Finite Group Databasebeta

← #14811337

Subgroups — #14811337

Order 2816.

842 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13078478C211yesmaximal, characteristic, complemented
2#13078465C211yesmaximal, characteristic, complemented
3#13077149C211yesmaximal, characteristic
4#13070162C211yesmaximal, characteristic, complemented
5#13069983C211yesmaximal, characteristic, complemented
6#13069981C211yesmaximal, characteristic, complemented
7#13069775C211yesmaximal, characteristic, complemented
8#13068893C211yesmaximal, characteristic, complemented
9#13068891C211yesmaximal, characteristic, complemented
10#13068849C211yesmaximal, characteristic, complemented
11#13068847C211yesmaximal, characteristic, complemented
12#13068680C211yesmaximal, characteristic, complemented
13#13068673C211yesmaximal, characteristic, complemented
14#13068575C211yesmaximal, characteristic
15C11×#2898C211yesmaximal, characteristic
16C2×C11×2−1+4C2211yescharacteristic
17C2×C11×#131C2211yescharacteristic
18C11×#520C2211yescharacteristic
19#10623358C2211yescharacteristic
20C2×#67001C2211yescharacteristic
21#10623050C2211yescharacteristic
22C11×#419—21no—
23C11×#419C2211yescharacteristic
24C11×#417—21no—
25C11×#417C2211yescharacteristic
26C11×#416—21no—
27C11×#416C2211yescharacteristic
28C2×C11×#103—21no—
29C2×C11×#103C2211yescharacteristic
30C2×#66901C2211yescharacteristic
31C2×#66901C2211yescharacteristic
32#10622846—21no—
33#10622846C2211yescharacteristic
34#10622845—21no—
35#10622845C2211yescharacteristic
36#10622845C2211yescharacteristic
37#10622845C2211yescharacteristic
38#10622843C2211yescharacteristic
39C2×#66900C2211yescharacteristic
40#10622838—21no—
41#10622838C2211yescharacteristic
42C2×#66899—21no—
43C2×#66899C2211yescharacteristic
44C2×#66886C2211yescharacteristic
45C2×#66883C2211yescharacteristic
46C2×#66882C2211yescharacteristic
47#10622715C2211yescharacteristic
48#10622635—21no—
49#10622633—21no—
50#10622622—21no—
51#10622621—21no—
52#10622612C2211yescharacteristic
53#10622608C2211yescharacteristic
54#10622590C2211yescharacteristic
55#10622577C2211yescharacteristic
56C2×#66872C2211yescharacteristic
57#10622409—21no—
58#10622407—21no—
59#10622379—21no—
60#10622372—21no—
61#10622193C2211yescharacteristic
62#10622192C2211yescharacteristic
63#10622192C2211yescharacteristic
64#10622188C2211yescharacteristic
65#10622170C2211yescharacteristic
66#10622168C2211yescharacteristic
67C11×2−1+4—22no—
68C11×2−1+4—22no—
69C11×2−1+4—41no—
70C11×2−1+4Q814yescomplemented
71C2×C11×#41—21no—
72C2×C11×#41—21no—
73C2×C11×#41—41no—
74C2×C11×#41C2311yescharacteristic
75C2×C11×#41C2311yescharacteristic
76C22×Q8×C11—21no—
77C22×Q8×C11—21no—
78C22×Q8×C11C2311yescharacteristic
79C11×#131—22no—
80C11×#131—41no—
81C2×C11×M16—21no—
82C2×C11×M16C2311yescharacteristic
83C22×C8×C11—21no—
84C22×C8×C11C2311yescharacteristic
85C4×Q8×C11—21no—
86C4×D4×C11—21no—
87C4×D4×C11—41no—
88C11×#117—21no—
89C11×#117C2311yescharacteristic
90C2×C11×C4⋊C4—21no—
91C2×C11×C4⋊C4C2311yescharacteristic
92#67001—22no—
93#67001—41no—
94Q8×Dic11—21no—
95D4×Dic11—21no—
96D4×Dic11—41no—
97#66979—21no—
98#66979C2311yescharacteristic
99C2×C4⋊Dic11—21no—
100C2×C4⋊Dic11C2311yescharacteristic
101C2×C4⋊Dic11C2311yescharacteristic
102C2×C4×Dic11C2311yescharacteristic
103C2×#4318—21no—
104C2×#4318C2311yescharacteristic
105C2×#4318C2311yescharacteristic
106C2×#4318C2311yescharacteristic
107C22×C11⋊C8—21no—
108C22×C11⋊C8C2311yescharacteristic
109#66943—21no—
110#66941—21no—
111#66931—41no—
112#66923—21no—
113#66922—21no—
114C11×#103—21no—
115C11×#103—21no—
116C11×#103—21no—
117C11×#103—21no—
118C11×#103—21no—
119C11×#103D411yescharacteristic
120C11×#103D411yescharacteristic
121C11×#102—21no—
122C11×#102D411yescharacteristic
123C11×#102D411yescharacteristic
124#66901—22no—
125#66901—22no—
126#66900—41no—
127#66899—21no—
128#66899—21no—
129#66899—21no—
130#66899—21no—
131#66899—21no—
132#66899—21no—
133#66899—21no—
134#66899D411yescharacteristic
135#66899D411yescharacteristic
136#66896—21no—
137#66896—21no—
138#66896—21no—
139#66896D411yescharacteristic
140#66896D411yescharacteristic
141C11⋊#95—41no—
142C11⋊#95—41no—
143#66886—41no—
144#66883—41no—
145#66882—21no—
146#66882—21no—
147Dic11⋊C8—21no—
148Dic11⋊C8—21no—
149#66872—21no—
150#66872—21no—
151#66872—21no—
152#66872—21no—
153#66871—21no—
154#66871—21no—
155C4⋊(C11⋊C8)—21no—
156C4⋊(C11⋊C8)—21no—
157#25985—111nomaximal, Sylow
158C11×#41—21no—
159C11×#41—21no—
160C11×#41—21no—
161C11×#41—21no—
162C11×#41—24no—
163C11×#41—24no—
164C11×#41—44no—
165C11×#41—42no—
166C11×#41—42no—
167C11×#41—44no—
168C11×#41C2×Q812yes—
169C11×#41C2×Q812yes—
170C11×#41#4112yes—
171C11×#41#4112yes—
172C2×Q8×C11—21no—
173C2×Q8×C11—24no—
174C2×Q8×C11—24no—
175C2×Q8×C11—21no—
176C2×Q8×C11—21no—
177C2×Q8×C11—21no—
178C2×Q8×C11—21no—
179C2×Q8×C11—41no—
180C2×Q8×C11—42no—
181C2×Q8×C11—42no—
182C2×Q8×C11C2×D411yescharacteristic
183C2×Q8×C11C2×D411yescharacteristic
184C2×Q8×C11C2×Q812yes—
185C2×D4×C11—21no—
186C2×D4×C11—21no—
187C2×D4×C11—41no—
188C2×D4×C11C2×D411yescharacteristic
189C2×D4×C11C2×D411yescharacteristic
190C22×C4×C11—21no—
191C22×C4×C11—21no—
192C22×C4×C11—21no—
193C22×C4×C11—21no—
194C22×C4×C11—41no—
195C22×C4×C11—41no—
196C22×C4×C11—41no—
197C22×C4×C11C2411yescharacteristic
198C22×Dic11—21no—
199C22×Dic11—21no—
200C22×Dic11—21no—
201C22×Dic11—41no—
202C2×Dic22—41no—
203C11×M16—21no—
204C11×M16—21no—
205C11×M16—42no—
206C2×C8×C11—21no—
207C2×C8×C11—21no—
208C2×C8×C11—21no—
209C2×C8×C11—42no—
210C2×C8×C11C2×D411yescharacteristic
211C2×C8×C11C2×D411yescharacteristic
212C11×C4⋊C4—21no—
213C11×C4⋊C4—21no—
214C11×C4⋊C4—41no—
215C11×C4⋊C4C2×D411yescharacteristic
216C11×C4⋊C4C2×D411yescharacteristic
217C11×C22⋊C4—21no—
218C11×C22⋊C4—41no—
219C42×C11—21no—
220C42×C11—41no—
221C22⋊Dic11—21no—
222C22⋊Dic11—41no—
223C4⋊Dic11—21no—
224C4⋊Dic11—21no—
225C4⋊Dic11—21no—
226C4⋊Dic11—21no—
227C4⋊Dic11—41no—
228C4⋊Dic11—41no—
229C4⋊Dic11C2×D411yescharacteristic
230C4⋊Dic11C2×D411yescharacteristic
231Dic11⋊C4—41no—
232Dic11⋊C4—41no—
233C4×Dic11—21no—
234C4×Dic11—21no—
235C4×Dic11—21no—
236C4×Dic11—41no—
237#4318—21no—
238#4318—21no—
239#4318—21no—
240#4318—21no—
241#4318—21no—
242#4318—21no—
243#4318—42no—
244C2×C11⋊C8—21no—
245C2×C11⋊C8—21no—
246C2×C11⋊C8—21no—
247C2×C11⋊C8—21no—
248C2×C11⋊C8—21no—
249C2×C11⋊C8—42no—
250C2×C11⋊C8C2×D411yescharacteristic
251C2×C11⋊C8C2×D411yescharacteristic
252#3084—111no—
253#3083—111no—
254#2898—111no—
255#2898D1111yescharacteristic
256#2097—111no—
257#2064—111no—
258#2064—111no—
259#2028—111no—
260#1866—111no—
261#1866—111no—
262#1860—111no—
263#1860—111no—
264#1816—111no—
265#1815—111no—
266#1793—111no—
267C23×C11—21no—
268C23×C11—21no—
269C23×C11H(5,𝔽2)11yescharacteristic
270Q8×C11—21no—
271Q8×C11—21no—
272Q8×C11—22no—
273Q8×C11—22no—
274Q8×C11—44no—
275Q8×C11—41no—
276Q8×C11—41no—
277Q8×C11C22⋊Q812yes—
278Q8×C11C22⋊Q812yes—
279D4×C11—22no—
280D4×C11—22no—
281D4×C11—22no—
282D4×C11—22no—
283D4×C11—21no—
284D4×C11—21no—
285D4×C11—44no—
286D4×C11C22⋊Q812yes—
287D4×C11C22⋊Q812yes—
288C2×C4×C11—21no—
289C2×C4×C11—21no—
290C2×C4×C11—22no—
291C2×C4×C11—22no—
292C2×C4×C11—22no—
293C2×C4×C11—22no—
294C2×C4×C11—22no—
295C2×C4×C11—21no—
296C2×C4×C11—21no—
297C2×C4×C11—44no—
298C2×C4×C11—44no—
299C2×C4×C11—41no—
300C2×C4×C11—41no—
301C2×C4×C11—41no—
302C2×C4×C11—81no—
303C2×C4×C11—161no—
304C2×C4×C11C22×D411yescharacteristic
305C2×C4×C11C22×D411yescharacteristic
306C2×C4×C11C22×Q811yescharacteristic
307C2×C4×C11C2×#4111yescharacteristic
308C2×C4×C11C2×#4111yescharacteristic
309C2×C4×C11H(5,𝔽2)11yescharacteristic
310C2×Dic11—21no—
311C2×Dic11—21no—
312C2×Dic11—41no—
313C2×Dic11—41no—
314C2×Dic11—41no—
315C2×Dic11—41no—
316C2×Dic11—41no—
317C2×Dic11—81no—
318C2×Dic11—81no—
319C2×Dic11—161no—
320Dic22—161no—
321C8×C11—21no—
322C8×C11—21no—
323C8×C11—41no—
324C11⋊C8—21no—
325C11⋊C8—21no—
326C11⋊C8—41no—
327C11⋊C8—41no—
328C11⋊C8—41no—
329C2×2−1+4C2×D1111yescharacteristic
330C2×#131—111no—
331C2×#131C2×D1111yescharacteristic
332#544—111no—
333#520—111no—
334#520C2×D1111yescharacteristic
335#479—221no—
336#479—221no—
337#478—221no—
338#478—221no—
339#477—221no—
340#476—221no—
341#475—221no—
342#474—221no—
343C2×#108—111no—
344C2×#108—111no—
345#428—111no—
346#428—111no—
347C2×#107—111no—
348C2×#106—111no—
349#424—111no—
350#424—111no—
351C2×#104—111no—
352C2×#104—111no—
353#419—21no—
354#419—111no—
355#419—221no—
356#419C2×D1111yescharacteristic
357#417—21no—
358#417—111no—
359#417—111no—
360#417—111no—
361#417—221no—
362#417C2×D1111yescharacteristic
363#416—21no—
364#416—111no—
365#416—221no—
366#416C2×D1111yescharacteristic
367C2×#103—21no—
368C2×#103—111no—
369C2×#103—221no—
370C2×#103C2×D1111yescharacteristic
371C2×#101—111no—
372#391—111no—
373#382—111no—
374#343—111no—
375#343—111no—
376#343—111no—
377#339—111no—
378#339—111no—
379#336—111no—
380C22×C11—21no—
381C22×C11—21no—
382C22×C11—22no—
383C22×C11—22no—
384C22×C11—41no—
385C22×C11—41no—
386C22×C11—41no—
387C22×C11—41no—
388C22×C11#54311yescharacteristic
389C22×C11D4211yescharacteristic
390C22×C11#55111yescharacteristic
391C4×C11—22no—
392C4×C11—22no—
393C4×C11—42no—
394C4×C11—41no—
395C4×C11—41no—
396C4×C11—81no—
397C4×C11C2×C22⋊Q811yescharacteristic
398C4×C11C2×C22⋊Q811yescharacteristic
399C4×C11#54611yescharacteristic
400C4×C11#54611yescharacteristic
401Dic11—41no—
402Dic11—41no—
403Dic11—81no—
404Dic11—81no—
405Dic11—81no—
4062−1+4—22no—
4072−1+4—22no—
4082−1+4—41no—
4092−1+4Dic2214yescomplemented
410C2×#41—21no—
411C2×#41—21no—
412C2×#41—41no—
413C2×#41C22×D1111yescharacteristic
414C2×#41C22×D1111yescharacteristic
415C22×Q8—21no—
416C22×Q8—21no—
417C22×Q8C22×D1111yescharacteristic
418#131—22no—
419#131—41no—
420#131—222no—
421#131—441no—
422C2×M16—21no—
423C2×M16—111no—
424C2×M16—111no—
425C2×M16—111no—
426C2×M16—221no—
427C2×M16C22×D1111yescharacteristic
428C22×C8—21no—
429C22×C8—111no—
430C22×C8—221no—
431C22×C8C22×D1111yescharacteristic
432C4⋊Q8—221no—
433C4⋊Q8—221no—
434#125—221no—
435#125—221no—
436C22⋊Q8—441no—
437C4×Q8—21no—
438C4×Q8—221no—
439C4×D4—21no—
440C4×D4—41no—
441C4×D4—221no—
442C4×D4—441no—
443#117—21no—
444#117—111no—
445#117—221no—
446#117C22×D1111yescharacteristic
447C2×C4⋊C4—21no—
448C2×C4⋊C4—111no—
449C2×C4⋊C4—111no—
450C2×C4⋊C4—221no—
451C2×C4⋊C4C22×D1111yescharacteristic
452C2×C42—111no—
453#108—441no—
454#108—441no—
455#107—221no—
456#107—221no—
457#107—221no—
458#107—221no—
459#106—221no—
460#106—221no—
461#106—221no—
462#106—221no—
463C4⋊C8—221no—
464C4⋊C8—221no—
465C4⋊C8—221no—
466C4⋊C8—221no—
467#104—222no—
468#104—222no—
469#103—21no—
470#103—21no—
471#103—21no—
472#103—21no—
473#103—21no—
474#103—111no—
475#103—111no—
476#103—221no—
477#103—221no—
478#103—221no—
479#103—221no—
480#103—221no—
481#103—221no—
482#103—221no—
483#103C22⋊D1111yescharacteristic
484#103C22⋊D1111yescharacteristic
485#102—21no—
486#102—111no—
487#102—111no—
488#102—221no—
489#102—221no—
490#102—221no—
491#102C22⋊D1111yescharacteristic
492#102C22⋊D1111yescharacteristic
493#101—441no—
494#95—441no—
495#95—441no—
496C2×C11—21no—
497C2×C11—21no—
498C2×C11—42no—
499C2×C11—42no—
500C2×C11#243711yescharacteristic
501C2×C11#328911yescharacteristic
502C2×C11#329011yescharacteristic
503#41—21no—
504#41—21no—
505#41—21no—
506#41—21no—
507#41—24no—
508#41—24no—
509#41—44no—
510#41—42no—
511#41—42no—
512#41—44no—
513#41C2×Dic2212yes—
514#41C2×Dic2212yes—
515#41#433912yes—
516#41#434112yes—
517C2×Q8—21no—
518C2×Q8—24no—
519C2×Q8—24no—
520C2×Q8—21no—
521C2×Q8—21no—
522C2×Q8—21no—
523C2×Q8—21no—
524C2×Q8—41no—
525C2×Q8—42no—
526C2×Q8—42no—
527C2×Q8—441no—
528C2×Q8C2×Dic2212yes—
529C2×Q8D4×D1111yescharacteristic
530C2×Q8C2×C22⋊D1111yescharacteristic
531C2×D4—21no—
532C2×D4—21no—
533C2×D4—41no—
534C2×D4D4×D1111yescharacteristic
535C2×D4C2×C22⋊D1111yescharacteristic
536C22×C4—21no—
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—41no—
541C22×C4—41no—
542C22×C4—41no—
543C22×C4—221no—
544C22×C4—221no—
545C22×C4—221no—
546C22×C4—441no—
547C22×C4C23×D1111yescharacteristic
548M16—21no—
549M16—21no—
550M16—42no—
551M16—221no—
552M16—221no—
553M16—221no—
554M16—221no—
555M16—221no—
556M16—221no—
557M16—442no—
558C2×C8—21no—
559C2×C8—21no—
560C2×C8—21no—
561C2×C8—42no—
562C2×C8—111no—
563C2×C8—111no—
564C2×C8—221no—
565C2×C8—221no—
566C2×C8—221no—
567C2×C8—221no—
568C2×C8—221no—
569C2×C8—442no—
570C2×C8C2×C22⋊D1111yescharacteristic
571C2×C8C2×C22⋊D1111yescharacteristic
572C4⋊C4—21no—
573C4⋊C4—21no—
574C4⋊C4—41no—
575C4⋊C4—111no—
576C4⋊C4—111no—
577C4⋊C4—221no—
578C4⋊C4—221no—
579C4⋊C4—221no—
580C4⋊C4—221no—
581C4⋊C4—441no—
582C4⋊C4—441no—
583C4⋊C4—441no—
584C4⋊C4—441no—
585C4⋊C4C2×C22⋊D1111yescharacteristic
586C4⋊C4C2×C22⋊D1111yescharacteristic
587C22⋊C4—21no—
588C22⋊C4—41no—
589C22⋊C4—221no—
590C22⋊C4—441no—
591C42—21no—
592C42—41no—
593C42—221no—
594C42—221no—
595C42—221no—
596C42—441no—
597C11#2598511yescharacteristic, complemented, Sylow
598C23—21no—
599C23—21no—
600C23#6703711yescharacteristic
601Q8—21no—
602Q8—21no—
603Q8—22no—
604Q8—22no—
605Q8—44no—
606Q8—41no—
607Q8—41no—
608Q8—1761no—
609Q8#6693112yes—
610Q8#6697712yes—
611D4—22no—
612D4—22no—
613D4—22no—
614D4—22no—
615D4—21no—
616D4—21no—
617D4—44no—
618D4#6693112yes—
619D4#6697712yes—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—81no—
635C2×C4—161no—
636C2×C4—221no—
637C2×C4—221no—
638C2×C4—441no—
639C2×C4—441no—
640C2×C4—441no—
641C2×C4—441no—
642C2×C4—441no—
643C2×C4—881no—
644C2×C4—881no—
645C2×C4—1761no—
646C2×C4C22×Dic2211yescharacteristic
647C2×C4C2×#433911yescharacteristic
648C2×C4C2×D4×D1111yescharacteristic
649C2×C4C2×#434111yescharacteristic
650C2×C4#6703711yescharacteristic
651C2×C4C22×C22⋊D1111yescharacteristic
652C8—21no—
653C8—21no—
654C8—41no—
655C8—221no—
656C8—221no—
657C8—441no—
658C8—441no—
659C8—441no—
660C22—21no—
661C22—21no—
662C22—22no—
663C22—22no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22#1062301411yescharacteristic
669C22#1062302611yescharacteristic
670C22D4×C22⋊D1111yescharacteristic
671C4—22no—
672C4—22no—
673C4—41no—
674C4—41no—
675C4—42no—
676C4—81no—
677C4—441no—
678C4—441no—
679C4—881no—
680C4—881no—
681C4—881no—
682C4C2×#6693111yescharacteristic
683C4#1062302111yescharacteristic
684C4C2×#6697711yescharacteristic
685C4#1062333711yescharacteristic
686C2—21no—
687C2—21no—
688C2—42no—
689C2—42no—
690C2#1307227711yescharacteristic
691C2#1308038911yescharacteristic
692C2#1308040211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#2898C2
socleC22×C11D42
prime radicalC2×C11×2−1+4C22
hypercenter#2898D11
hypocenterC11#25985
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2898

Complements

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

NH ≅ G/Ntype
1. #13078478C2semidirect
2. #13078465C2semidirect
4. #13070162C2semidirect
5. #13069983C2semidirect
6. #13069981C2semidirect
7. #13069775C2semidirect
8. #13068893C2semidirect
9. #13068891C2semidirect
10. #13068849C2semidirect
11. #13068847C2semidirect
12. #13068680C2semidirect
13. #13068673C2semidirect
70. C11×2−1+4Q8semidirect
409. 2−1+4Dic22semidirect
597. C11#25985semidirect

Maximal subgroups

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

subgroupindex
#130784782
#130784652
#130771492
#130701622
#130699832
#130699812
#130697752
#130688932
#130688912
#130688492
#130688472
#130686802
#130686732
#130685752
C11×#28982
#2598511