Finite Group Databasebeta

← #14754751

Subgroups — #14754751

Order 2816.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623053C211yesmaximal, characteristic
2C2×#10622608C211yesmaximal, characteristic, complemented
3C2×#10622746C211yesmaximal, characteristic
4C2×#10622579C211yesmaximal, characteristic
5C2×#10622598C211yesmaximal, characteristic, complemented
6#13068885C212yesmaximal, complemented
7#13068878C212yesmaximal, complemented
8#13068871C212yesmaximal, complemented
9#13068864C212yesmaximal, complemented
10#13068309C211yesmaximal, characteristic, complemented
11C11×#1736C211yesmaximal, characteristic, complemented
12C22×#4339C2211yescharacteristic
13C22×C11×M16C2211yescharacteristic
14C2×C11×#117C2211yescharacteristic
15C22×C11×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic11—21no—
17C22×C4⋊Dic11C2211yescharacteristic
18C22×C4×Dic11C2211yescharacteristic
19C22×#4318C2211yescharacteristic
20#10623053—24no—
21#10623053—24no—
22C2×#66946C2211yescharacteristic
23C2×#66946C2211yescharacteristic
24C2×#66940C2211yescharacteristic
25C2×#66940C2211yescharacteristic
26C2×(C22⋊D11)⋊C4—21no—
27C2×(C22⋊D11)⋊C4—21no—
28C2×#66925—21no—
29C2×C42×D11—21no—
30C2×C11⋊#95—21no—
31#10622746—24no—
32C2×#66885C2212yes—
33C2×#66880C2212yes—
34#10622608—24no—
35#10622598—21no—
36#10622598—21no—
37#10622598C2212yes—
38#10622598C2212yes—
39C2×#66874C2212yes—
40#10622579C418yescomplemented
41C2×#66873C2212yes—
42C2×#66872C2212yes—
43C2×#66871C2212yes—
44C11×#340C2212yes—
45C11×#340C2212yes—
46#10622188C2212yes—
47#10622188C2212yes—
48C23×C4×C11—21no—
49C23×C4×C11C2311yescharacteristic
50C22×C22⋊D11—21no—
51C23×Dic11—21no—
52C23×Dic11D411yescharacteristic
53C23×Dic11D411yescharacteristic
54C2×#4339—22no—
55C2×#4339—22no—
56C2×#4339—22no—
57C2×#4339C2×C414yes—
58C2×#4339D414yes—
59C2×#4339Q814yes—
60C22×D44C2311yescharacteristic
61C22×C4×D11—21no—
62C22×C4×D11—21no—
63C22×C4×D11—21no—
64C22×Dic22C2311yescharacteristic
65C2×C11×M16—22no—
66C2×C11×M16—22no—
67C2×C11×M16—22no—
68C22×C8×C11C2312yes—
69C11×#117—22no—
70C11×#117D414yes—
71C2×C11×C4⋊C4—21no—
72C2×C11×C4⋊C4—22no—
73C2×C11×C4⋊C4C2×C414yescomplemented
74C2×C11×C4⋊C4C2312yes—
75C2×C11×C4⋊C4C2311yescharacteristic
76C2×C11×C4⋊C4C2311yescharacteristic
77C2×C11×C22⋊C4—21no—
78C2×C42×C11—21no—
79C2×(C2×D11)⋊C4—21no—
80C2×(C2×D11)⋊C4—21no—
81C2×C4⋊Dic11—22no—
82C2×C4⋊Dic11—22no—
83C2×C4⋊Dic11—21no—
84C2×C4⋊Dic11—22no—
85C2×C4⋊Dic11—22no—
86C2×C4⋊Dic11—44no—
87C2×C4⋊Dic11C2312yes—
88C2×Dic11⋊C4—21no—
89C2×Dic11⋊C4—21no—
90C2×C4×Dic11—22no—
91C2×C4×Dic11—22no—
92C2×C4×Dic11—21no—
93C2×C4×Dic11—21no—
94C2×C4×Dic11D411yescharacteristic
95C2×C4×Dic11D411yescharacteristic
96C2×C4×Dic11C2311yescharacteristic
97C2×C4×Dic11C2311yescharacteristic
98C2×#4318—22no—
99C2×#4318—22no—
100C2×#4318C2×C414yescomplemented
101C22×C11⋊C8C2312yes—
102#66946—22no—
103#66946—22no—
104#66946—22no—
105#66946—22no—
106#66940—22no—
107#66940—22no—
108#66940—22no—
109#66940—22no—
110(C22⋊D11)⋊C4—28no—
111(C22⋊D11)⋊C4—28no—
112#66925—28no—
113C42×D11—28no—
114C11⋊#95—22no—
115C11⋊#95—22no—
116#66885—24no—
117#66880—24no—
118#66874C2×C418yescomplemented
119#66873C2×C418yescomplemented
120#66872—24no—
121#66871—24no—
122#20875—111nomaximal, Sylow
123C24×C11C2×D411yescharacteristic
124C23×D11—21no—
125C22×C4×C11—24no—
126C22×C4×C11—22no—
127C22×C4×C11—21no—
128C22×C4×C11—21no—
129C22×C4×C11—42no—
130C22×C4×C11—42no—
131C22×C4×C11C22×C412yes—
132C22×C4×C11C2×D412yes—
133C22×C4×C11C2×D411yescharacteristic
134C22×C4×C11C2×D412yes—
135C22×C4×C11C2×Q812yes—
136C22×C4×C11#4112yes—
137C22×C4×C11#4112yes—
138C22×C4×C11C2411yescharacteristic
139C2×C22⋊D11—24no—
140C2×C22⋊D11—24no—
141C2×C22⋊D11—24no—
142C22×Dic11—22no—
143C22×Dic11—22no—
144C22×Dic11—22no—
145C22×Dic11—22no—
146C22×Dic11—21no—
147C22×Dic11—22no—
148C22×Dic11—41no—
149C22×Dic11—41no—
150C22×Dic11—42no—
151C22×Dic11—42no—
152C22×Dic11—42no—
153C22×Dic11C2×D411yescharacteristic
154C22×Dic11C2×D411yescharacteristic
155C22×Dic11C2×D411yescharacteristic
156C22×Dic11C2×D411yescharacteristic
157#4339—28no—
158#4339—28no—
159#4339—28no—
160#4339C4⋊C4116yescomplemented
161C2×D44—21no—
162C2×D44—21no—
163C2×D44—21no—
164C2×D44C22×C412yes—
165C2×D44C2×D412yes—
166C2×D44C2×Q812yes—
167C2×C4×D11—24no—
168C2×C4×D11—24no—
169C2×C4×D11—24no—
170C2×C4×D11—24no—
171C2×C4×D11—24no—
172C2×C4×D11—42no—
173C2×C4×D11—42no—
174C2×C4×D11—42no—
175C2×C4×D11—42no—
176C2×Dic22—21no—
177C2×Dic22—21no—
178C2×Dic22—21no—
179C2×Dic22C22×C412yes—
180C2×Dic22C2×D412yes—
181C2×Dic22C2×Q812yes—
182C11×M16—44no—
183C2×C8×C11—22no—
184C2×C8×C11—22no—
185C2×C8×C11—22no—
186C11×C4⋊C4—24no—
187C11×C4⋊C4—21no—
188C11×C4⋊C4—21no—
189C11×C4⋊C4—22no—
190C11×C4⋊C4C22×C414yes—
191C11×C4⋊C4C2×D412yes—
192C11×C4⋊C4C2×D412yes—
193C11×C22⋊C4—24no—
194C42×C11—24no—
195(C2×D11)⋊C4—24no—
196(C2×D11)⋊C4—24no—
197C4⋊Dic11—22no—
198C4⋊Dic11—22no—
199C4⋊Dic11—44no—
200C4⋊Dic11—44no—
201C4⋊Dic11—42no—
202Dic11⋊C4—24no—
203Dic11⋊C4—24no—
204C4×Dic11—24no—
205C4×Dic11—24no—
206C4×Dic11—22no—
207C4×Dic11—22no—
208C4×Dic11—21no—
209C4×Dic11—21no—
210C4×Dic11—21no—
211C4×Dic11—21no—
212#4318—28no—
213C2×C11⋊C8—22no—
214C2×C11⋊C8—22no—
215C2×C11⋊C8C22×C414yes—
216C2×C4×#41—111no—
217C2×#428—111no—
218C2×#417—111no—
219C2×#417—111no—
220C2×#389—111no—
221#1865—112no—
222#1864—112no—
223#1863—112no—
224#1862—112no—
225#1736D1111yescharacteristic, complemented
226#1728—111no—
227C23×C11—21no—
228C23×C11—21no—
229C23×C11—21no—
230C23×C11—21no—
231C23×C11C4×D412yes—
232C23×C11#12012yes—
233C23×C11C22⋊Q812yes—
234C23×C11C22×D411yescharacteristic
235C22×D11—22no—
236C22×D11—22no—
237C22×D11—22no—
238C22×D11—41no—
239C22×D11—41no—
240C22×D11—41no—
241C22×D11—41no—
242C2×C4×C11—24no—
243C2×C4×C11—22no—
244C2×C4×C11—22no—
245C2×C4×C11—48no—
246C2×C4×C11—42no—
247C2×C4×C11—41no—
248C2×C4×C11—41no—
249C2×C4×C11—41no—
250C2×C4×C11—41no—
251C2×C4×C11—42no—
252C2×C4×C11C2×C4⋊C414yes—
253C2×C4×C11C4×D414yes—
254C2×C4×C11C4×D412yes—
255C2×C4×C11#12012yes—
256C2×C4×C11C22⋊Q812yes—
257C2×C4×C11C22⋊Q814yes—
258C2×C4×C11#12314yes—
259C2×C4×C11C23×C411yescharacteristic
260C2×C4×C11C22×D411yescharacteristic
261C2×C4×C11C22×D411yescharacteristic
262C2×C4×C11C22×Q811yescharacteristic
263C2×C4×C11C2×#4111yescharacteristic
264C2×C4×C11C2×#4111yescharacteristic
265C22⋊D11—216no—
266C2×Dic11—22no—
267C2×Dic11—21no—
268C2×Dic11—21no—
269C2×Dic11—22no—
270C2×Dic11—22no—
271C2×Dic11—21no—
272C2×Dic11—21no—
273C2×Dic11—21no—
274C2×Dic11—21no—
275C2×Dic11—21no—
276C2×Dic11—21no—
277C2×Dic11—21no—
278C2×Dic11—21no—
279C2×Dic11—44no—
280C2×Dic11—44no—
281C2×Dic11—42no—
282C2×Dic11—42no—
283C2×Dic11—42no—
284C2×Dic11—84no—
285C2×Dic11AGL(1,ℤ/8)12yes—
286C2×Dic11#13712yes—
287D44—22no—
288D44—22no—
289D44—22no—
290D44C2×C4⋊C414yescomplemented
291C4×D11—216no—
292C4×D11—48no—
293C4×D11—48no—
294Dic22—22no—
295Dic22—22no—
296Dic22—22no—
297Dic22C2×C4⋊C414yescomplemented
298C8×C11—42no—
299C11⋊C8—24no—
300C22×#41—111no—
301C22×M16—111no—
302C22×M16C2×D1111yescharacteristic
303C4×#41—224no—
304C4×#41—224no—
305C2×C4×Q8—111no—
306C2×C4×Q8—111no—
307C2×C4×D4—111no—
308C2×C4×D4—111no—
309C2×C4×D4—221no—
310C2×C4×D4—221no—
311C2×#117—221no—
312C2×#117C2×D1111yescharacteristic
313C22×C4⋊C4—111no—
314C22×C4⋊C4—221no—
315C22×C4⋊C4C2×D1111yescharacteristic
316C22×C42—111no—
317C22×C42—221no—
318#428—224no—
319C2×#107—112no—
320C2×#106—112no—
321#417—118no—
322#417—224no—
323C2×#103—112no—
324C2×#103—112no—
325C2×#102—112no—
326C2×#102—112no—
327#389—112no—
328#389—112no—
329#389—221no—
330#389—221no—
331C2×#95—221no—
332#340C2×D1112yes—
333#340C2×D1112yes—
334#336—112no—
335#336—112no—
336C22×C11—22no—
337C22×C11—22no—
338C22×C11—22no—
339C22×C11—22no—
340C22×C11—22no—
341C22×C11—22no—
342C22×C11#38514yes—
343C22×C11C2×C4×D411yescharacteristic
344C22×C11C2×#12011yescharacteristic
345C22×C11C2×C22⋊Q811yescharacteristic
346C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
347C22×C11C2×#13711yescharacteristic
348C22×C11#57511yescharacteristic
349C22×C11#57511yescharacteristic
350C2×D11—24no—
351C2×D11—42no—
352C2×D11—42no—
353C2×D11—42no—
354C2×D11—42no—
355C2×D11—42no—
356C2×D11—42no—
357C4×C11—44no—
358C4×C11—42no—
359C4×C11—42no—
360C4×C11#38514yes—
361C4×C11C22×C4⋊C411yescharacteristic
362C4×C11C2×C4×D411yescharacteristic
363C4×C11C2×C22⋊Q811yescharacteristic
364C4×C11C2×#12311yescharacteristic
365Dic11—22no—
366Dic11—22no—
367Dic11—24no—
368Dic11—41no—
369Dic11—41no—
370Dic11—82no—
371C2×#41—114no—
372C2×#41—114no—
373C2×#41—114no—
374C2×#41—222no—
375C2×#41—222no—
376C2×#41—222no—
377C22×Q8—111no—
378C22×D4—111no—
379C22×D4—221no—
380C23×C4—21no—
381C23×C4—111no—
382C23×C4—111no—
383C23×C4—221no—
384C23×C4—221no—
385C23×C4—221no—
386C23×C4—221no—
387C23×C4C22×D1111yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—22no—
391C2×M16—114no—
392C2×M16—222no—
393C2×M16—222no—
394C22×C8—112no—
395C22×C8C22×D1112yes—
396C4×Q8—222no—
397C4×Q8—222no—
398C4×Q8—222no—
399C4×Q8—222no—
400C4×D4—228no—
401C4×D4—228no—
402C4×D4—222no—
403C4×D4—222no—
404C4×D4—222no—
405C4×D4—222no—
406#117—22no—
407#117—228no—
408#117D4414yes—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—112no—
412C2×C4⋊C4—221no—
413C2×C4⋊C4—221no—
414C2×C4⋊C4—222no—
415C2×C4⋊C4—222no—
416C2×C4⋊C4—222no—
417C2×C4⋊C4—222no—
418C2×C4⋊C4—221no—
419C2×C4⋊C4—444no—
420C2×C4⋊C4C4×D1114yescomplemented
421C2×C4⋊C4C22×D1112yes—
422C2×C4⋊C4C22×D1111yescharacteristic
423C2×C4⋊C4C22×D1111yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—221no—
426C2×C22⋊C4—221no—
427C2×C42—21no—
428C2×C42—111no—
429C2×C42—111no—
430C2×C42—111no—
431C2×C42—111no—
432C2×C42—228no—
433C2×C42—221no—
434C2×C42—221no—
435C2×C42—222no—
436C2×C42—222no—
437#107—224no—
438#106—224no—
439#103—118no—
440#103—224no—
441#102—118no—
442#102—224no—
443#95—222no—
444#95—222no—
445C2×C11—24no—
446C2×C11C2×#38511yescharacteristic
447C2×C11C4×AGL(1,ℤ/8)11yescharacteristic
448C2×C11C4×#13711yescharacteristic
449C2×C11#300611yescharacteristic
450C2×C11#300711yescharacteristic
451C2×C11#307911yescharacteristic
452C2×C11#308011yescharacteristic
453D11—44no—
454C24—221no—
455C24D4×D1111yescharacteristic
456#41—1116no—
457#41—228no—
458#41—228no—
459#41—228no—
460C2×Q8—112no—
461C2×Q8—112no—
462C2×Q8—112no—
463C2×Q8—221no—
464C2×Q8—221no—
465C2×Q8—221no—
466C2×D4—112no—
467C2×D4—112no—
468C2×D4—112no—
469C2×D4—224no—
470C2×D4—224no—
471C2×D4—224no—
472C2×D4—221no—
473C2×D4—221no—
474C2×D4—221no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—111no—
482C22×C4—111no—
483C22×C4—111no—
484C22×C4—111no—
485C22×C4—224no—
486C22×C4—224no—
487C22×C4—222no—
488C22×C4—222no—
489C22×C4—221no—
490C22×C4—224no—
491C22×C4—224no—
492C22×C4—224no—
493C22×C4—222no—
494C22×C4—222no—
495C22×C4—222no—
496C22×C4—442no—
497C22×C4—442no—
498C22×C4—442no—
499C22×C4—442no—
500C22×C4—441no—
501C22×C4—441no—
502C22×C4—442no—
503C22×C4—442no—
504C22×C4—442no—
505C22×C4C2×C4×D1112yes—
506C22×C4C2×D4412yes—
507C22×C4#433912yes—
508C22×C4D4×D1111yescharacteristic
509C22×C4D4×D1112yes—
510C22×C4#434112yes—
511C22×C4Q8×D1112yes—
512C22×C4C23×D1111yescharacteristic
513M16—44no—
514M16—228no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—114no—
519C2×C8—222no—
520C2×C8—222no—
521C4⋊C4—24no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—224no—
526C4⋊C4—224no—
527C4⋊C4—222no—
528C4⋊C4—222no—
529C4⋊C4—444no—
530C4⋊C4—444no—
531C4⋊C4—442no—
532C4⋊C4C2×C4×D1114yes—
533C4⋊C4C2×D4412yes—
534C4⋊C4C2×D4412yes—
535C22⋊C4—24no—
536C22⋊C4—224no—
537C22⋊C4—224no—
538C42—24no—
539C42—224no—
540C42—224no—
541C42—222no—
542C42—222no—
543C42—221no—
544C42—221no—
545C42—221no—
546C42—221no—
547C11#2087511yescharacteristic, complemented, Sylow
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—222no—
553C23—222no—
554C23—222no—
555C23—441no—
556C23—441no—
557C23—441no—
558C23—441no—
559C23(C22⋊D11)⋊C412yes—
560C23#6693512yes—
561C23#6694912yes—
562C23C2×D4×D1111yescharacteristic
563Q8—114no—
564Q8—222no—
565Q8—222no—
566Q8—222no—
567D4—114no—
568D4—2216no—
569D4—222no—
570D4—222no—
571D4—222no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—112no—
583C2×C4—112no—
584C2×C4—222no—
585C2×C4—221no—
586C2×C4—221no—
587C2×C4—222no—
588C2×C4—222no—
589C2×C4—221no—
590C2×C4—221no—
591C2×C4—2216no—
592C2×C4—221no—
593C2×C4—221no—
594C2×C4—221no—
595C2×C4—221no—
596C2×C4—221no—
597C2×C4—221no—
598C2×C4—448no—
599C2×C4—448no—
600C2×C4—444no—
601C2×C4—444no—
602C2×C4—442no—
603C2×C4—442no—
604C2×C4—442no—
605C2×C4—884no—
606C2×C4C4×D4414yes—
607C2×C4(C22⋊D11)⋊C412yes—
608C2×C4#6693512yes—
609C2×C4#6693614yes—
610C2×C4C4⋊C4×D1114yes—
611C2×C4#6694912yes—
612C2×C4#6695014yes—
613C2×C4C22×C4×D1111yescharacteristic
614C2×C4C22×D4411yescharacteristic
615C2×C4C2×#433911yescharacteristic
616C2×C4C2×D4×D1111yescharacteristic
617C2×C4C2×#434111yescharacteristic
618C2×C4C2×Q8×D1111yescharacteristic
619C8—42no—
620C8—224no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—224no—
628C22—442no—
629C22—442no—
630C22—442no—
631C22—442no—
632C22—442no—
633C22—442no—
634C22#1062228114yes—
635C22C2×(C22⋊D11)⋊C411yescharacteristic
636C22C2×#6693511yescharacteristic
637C22C2×#6694911yescharacteristic
638C22D11×AGL(1,ℤ/8)11yescharacteristic
639C22#1062330411yescharacteristic
640C22D11×#13711yescharacteristic
641C22#1062330811yescharacteristic
642C4—42no—
643C4—42no—
644C4—44no—
645C4—222no—
646C4—222no—
647C4—224no—
648C4—441no—
649C4—441no—
650C4—882no—
651C4#1062228114yes—
652C4C2×C4×D4411yescharacteristic
653C4C2×#6693611yescharacteristic
654C4C2×C4⋊C4×D1111yescharacteristic
655C4C2×#6695011yescharacteristic
656C2—24no—
657C2—444no—
658C2C2×#1062228111yescharacteristic
659C2#1307742011yescharacteristic
660C2#1307743111yescharacteristic
661C2#1307786911yescharacteristic
662C2#1307787711yescharacteristic
663C2#1307842911yescharacteristic
664C2#1307843711yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1736C2
socleC23×C11C22×D4
prime radicalC22×#4339C22
cosocleC22×C4×C11C24
hypercenter#1736D11
hypocenterC11#20875
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#10622608C2semidirect
5. C2×#10622598C2semidirect
6. #13068885C2semidirect
7. #13068878C2semidirect
8. #13068871C2semidirect
9. #13068864C2semidirect
10. #13068309C2semidirect
11. C11×#1736C2semidirect
40. #10622579C4semidirect
73. C2×C11×C4⋊C4C2×C4semidirect
100. C2×#4318C2×C4semidirect
118. #66874C2×C4semidirect
119. #66873C2×C4semidirect
160. #4339C4⋊C4semidirect
225. #1736D11semidirect
290. D44C2×C4⋊C4semidirect
297. Dic22C2×C4⋊C4semidirect
420. C2×C4⋊C4C4×D11semidirect
547. C11#20875semidirect

Maximal subgroups

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

subgroupindex
C2×#106230532
C2×#106226082
C2×#106227462
C2×#106225792
C2×#106225982
#130688852
#130688782
#130688712
#130688642
#130683092
C11×#17362
#2087511