Finite Group Databasebeta

← #14756928

Subgroups — #14756928

Order 2816.

862 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×SD16×Dic11C211yesmaximal, characteristic
2#13071253C211yesmaximal, characteristic
3#13070981C211yesmaximal, characteristic, complemented
4#13069246C211yesmaximal, characteristic
5#13069240C211yesmaximal, characteristic
6#13069202C211yesmaximal, characteristic, complemented
7#13069200C211yesmaximal, characteristic, complemented
8#13068973C211yesmaximal, characteristic
9#13068884C211yesmaximal, characteristic, complemented
10#13068882C211yesmaximal, characteristic, complemented
11#13068504C211yesmaximal, characteristic, complemented
12#13068448C211yesmaximal, characteristic, complemented
13#13068443C211yesmaximal, characteristic
14#13068366C211yesmaximal, characteristic, complemented
15C11×#1880C211yesmaximal, characteristic
16C22×C11×SD16C2211yescharacteristic
17C2×C4×Q8×C11C2211yescharacteristic
18C2×Q8×Dic11C2211yescharacteristic
19C2×Q8×Dic11C2211yescharacteristic
20C2×D4×Dic11C2211yescharacteristic
21C2×C11×#103C2211yescharacteristic
22C2×C11×#102C2211yescharacteristic
23C11×#390C2211yescharacteristic
24C2×#66899C2211yescharacteristic
25C2×#66899C2211yescharacteristic
26#10622834C2211yescharacteristic
27C2×#66896C2211yescharacteristic
28C2×#66896C2211yescharacteristic
29SD16×Dic11C418yescomplemented
30C2×#66881C2211yescharacteristic
31C2×#66879C2211yescharacteristic
32C2×C8×Dic11C2211yescharacteristic
33C4⋊C4×Dic11—21no—
34C4⋊C4×Dic11—21no—
35C4⋊C4×Dic11C2211yescharacteristic
36C4⋊C4×Dic11C2211yescharacteristic
37C4⋊C4×Dic11C2211yescharacteristic
38#10622586C2211yescharacteristic
39C2×#66872C2211yescharacteristic
40C2×#66872C2211yescharacteristic
41#10622558—21no—
42#10622557—21no—
43#10622556—21no—
44#10622555—21no—
45C4×C4⋊Dic11—21no—
46C4×C4⋊Dic11C2211yescharacteristic
47#10622545—21no—
48#10622545C2211yescharacteristic
49#10622541C2211yescharacteristic
50C11×#340C2211yescharacteristic
51C11×#336C2211yescharacteristic
52#10622189C2211yescharacteristic
53#10622189C2211yescharacteristic
54#10622189C2211yescharacteristic
55#10622188C2211yescharacteristic
56#10622188C2211yescharacteristic
57#10622167C2211yescharacteristic
58#10622167C2211yescharacteristic
59#10622162C2211yescharacteristic
60C22×Q8×C11C2311yescharacteristic
61C22×D4×C11C2311yescharacteristic
62C23×Dic11—21no—
63C2×C11×SD16—22no—
64C2×C11×SD16—22no—
65C2×C11×SD16C2×C414yes—
66C22×C8×C11C2311yescharacteristic
67C22×C8×C11C2311yescharacteristic
68C4×Q8×C11—22no—
69C4×Q8×C11—22no—
70C2×C11×C4⋊C4—21no—
71C2×C11×C4⋊C4C2311yescharacteristic
72C2×C11×C4⋊C4C2311yescharacteristic
73C2×C11×C22⋊C4—21no—
74C2×C11×C22⋊C4—21no—
75C2×C42×C11—21no—
76C2×C42×C11C2311yescharacteristic
77C2×C22⋊Dic11—21no—
78C2×C22⋊Dic11—21no—
79C2×C22⋊Dic11—21no—
80Q8×Dic11—22no—
81Q8×Dic11—22no—
82Q8×Dic11—22no—
83Q8×Dic11C2×C414yes—
84D4×Dic11—22no—
85D4×Dic11C2×C414yes—
86C2×C4⋊Dic11—21no—
87C2×C4⋊Dic11—21no—
88C2×C4⋊Dic11C2311yescharacteristic
89C2×C4⋊Dic11C2311yescharacteristic
90C2×C4⋊Dic11C2311yescharacteristic
91C2×C4⋊Dic11C2311yescharacteristic
92C2×Dic11⋊C4—21no—
93C2×Dic11⋊C4—21no—
94C2×C4×Dic11—21no—
95C2×C4×Dic11—21no—
96C2×C4×Dic11—21no—
97C2×C4×Dic11—21no—
98C2×C4×Dic11—21no—
99C2×C4×Dic11—21no—
100C2×C4×Dic11C2311yescharacteristic
101C2×C4×Dic11C2311yescharacteristic
102C22×C11⋊C8C2311yescharacteristic
103C22×C11⋊C8C2311yescharacteristic
104C11×#103—22no—
105C11×#102—22no—
106#66899—22no—
107#66899C2×C414yes—
108#66896—22no—
109#66896C2×C414yes—
110C11⋊#95—21no—
111C11⋊#95—21no—
112C11⋊#95—21no—
113C11⋊#95—21no—
114C11⋊#95—21no—
115C11⋊#95—21no—
116C11⋊#95D411yescharacteristic
117C11⋊#95D411yescharacteristic
118C11⋊#95D411yescharacteristic
119C11⋊#95D411yescharacteristic
120#66881C2×C414yes—
121#66879—22no—
122C8×Dic11C2×C414yes—
123#66872—22no—
124#66872—22no—
125#21131—111nomaximal, Sylow
126C24×C11—21no—
127C2×Q8×C11—21no—
128C2×Q8×C11—21no—
129C2×Q8×C11—21no—
130C2×Q8×C11C22×C412yes—
131C2×Q8×C11#4112yes—
132C2×Q8×C11#4112yes—
133C2×D4×C11—21no—
134C2×D4×C11—21no—
135C2×D4×C11—21no—
136C2×D4×C11—21no—
137C2×D4×C11—21no—
138C2×D4×C11C22×C412yes—
139C22×C4×C11—21no—
140C22×C4×C11—21no—
141C22×C4×C11—21no—
142C22×C4×C11C2×D411yescharacteristic
143C22×C4×C11C2×D411yescharacteristic
144C22×C4×C11C2411yescharacteristic
145C22×Dic11—24no—
146C22×Dic11—21no—
147C22×Dic11—21no—
148C22×Dic11—21no—
149C22×Dic11—21no—
150C22×Dic11—81no—
151C22×Dic11—81no—
152C22×Dic11C2×D411yescharacteristic
153C22×Dic11C2×D411yescharacteristic
154C22×Dic11C2×D411yescharacteristic
155C22×Dic11C2×D411yescharacteristic
156C11×SD16—28no—
157C2×C8×C11—21no—
158C2×C8×C11—21no—
159C2×C8×C11—21no—
160C2×C8×C11—21no—
161C2×C8×C11—21no—
162C2×C8×C11C22×C412yes—
163C11×C4⋊C4—24no—
164C11×C4⋊C4—21no—
165C11×C4⋊C4—21no—
166C11×C4⋊C4—21no—
167C11×C4⋊C4#4112yes—
168C11×C22⋊C4—81no—
169C11×C22⋊C4—81no—
170C42×C11—24no—
171C42×C11—21no—
172C42×C11—21no—
173C22⋊Dic11—24no—
174C22⋊Dic11—81no—
175C22⋊Dic11—81no—
176C4⋊Dic11—24no—
177C4⋊Dic11—24no—
178C4⋊Dic11—21no—
179C4⋊Dic11—21no—
180C4⋊Dic11—21no—
181C4⋊Dic11—21no—
182C4⋊Dic11—21no—
183C4⋊Dic11C22×C412yes—
184C4⋊Dic11C22×C412yes—
185C4⋊Dic11#4112yes—
186Dic11⋊C4—81no—
187Dic11⋊C4—81no—
188C4×Dic11—24no—
189C4×Dic11—24no—
190C4×Dic11—21no—
191C4×Dic11—21no—
192C4×Dic11—81no—
193C4×Dic11—81no—
194C4×Dic11—81no—
195C4×Dic11—81no—
196C4×Dic11C22×C412yes—
197C4×Dic11#4112yes—
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×C4×SD16—111no—
205#2319—111no—
206#2276—111no—
207#1926—111no—
208#1926—111no—
209#1920—111no—
210#1920—111no—
211#1880—111no—
212#1880D1111yescharacteristic
213#1865—111no—
214#1865—111no—
215C4×#106—111no—
216#1763—111no—
217#1762—111no—
218#1746—111no—
219C23×C11—22no—
220C23×C11—41no—
221C23×C11—41no—
222C23×C11—41no—
223C23×C11—41no—
224C23×C11—41no—
225C23×C11—41no—
226C23×C11C22×D411yescharacteristic
227Q8×C11—22no—
228Q8×C11—22no—
229Q8×C11—22no—
230Q8×C11#11714yes—
231D4×C11—22no—
232D4×C11—22no—
233D4×C11—41no—
234D4×C11—41no—
235C2×C4×C11—22no—
236C2×C4×C11—22no—
237C2×C4×C11—22no—
238C2×C4×C11—21no—
239C2×C4×C11—21no—
240C2×C4×C11—22no—
241C2×C4×C11—21no—
242C2×C4×C11—21no—
243C2×C4×C11—21no—
244C2×C4×C11—21no—
245C2×C4×C11—41no—
246C2×C4×C11—41no—
247C2×C4×C11—41no—
248C2×C4×C11—41no—
249C2×C4×C11—41no—
250C2×C4×C11C23×C411yescharacteristic
251C2×C4×C11C2×#4111yescharacteristic
252C2×C4×C11C2×#4111yescharacteristic
253C2×C4×C11C2×#4111yescharacteristic
254C2×C4×C11C2×#4111yescharacteristic
255C2×C4×C11C2×#4111yescharacteristic
256C2×Dic11—21no—
257C2×Dic11—21no—
258C2×Dic11—21no—
259C2×Dic11—21no—
260C2×Dic11—22no—
261C2×Dic11—22no—
262C2×Dic11—21no—
263C2×Dic11—21no—
264C2×Dic11—22no—
265C2×Dic11—21no—
266C2×Dic11—21no—
267C2×Dic11—21no—
268C2×Dic11—21no—
269C2×Dic11—41no—
270C2×Dic11—41no—
271C2×Dic11—41no—
272C2×Dic11—41no—
273C2×Dic11—41no—
274C2×Dic11—41no—
275C2×Dic11—41no—
276C2×Dic11—41no—
277C2×Dic11—41no—
278C2×Dic11—84no—
279C2×Dic11C4×D412yes—
280C2×Dic11C4×D412yes—
281C8×C11—22no—
282C8×C11—41no—
283C11⋊C8—22no—
284C11⋊C8—41no—
285C22×SD16C2×D1111yescharacteristic
286C2×C4×Q8—111no—
287C2×C4×Q8—111no—
288C2×C4×Q8C2×D1111yescharacteristic
289C2×C4×D4—111no—
290C4×SD16—118no—
291C2×#106—111no—
292C2×#106—111no—
293C2×#106—111no—
294C2×#103—111no—
295C2×#103—111no—
296C2×#103C2×D1111yescharacteristic
297C2×#102—111no—
298C2×#102—111no—
299C2×#102C2×D1111yescharacteristic
300C2×#97—111no—
301C2×C4×C8—111no—
302#390—111no—
303#390C2×D1111yescharacteristic
304#389—111no—
305#389—111no—
306#388—221no—
307#388—221no—
308#381—221no—
309#380—221no—
310C4×C4⋊C4—111no—
311C4×C4⋊C4—111no—
312C4×C4⋊C4—111no—
313C4×C4⋊C4—111no—
314C4×C4⋊C4—221no—
315C4×C4⋊C4—221no—
316C4×C4⋊C4—221no—
317#376—111no—
318#376—221no—
319#340—111no—
320#340—111no—
321#340—111no—
322#340—111no—
323#340—111no—
324#340C2×D1111yescharacteristic
325#336—111no—
326#336—111no—
327#336—111no—
328#336C2×D1111yescharacteristic
329C22×C11—42no—
330C22×C11—42no—
331C22×C11—42no—
332C22×C11—81no—
333C22×C11—81no—
334C22×C11—81no—
335C22×C11—81no—
336C22×C11C2×C4×D411yescharacteristic
337C22×C11#52511yescharacteristic
338C22×C11#52511yescharacteristic
339C22×C11#57711yescharacteristic
340C22×C11#57711yescharacteristic
341C22×C11#57711yescharacteristic
342C22×C11#57711yescharacteristic
343C4×C11—24no—
344C4×C11—41no—
345C4×C11—42no—
346C4×C11—41no—
347C4×C11—81no—
348C4×C11C2×#11711yescharacteristic
349C4×C11C4×#4111yescharacteristic
350C4×C11#52911yescharacteristic
351C4×C11#52911yescharacteristic
352Dic11—22no—
353Dic11—22no—
354Dic11—41no—
355Dic11—42no—
356Dic11—41no—
357Dic11—42no—
358Dic11—42no—
359Dic11—81no—
360C22×Q8C22×D1111yescharacteristic
361C22×D4C22×D1111yescharacteristic
362C23×C4—221no—
363C2×SD16—22no—
364C2×SD16—22no—
365C2×SD16C4×D1114yes—
366C22×C8—111no—
367C22×C8—111no—
368C22×C8C22×D1111yescharacteristic
369C22×C8C22×D1111yescharacteristic
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—114no—
373C4×Q8—222no—
374C4×Q8—222no—
375C4×Q8—222no—
376C4×D4—114no—
377C4×D4—222no—
378C2×C4⋊C4—21no—
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⋊C4C22×D1111yescharacteristic
388C2×C4⋊C4C22×D1111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—221no—
392C2×C22⋊C4—221no—
393C2×C22⋊C4—221no—
394C2×C42—21no—
395C2×C42—111no—
396C2×C42—111no—
397C2×C42—221no—
398C2×C42—221no—
399C2×C42—221no—
400C2×C42—221no—
401C2×C42—221no—
402C2×C42—221no—
403C2×C42C22×D1111yescharacteristic
404#106—114no—
405#106—222no—
406#106—222no—
407#103—22no—
408#103—114no—
409#103—222no—
410#102—22no—
411#102—114no—
412#102—222no—
413#97—222no—
414C4×C8—114no—
415#95—111no—
416#95—111no—
417#95—111no—
418#95—111no—
419#95—221no—
420#95—221no—
421#95—221no—
422#95—221no—
423#95—221no—
424#95—221no—
425C2×C11—82no—
426C2×C11#229011yescharacteristic
427C2×C11#294911yescharacteristic
428C2×C11#295211yescharacteristic
429C2×C11#312511yescharacteristic
430C2×C11#312511yescharacteristic
431C2×C11#312611yescharacteristic
432C2×C11#312611yescharacteristic
433C24—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8C2×C4×D1112yes—
438C2×Q8#434112yes—
439C2×Q8#434112yes—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C2×C4×D1112yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—111no—
450C22×C4—111no—
451C22×C4—111no—
452C22×C4—111no—
453C22×C4—224no—
454C22×C4—221no—
455C22×C4—221no—
456C22×C4—221no—
457C22×C4—221no—
458C22×C4—881no—
459C22×C4—881no—
460C22×C4D4×D1111yescharacteristic
461C22×C4D4×D1111yescharacteristic
462C22×C4C23×D1111yescharacteristic
463SD16—28no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—112no—
470C2×C8—221no—
471C2×C8—221no—
472C2×C8—221no—
473C2×C8—221no—
474C2×C8—221no—
475C2×C8C2×C4×D1112yes—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—112no—
481C4⋊C4—112no—
482C4⋊C4—112no—
483C4⋊C4—224no—
484C4⋊C4—224no—
485C4⋊C4—221no—
486C4⋊C4—221no—
487C4⋊C4—221no—
488C4⋊C4—221no—
489C4⋊C4—221no—
490C4⋊C4—881no—
491C4⋊C4—881no—
492C4⋊C4#434112yes—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—224no—
496C22⋊C4—881no—
497C22⋊C4—881no—
498C42—24no—
499C42—21no—
500C42—21no—
501C42—112no—
502C42—112no—
503C42—224no—
504C42—224no—
505C42—221no—
506C42—221no—
507C42—881no—
508C42—881no—
509C42—881no—
510C42—881no—
511C11#2113111yescharacteristic, complemented, Sylow
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C2×D4×D1111yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#6693014yes—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—112no—
544C2×C4—112no—
545C2×C4—221no—
546C2×C4—221no—
547C2×C4—221no—
548C2×C4—221no—
549C2×C4—221no—
550C2×C4—221no—
551C2×C4—221no—
552C2×C4—221no—
553C2×C4—222no—
554C2×C4—222no—
555C2×C4—221no—
556C2×C4—222no—
557C2×C4—221no—
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×C4C2×#434111yescharacteristic
570C2×C4C2×#434111yescharacteristic
571C2×C4C2×#434111yescharacteristic
572C2×C4#41×D1111yescharacteristic
573C2×C4#41×D1111yescharacteristic
574C8—22no—
575C8—41no—
576C8—222no—
577C8—441no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C4×D4×D1111yescharacteristic
586C22#1062311711yescharacteristic
587C22#1062313711yescharacteristic
588C22#1062329311yescharacteristic
589C22#1062330111yescharacteristic
590C22#1062330611yescharacteristic
591C22#1062330911yescharacteristic
592C4—24no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—81no—
597C4—222no—
598C4—222no—
599C4—441no—
600C4—442no—
601C4—441no—
602C4—442no—
603C4—442no—
604C4—881no—
605C4C2×#6693011yescharacteristic
606C4C4×#434111yescharacteristic
607C4#1062313111yescharacteristic
608C4#1062318011yescharacteristic
609C2—82no—
610C2#1307105611yescharacteristic
611C2#1307747211yescharacteristic
612C2#1307749411yescharacteristic
613C2#1307889211yescharacteristic
614C2#1307889411yescharacteristic
615C2#1307890111yescharacteristic
616C2#1307890511yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1880C2
socleC23×C11C22×D4
prime radicalC22×D4×C11C23
cosocleC22×C4×C11C24
hypercenter#1880D11
hypocenterC11#21131
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
3. #13070981C2semidirect
6. #13069202C2semidirect
7. #13069200C2semidirect
9. #13068884C2semidirect
10. #13068882C2semidirect
11. #13068504C2semidirect
12. #13068448C2semidirect
14. #13068366C2semidirect
29. SD16×Dic11C4semidirect
511. C11#21131semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic112
#130712532
#130709812
#130692462
#130692402
#130692022
#130692002
#130689732
#130688842
#130688822
#130685042
#130684482
#130684432
#130683662
C11×#18802
#2113111