C22⋊C8—111no—
301C2×#97—111no—
302C2×C4×C8—111no—
303#391—111no—
304#389—111no—
305#389—111no—
306#389—111no—
307#389—111no—
308#389—221no—
309#389C2×D1111yescharacteristic
310#388—221no—
311#388—221no—
312#387—221no—
313#385—221no—
314#385—221no—
315#384—221no—
316#382—221no—
317#380—111no—
318C4×C4⋊C4—111no—
319#340—111no—
320#340—111no—
321#340—111no—
322#340—111no—
323#340—111no—
324#340—111no—
325#340—111no—
326#340C2×D1111yescharacteristic
327#336—111no—
328#336C2×D1111yescharacteristic
329C22×C11C2×C4×D411yescharacteristic
330C22×C11#53411yescharacteristic
331C22×C11#53611yescharacteristic
332C22×C11#57611yescharacteristic
333C22×C11#57711yescharacteristic
334C22×C11#57811yescharacteristic
335C22×C11#57811yescharacteristic
336C2×D11—22no—
337C2×D11—22no—
338C2×D11—22no—
339C2×D11—41no—
340C2×D11—41no—
341C2×D11—41no—
342C2×D11—41no—
343C4×C11—42no—
344C4×C11—41no—
345C4×C11—41no—
346C4×C11—42no—
347C4×C11—81no—
348C4×C11—81no—
349C4×C11#52011yescharacteristic
350C4×C11#52011yescharacteristic
351C4×C11#53911yescharacteristic
352C4×C11#53911yescharacteristic
353Dic11—22no—
354Dic11—41no—
355Dic11—41no—
356Dic11—42no—
357Dic11—42no—
358Dic11—81no—
359Dic11—81no—
360C22×Q8C22×D1111yescharacteristic
361C23×C4—111no—
362C23×C4—221no—
363C22×C8—111no—
364C22×C8—111no—
365C22×C8C22×D1111yescharacteristic
366C22×C8C22×D1111yescharacteristic
367C22⋊Q8—114no—
368C22⋊Q8—222no—
369C4×Q8—41no—
370C4×Q8—41no—
371#117—114no—
372#117—222no—
373#117—441no—
374#117—441no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—111no—
378C2×C4⋊C4—111no—
379C2×C4⋊C4—111no—
380C2×C4⋊C4—111no—
381C2×C4⋊C4—111no—
382C2×C4⋊C4—111no—
383C2×C4⋊C4—221no—
384C2×C4⋊C4—221no—
385C2×C4⋊C4—221no—
386C2×C4⋊C4—221no—
387C2×C4⋊C4—221no—
388C2×C4⋊C4—221no—
389C2×C4⋊C4—441no—
390C2×C4⋊C4—441no—
391C2×C4⋊C4C22×D1111yescharacteristic
392C2×C4⋊C4C22×D1111yescharacteristic
393C2×C4⋊C4C22×D1111yescharacteristic
394C2×C22⋊C4—111no—
395C2×C22⋊C4—111no—
396C2×C22⋊C4—221no—
397C2×C22⋊C4—221no—
398C2×C22⋊C4—221no—
399C2×C42—21no—
400C2×C42—111no—
401C2×C42—221no—
402C2×C42—221no—
403C2×C42C22×D1111yescharacteristic
404#107—41no—
405#107—114no—
406#107—441no—
407#106—114no—
408#103—41no—
409#103—114no—
410#103—441no—
411#103C4×D1114yescomplemented
412C22⋊C8—114no—
413C22⋊C8—222no—
414#97—441no—
415C4×C8—441no—
416#95—21no—
417#95—221no—
418#95—221no—
419#95—221no—
420#95—221no—
421#95—221no—
422#95—221no—
423#95—221no—
424#95—221no—
425C2×C11#232211yescharacteristic
426C2×C11#294711yescharacteristic
427C2×C11#295111yescharacteristic
428C2×C11#319811yescharacteristic
429C2×C11#320211yescharacteristic
430C2×C11#324211yescharacteristic
431C2×C11#324311yescharacteristic
432D11—42no—
433C24—111no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C2×C4×D1112yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—111no—
445C22×C4—112no—
446C22×C4—112no—
447C22×C4—111no—
448C22×C4—111no—
449C22×C4—221no—
450C22×C4—221no—
451C22×C4—221no—
452C22×C4—221no—
453C22×C4—221no—
454C22×C4—221no—
455C22×C4—221no—
456C22×C4—221no—
457C22×C4—442no—
458C22×C4—881no—
459C22×C4—881no—
460C22×C4D4×D1111yescharacteristic
461C22×C4D4×D1111yescharacteristic
462C22×C4C23×D1111yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—112no—
469C2×C8—221no—
470C2×C8—221no—
471C2×C8—221no—
472C2×C8—221no—
473C2×C8—221no—
474C2×C8C2×C4×D1112yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—112no—
483C4⋊C4—112no—
484C4⋊C4—224no—
485C4⋊C4—221no—
486C4⋊C4—221no—
487C4⋊C4—221no—
488C4⋊C4—221no—
489C4⋊C4—221no—
490C4⋊C4—221no—
491C4⋊C4—441no—
492C4⋊C4—441no—
493C4⋊C4—442no—
494C4⋊C4—881no—
495C4⋊C4—881no—
496C4⋊C4—881no—
497C4⋊C4—881no—
498C4⋊C4C2×C4×D1112yes—
499C22⋊C4—224no—
500C22⋊C4—224no—
501C22⋊C4—441no—
502C22⋊C4—441no—
503C22⋊C4—881no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—224no—
508C42—221no—
509C42—221no—
510C42—881no—
511C11#2209411yescharacteristic, complemented, Sylow
512C23—112no—
513C23—221no—
514C23—221no—
515C23—221no—
516C23—221no—
517C23—221no—
518C23—221no—
519C23C2×D4×D1111yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—112no—
543C2×C4—221no—
544C2×C4—221no—
545C2×C4—222no—
546C2×C4—222no—
547C2×C4—222no—
548C2×C4—222no—
549C2×C4—221no—
550C2×C4—221no—
551C2×C4—221no—
552C2×C4—221no—
553C2×C4—222no—
554C2×C4—222no—
555C2×C4—221no—
556C2×C4—221no—
557C2×C4—441no—
558C2×C4—441no—
559C2×C4—441no—
560C2×C4—441no—
561C2×C4—441no—
562C2×C4—441no—
563C2×C4—441no—
564C2×C4—441no—
565C2×C4—441no—
566C2×C4—441no—
567C2×C4—884no—
568C2×C4C22×C4×D1111yescharacteristic
569C2×C4#6703711yescharacteristic
570C2×C4#6703711yescharacteristic
571C2×C4#6704011yescharacteristic
572C2×C4#41×D1111yescharacteristic
573C2×C4#6704311yescharacteristic
574C8—22no—
575C8—41no—
576C8—222no—
577C8—441no—
578C22—222no—
579C22—222no—
580C22—222no—
581C22—441no—
582C22—441no—
583C22—441no—
584C22—441no—
585C22C4×D4×D1111yescharacteristic
586C22#1062312011yescharacteristic
587C22#1062315911yescharacteristic
588C22#1062329711yescharacteristic
589C22#1062330111yescharacteristic
590C22#1062330511yescharacteristic
591C22#1062331011yescharacteristic
592C4—41no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—222no—
599C4—441no—
600C4—442no—
601C4—441no—
602C4—442no—
603C4—881no—
604C4—881no—
605C4#1062303511yescharacteristic
606C4#1062307711yescharacteristic
607C4#1062316911yescharacteristic
608C4#1062319111yescharacteristic
609C2—442no—
610C2#1307128311yescharacteristic
611C2#1307745411yescharacteristic
612C2#1307748811yescharacteristic
613C2#1307946711yescharacteristic
614C2#1307950911yescharacteristic
615C2#1307988111yescharacteristic
616C2#1307990411yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1863C2
socleC23×C11C22×D4
prime radicalC23×D11C2×D4
cosocleC22×C4×C11C24
hypercenter#1863D11
hypocenterC11#22094
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
4. #13069357C2semidirect
5. #13069348C2semidirect
6. #13069308C2semidirect
7. #13069289C2semidirect
8. #13069195C2semidirect
9. #13069184C2semidirect
11. #13068870C2semidirect
15. C11×#1863C2semidirect
47. #10622428C4semidirect
106. C11×#103C2×C4semidirect
109. #66899C2×C4semidirect
120. #66881C2×C4semidirect
124. #66871C2×C4semidirect
215. #1863D11semidirect
411. #103C4×D11semidirect
511. C11#22094semidirect

Maximal subgroups

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

subgroupindex
C2×#106224282
#130712922
#130712382
#130693572
#130693482
#130693082
#130692892
#130691952
#130691842
#130689202
#130688702
#130688022
#130687102
#130683812
C11×#18632
#2209411