Finite Group Databasebeta

← #14675764

Subgroups — #14675764

Order 2816.

2222 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D11C211yesmaximal, characteristic
2C2×#10622549C211yesmaximal, characteristic, complemented
3C2×#10622590C211yesmaximal, characteristic, complemented
4C2×#10622600C211yesmaximal, characteristic
5C2×C4×(C2×D11)⋊C4C211yesmaximal, characteristic
6#13066871C214yesmaximal, complemented
7#13066845C211yesmaximal, characteristic, complemented
8(C2×D11)⋊#95C214yesmaximal
9C11×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D11C2212yes—
11C23×C4×D11C2211yescharacteristic
12C22×C11×C4⋊C4C2211yescharacteristic
13C22×C11×C4⋊C4C2211yescharacteristic
14C22×C42×C11C2211yescharacteristic
15C22×(C2×D11)⋊C4C2212yes—
16C22×(C2×D11)⋊C4C2212yes—
17C22×C4⋊Dic11C2211yescharacteristic
18C22×C4⋊Dic11C2211yescharacteristic
19C22×Dic11⋊C4C2212yes—
20C22×C4×Dic11C2211yescharacteristic
21C2×C4⋊C4×D11—24no—
22C2×C4⋊C4×D11—24no—
23C2×C4⋊C4×D11—24no—
24C2×C11×#95C2214yes—
25C2×C11⋊#95C2214yes—
26C2×C11⋊#95C2214yes—
27C2×C11⋊#95C2212yes—
28C2×C11⋊#95C2214yes—
29C2×C11⋊#95C2212yes—
30#10622600—24no—
31#10622590—24no—
32C4×(C2×D11)⋊C4C418yescomplemented
33#10622549—24no—
34C24×D11C2311yescharacteristic
35C23×C4×C11C2312yes—
36C23×C4×C11C2312yes—
37C23×C4×C11C2312yes—
38C23×C4×C11C2311yescharacteristic
39C23×Dic11C2312yes—
40C23×Dic11C2312yes—
41C23×Dic11C2312yes—
42C23×Dic11C2311yescharacteristic
43C22×C4×D11—22no—
44C22×C4×D11—22no—
45C22×C4×D11—22no—
46C22×C4×D11—22no—
47C22×C4×D11—24no—
48C22×C4×D11—24no—
49C22×C4×D11—24no—
50C22×C4×D11—28no—
51C22×C4×D11—44no—
52C22×C4×D11C2×C414yes—
53C22×C4×D11D414yes—
54C22×C4×D11Q814yes—
55C2×C11×C4⋊C4—22no—
56C2×C11×C4⋊C4—22no—
57C2×C11×C4⋊C4—22no—
58C2×C11×C4⋊C4—22no—
59C2×C11×C4⋊C4—22no—
60C2×C11×C4⋊C4D414yes—
61C2×C42×C11—22no—
62C2×C42×C11—22no—
63C2×C42×C11C2×C414yescomplemented
64C2×(C2×D11)⋊C4—24no—
65C2×(C2×D11)⋊C4—24no—
66C2×(C2×D11)⋊C4—24no—
67C2×(C2×D11)⋊C4—28no—
68C2×(C2×D11)⋊C4C2×C418yes—
69C2×C4⋊Dic11—22no—
70C2×C4⋊Dic11—22no—
71C2×C4⋊Dic11—22no—
72C2×C4⋊Dic11—22no—
73C2×C4⋊Dic11—22no—
74C2×C4⋊Dic11D414yes—
75C2×Dic11⋊C4—24no—
76C2×Dic11⋊C4—24no—
77C2×Dic11⋊C4—24no—
78C2×C4×Dic11—22no—
79C2×C4×Dic11—22no—
80C2×C4×Dic11C2×C414yescomplemented
81C4⋊C4×D11—232no—
82C11×#95—44no—
83C11⋊#95—28no—
84C11⋊#95—28no—
85C11⋊#95—24no—
86C11⋊#95—44no—
87C11⋊#95C2×C418yescomplemented
88#13881—111nomaximal, Sylow
89C24×C11C2411yescharacteristic
90C23×D11—21no—
91C23×D11—21no—
92C23×D11—21no—
93C23×D11—21no—
94C23×D11—22no—
95C23×D11—22no—
96C23×D11C22×C412yes—
97C23×D11C2×D412yes—
98C23×D11C2×Q812yes—
99C23×D11#4114yes—
100C23×D11#4114yes—
101C22×C4×C11—22no—
102C22×C4×C11—22no—
103C22×C4×C11—22no—
104C22×C4×C11—22no—
105C22×C4×C11—22no—
106C22×C4×C11—22no—
107C22×C4×C11—24no—
108C22×C4×C11—22no—
109C22×C4×C11—24no—
110C22×C4×C11—22no—
111C22×C4×C11—22no—
112C22×C4×C11—24no—
113C22×C4×C11—22no—
114C22×C4×C11—22no—
115C22×C4×C11—24no—
116C22×C4×C11C22×C414yes—
117C22×C4×C11C22×C412yes—
118C22×C4×C11C2×D414yes—
119C22×C4×C11C2×D412yes—
120C22×C4×C11C2×D412yes—
121C22×C4×C11C2×D412yes—
122C22×C4×C11C2×Q812yes—
123C22×C4×C11#4112yes—
124C22×C4×C11#4112yes—
125C22×Dic11—21no—
126C22×Dic11—21no—
127C22×Dic11—22no—
128C22×Dic11—22no—
129C22×Dic11—22no—
130C22×Dic11—22no—
131C22×Dic11—22no—
132C22×Dic11—22no—
133C22×Dic11—21no—
134C22×Dic11—21no—
135C22×Dic11—22no—
136C22×Dic11—22no—
137C22×Dic11—22no—
138C22×Dic11—24no—
139C22×Dic11—24no—
140C22×Dic11—22no—
141C22×Dic11—24no—
142C22×Dic11—22no—
143C22×Dic11—24no—
144C22×Dic11C22×C414yes—
145C22×Dic11C22×C412yes—
146C22×Dic11C2×D414yes—
147C22×Dic11C2×D412yes—
148C22×Dic11C2×Q812yes—
149C2×C4×D11—216no—
150C2×C4×D11—28no—
151C2×C4×D11—28no—
152C2×C4×D11—28no—
153C2×C4×D11—28no—
154C2×C4×D11—28no—
155C2×C4×D11—28no—
156C2×C4×D11—416no—
157C2×C4×D11—416no—
158C2×C4×D11—48no—
159C2×C4×D11—48no—
160C2×C4×D11C4⋊C4116yes—
161C11×C4⋊C4—28no—
162C11×C4⋊C4—44no—
163C42×C11—28no—
164(C2×D11)⋊C4—216no—
165(C2×D11)⋊C4—48no—
166C4⋊Dic11—28no—
167C4⋊Dic11—44no—
168Dic11⋊C4—216no—
169C4×Dic11—28no—
170C23×C4⋊C4—111no—
171C2×#382—111no—
172C2×#382—111no—
173C2×#380—111no—
174C2×C4×C22⋊C4—111no—
175#1442—114no—
176C4⋊#95—111no—
177C4⋊#95D1111yescharacteristic, complemented
178C22⋊#95—114no—
179C23×C11C23×C411yescharacteristic
180C23×C11C22×D411yescharacteristic
181C23×C11C22×D411yescharacteristic
182C23×C11C22×D411yescharacteristic
183C23×C11C22×Q811yescharacteristic
184C23×C11C2×#4111yescharacteristic
185C23×C11C2×#4111yescharacteristic
186C23×C11C2×#4112yes—
187C23×C11C2×#4112yes—
188C23×C11H(5,𝔽2)12yes—
189C23×C112−1+412yes—
190C22×D11—22no—
191C22×D11—22no—
192C22×D11—24no—
193C22×D11—22no—
194C22×D11—22no—
195C22×D11—22no—
196C22×D11—22no—
197C22×D11—22no—
198C22×D11—22no—
199C22×D11—24no—
200C22×D11—24no—
201C22×D11—24no—
202C22×D11—24no—
203C22×D11—24no—
204C22×D11—24no—
205C22×D11—24no—
206C22×D11—24no—
207C22×D11—24no—
208C22×D11C2×C4⋊C414yes—
209C22×D11#11718yes—
210C22×D11#12414yes—
211C22×D11#12414yes—
212C22×D11#12518yes—
213C2×C4×C11—24no—
214C2×C4×C11—24no—
215C2×C4×C11—24no—
216C2×C4×C11—24no—
217C2×C4×C11—28no—
218C2×C4×C11—42no—
219C2×C4×C11—42no—
220C2×C4×C11—44no—
221C2×C4×C11—42no—
222C2×C4×C11—42no—
223C2×C4×C11—42no—
224C2×C4×C11—42no—
225C2×C4×C11—42no—
226C2×C4×C11—42no—
227C2×C4×C11—42no—
228C2×C4×C11—44no—
229C2×C4×C11—44no—
230C2×C4×C11C2×C4⋊C414yes—
231C2×C4×C11C4×D414yes—
232C2×C4×C11C4×D414yes—
233C2×C4×C11#12014yes—
234C2×C4×C11C22⋊Q814yes—
235C2×C4×C11C22⋊Q814yes—
236C2×C4×C11#12314yes—
237C2×Dic11—22no—
238C2×Dic11—22no—
239C2×Dic11—22no—
240C2×Dic11—22no—
241C2×Dic11—22no—
242C2×Dic11—22no—
243C2×Dic11—24no—
244C2×Dic11—24no—
245C2×Dic11—24no—
246C2×Dic11—24no—
247C2×Dic11—28no—
248C2×Dic11—42no—
249C2×Dic11—42no—
250C2×Dic11—44no—
251C2×Dic11—42no—
252C2×Dic11—42no—
253C2×Dic11—42no—
254C2×Dic11—42no—
255C2×Dic11—42no—
256C2×Dic11—42no—
257C2×Dic11—42no—
258C2×Dic11—44no—
259C2×Dic11—44no—
260C2×Dic11C2×C4⋊C414yes—
261C4×D11—232no—
262C4×D11—432no—
263C24×C4—111no—
264C24×C4—112no—
265C22×C4⋊C4—111no—
266C22×C4⋊C4—112no—
267C22×C4⋊C4—111no—
268C22×C4⋊C4—224no—
269C22×C4⋊C4—224no—
270C22×C4⋊C4—224no—
271C22×C4⋊C4C2×D1111yescharacteristic
272C22×C4⋊C4C2×D1111yescharacteristic
273C22×C22⋊C4—112no—
274C22×C22⋊C4—112no—
275C22×C42—111no—
276C22×C42C2×D1111yescharacteristic
277#382—224no—
278#382—224no—
279#380—224no—
280C4×C22⋊C4—118no—
281C2×#95—112no—
282C2×#95—114no—
283C2×#95—114no—
284C2×#95—112no—
285C2×#95—114no—
286C2×#95C2×D1114yes—
287C22×C11C22×C4⋊C411yescharacteristic
288C22×C11C2×#11712yes—
289C22×C11C2×C4×D411yescharacteristic
290C22×C11C2×C4×D411yescharacteristic
291C22×C11#52012yes—
292C22×C11C2×#12011yescharacteristic
293C22×C11C2×C22⋊Q811yescharacteristic
294C22×C11C2×C22⋊Q811yescharacteristic
295C22×C11C2×#12311yescharacteristic
296C22×C11C2×#12411yescharacteristic
297C22×C11C2×#12411yescharacteristic
298C22×C11C2×#12512yes—
299C22×C11#53511yescharacteristic
300C22×C11#53611yescharacteristic
301C22×C11#54412yes—
302C22×C11#54612yes—
303C22×C11#54612yes—
304C22×C11#54812yes—
305C22×C11#54812yes—
306C22×C11#55212yes—
307C22×C11#55212yes—
308C22×C11#55312yes—
309C22×C11#55312yes—
310C2×D11—24no—
311C2×D11—28no—
312C2×D11—28no—
313C2×D11—24no—
314C2×D11—24no—
315C2×D11—28no—
316C2×D11—28no—
317C2×D11—24no—
318C2×D11—24no—
319C2×D11—24no—
320C2×D11#38218yescomplemented
321C4×C11—44no—
322C4×C11—44no—
323C4×C11—82no—
324C4×C11#38518yes—
325Dic11—24no—
326Dic11—44no—
327Dic11—44no—
328Dic11—82no—
329C25—111no—
330C23×C4—111no—
331C23×C4—114no—
332C23×C4—114no—
333C23×C4—114no—
334C23×C4—112no—
335C23×C4—112no—
336C23×C4—112no—
337C23×C4—224no—
338C23×C4—224no—
339C23×C4—228no—
340C23×C4—222no—
341C23×C4—222no—
342C23×C4—222no—
343C23×C4—222no—
344C23×C4—224no—
345C23×C4—444no—
346C23×C4C22×D1112yes—
347C23×C4C22×D1112yes—
348C23×C4C22×D1112yes—
349C23×C4C22×D1111yescharacteristic
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—114no—
356C2×C4⋊C4—2232no—
357C2×C4⋊C4—224no—
358C2×C4⋊C4—224no—
359C2×C4⋊C4—224no—
360C2×C4⋊C4—222no—
361C2×C4⋊C4—222no—
362C2×C4⋊C4—222no—
363C2×C4⋊C4—222no—
364C2×C4⋊C4—222no—
365C2×C4⋊C4D4414yes—
366C2×C22⋊C4—118no—
367C2×C22⋊C4—224no—
368C2×C22⋊C4—224no—
369C2×C22⋊C4—224no—
370C2×C22⋊C4—228no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—114no—
374C2×C42—222no—
375C2×C42—222no—
376C2×C42C4×D1114yescomplemented
377#95—44no—
378#95—118no—
379#95—224no—
380#95—228no—
381#95—228no—
382#95—444no—
383C2×C11C2×#38211yescharacteristic
384C2×C11C2×#38511yescharacteristic
385C2×C11#231711yescharacteristic
386C2×C11#235212yes—
387C2×C11#235212yes—
388C2×C11#243511yescharacteristic
389C2×C11#243611yescharacteristic
390C2×C11#244212yes—
391C2×C11#244212yes—
392C2×C11#250311yescharacteristic
393C2×C11#250411yescharacteristic
394D11—28no—
395C24—112no—
396C24—112no—
397C24—112no—
398C24—114no—
399C24—114no—
400C24—221no—
401C24—221no—
402C24—221no—
403C24—221no—
404C24—222no—
405C24—222no—
406C24C23×D1111yescharacteristic
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—112no—
423C22×C4—112no—
424C22×C4—112no—
425C22×C4—1116no—
426C22×C4—114no—
427C22×C4—114no—
428C22×C4—2216no—
429C22×C4—221no—
430C22×C4—221no—
431C22×C4—221no—
432C22×C4—221no—
433C22×C4—228no—
434C22×C4—228no—
435C22×C4—228no—
436C22×C4—228no—
437C22×C4—228no—
438C22×C4—228no—
439C22×C4—222no—
440C22×C4—222no—
441C22×C4—222no—
442C22×C4—222no—
443C22×C4—222no—
444C22×C4—224no—
445C22×C4—222no—
446C22×C4—222no—
447C22×C4—222no—
448C22×C4—224no—
449C22×C4—222no—
450C22×C4—222no—
451C22×C4—222no—
452C22×C4—224no—
453C22×C4—224no—
454C22×C4—4416no—
455C22×C4—4416no—
456C22×C4—448no—
457C22×C4—448no—
458C22×C4C2×C4×D1114yes—
459C22×C4C2×C4×D1112yes—
460C22×C4C2×D4414yes—
461C22×C4C2×D4412yes—
462C22×C4#433912yes—
463C22×C4D4×D1112yes—
464C22×C4D4×D1112yes—
465C22×C4#434112yes—
466C22×C4Q8×D1112yes—
467C4⋊C4—28no—
468C4⋊C4—44no—
469C4⋊C4—2216no—
470C4⋊C4—228no—
471C4⋊C4—444no—
472C22⋊C4—2216no—
473C22⋊C4—448no—
474C42—28no—
475C42—228no—
476C11#1388111yescharacteristic, complemented, Sylow
477C23—114no—
478C23—118no—
479C23—114no—
480C23—114no—
481C23—118no—
482C23—222no—
483C23—222no—
484C23—222no—
485C23—224no—
486C23—222no—
487C23—222no—
488C23—222no—
489C23—224no—
490C23—222no—
491C23—222no—
492C23—224no—
493C23—224no—
494C23—224no—
495C23—224no—
496C23—224no—
497C23—224no—
498C23—224no—
499C23—224no—
500C23C22×C4×D1111yescharacteristic
501C23C22×D4411yescharacteristic
502C23C2×#433911yescharacteristic
503C23C2×D4×D1111yescharacteristic
504C23C2×D4×D1111yescharacteristic
505C23C2×#434111yescharacteristic
506C23#6703712yes—
507C23C2×Q8×D1111yescharacteristic
508C23#6704012yes—
509C23#41×D1112yes—
510C23#41×D1112yes—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—114no—
529C2×C4—222no—
530C2×C4—222no—
531C2×C4—222no—
532C2×C4—222no—
533C2×C4—222no—
534C2×C4—222no—
535C2×C4—2232no—
536C2×C4—224no—
537C2×C4—224no—
538C2×C4—224no—
539C2×C4—224no—
540C2×C4—228no—
541C2×C4—4432no—
542C2×C4—442no—
543C2×C4—444no—
544C2×C4—442no—
545C2×C4—442no—
546C2×C4—442no—
547C2×C4—442no—
548C2×C4—442no—
549C2×C4—442no—
550C2×C4—442no—
551C2×C4—444no—
552C2×C4—444no—
553C2×C4—442no—
554C2×C4C4×D4414yes—
555C2×C4(C22⋊D11)⋊C414yes—
556C2×C4#6693514yes—
557C2×C4#6693614yes—
558C2×C4C4⋊C4×D1114yes—
559C2×C4#6694914yes—
560C2×C4#6695014yes—
561C22—118no—
562C22—224no—
563C22—224no—
564C22—224no—
565C22—228no—
566C22—224no—
567C22—228no—
568C22—224no—
569C22—228no—
570C22—224no—
571C22—228no—
572C22C2×C4×D4411yescharacteristic
573C22C2×(C22⋊D11)⋊C411yescharacteristic
574C22C2×#6693511yescharacteristic
575C22C2×#6693611yescharacteristic
576C22C2×C4⋊C4×D1111yescharacteristic
577C22#1062303512yes—
578C22C2×#6694911yescharacteristic
579C22C2×#6695011yescharacteristic
580C22D11×#11712yes—
581C22#1062308512yes—
582C22#1062308912yes—
583C22#1062310712yes—
584C22#1062311212yes—
585C22#1062314312yes—
586C22#1062314512yes—
587C22#1062316412yes—
588C22#1062317012yes—
589C22D11×#12411yescharacteristic
590C22D11×#12411yescharacteristic
591C22D11×#12512yes—
592C22#1062324611yescharacteristic
593C22#1062325511yescharacteristic
594C22#1062326012yes—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—224no—
599C4—444no—
600C4—444no—
601C4—882no—
602C4#1062228118yes—
603C2—228no—
604C2D11×#38211yescharacteristic
605C2C2×#1062228111yescharacteristic
606C2#1307123711yescharacteristic
607C2#1307153812yes—
608C2#1307154812yes—
609C2#1307226611yescharacteristic
610C2#1307227311yescharacteristic
611C2#1307233812yes—
612C2#1307234212yes—
613C2#1307310011yescharacteristic
614C2#1307310711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC11×C4⋊#95C2
socle, cosocleC24×C11C24
prime radicalC24×D11C23
hypercenterC4⋊#95D11
hypocenterC11#13881
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#10622549C2semidirect
3. C2×#10622590C2semidirect
6. #13066871C2semidirect
7. #13066845C2semidirect
9. C11×C4⋊#95C2semidirect
32. C4×(C2×D11)⋊C4C4semidirect
63. C2×C42×C11C2×C4semidirect
80. C2×C4×Dic11C2×C4semidirect
87. C11⋊#95C2×C4semidirect
177. C4⋊#95D11semidirect
320. C2×D11#382semidirect
376. C2×C42C4×D11semidirect
476. C11#13881semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D112
C2×#106225492
C2×#106225902
C2×#106226002
C2×C4×(C2×D11)⋊C42
#130668712
#130668452
(C2×D11)⋊#952
C11×C4⋊#952
#1388111