Finite Group Databasebeta

← #14788092

Subgroups — #14788092

Order 2816.

1198 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
1C2×#10623001C211yesmaximal, characteristic, complemented
2C2×#10622455C211yesmaximal, characteristic, complemented
3C2×#10622454C211yesmaximal, characteristic, complemented
4C4×#66983C211yesmaximal, characteristic
5#13069813C212yesmaximal, complemented
6#13069621C212yesmaximal
7#13069563C211yesmaximal, characteristic, complemented
8#13069538C211yesmaximal, characteristic, complemented
9#13068970C212yesmaximal, complemented
10#13068887C212yesmaximal, complemented
11C11×#1797C211yesmaximal, characteristic, complemented
12C22×C42×C11C2211yescharacteristic
13C2×#66983C2211yescharacteristic
14C2×C4×C22⋊D11—21no—
15C2×#66979—21no—
16C2×#66977—21no—
17C2×#66977C2212yes—
18C22×#4314C2212yes—
19C2×C4×D44C2211yescharacteristic
20#10623001—24no—
21#10623001—24no—
22C2×#66923—21no—
23C2×#66922C2211yescharacteristic
24C2×#66922C2211yescharacteristic
25C2×C4×Dic22—21no—
26C2×C4×Dic22C2212yes—
27C2×C11×C4⋊C8C2211yescharacteristic
28C2×C11×C4⋊C8C2211yescharacteristic
29C2×C11×C22⋊C8C2212yes—
30#10622826—21no—
31#10622825C2211yescharacteristic
32C2×#66885C2212yes—
33C2×#66881C2212yes—
34C2×#66880C2212yes—
35C2×#66880C2212yes—
36C2×#66880C2212yes—
37#10622553—21no—
38#10622552C2211yescharacteristic
39C4×(C2×D11)⋊C4—21no—
40C4×C4⋊Dic11C2211yescharacteristic
41#10622541C2211yescharacteristic
42#10622455—24no—
43#10622454—24no—
44C11×#336C2212yes—
45#10622189C2212yes—
46#10622189C2212yes—
47C23×C4×C11D411yescharacteristic
48C23×C4×C11D411yescharacteristic
49C23×C4×C11C2311yescharacteristic
50C22×C22⋊D11—21no—
51C22×D44C2311yescharacteristic
52C22×C4×D11—21no—
53C22×Dic22—21no—
54C22×Dic22C2312yes—
55C22×C8×C11C2312yes—
56C22×C8×C11C2312yes—
57C2×C42×C11—22no—
58C2×C42×C11—22no—
59C2×C42×C11D411yescharacteristic
60C2×C42×C11D411yescharacteristic
61C2×C42×C11C2311yescharacteristic
62C2×C42×C11C2311yescharacteristic
63C2×C22⋊Dic11—22no—
64C2×C22⋊Dic11—21no—
65#66983—22no—
66#66983—22no—
67C4×C22⋊D11—44no—
68C2×(C2×D11)⋊C4—21no—
69C2×(C2×D11)⋊C4—21no—
70C2×(C2×D11)⋊C4—21no—
71C2×(C2×D11)⋊C4—21no—
72#66979—28no—
73C2×C4⋊Dic11—21no—
74C2×C4⋊Dic11—21no—
75C2×C4⋊Dic11—21no—
76C2×C4⋊Dic11C2311yescharacteristic
77C2×C4⋊Dic11C2311yescharacteristic
78C2×C4⋊Dic11C2312yes—
79C2×C4⋊Dic11C2311yescharacteristic
80#66977—28no—
81#66977—24no—
82#66977—24no—
83C2×Dic11⋊C4—22no—
84C2×Dic11⋊C4—22no—
85C2×Dic11⋊C4—21no—
86C2×C4×Dic11—22no—
87C2×C4×Dic11—21no—
88C2×#4314—24no—
89C2×#4314—24no—
90C2×#4314—42no—
91C4×D44—22no—
92C4×D44—22no—
93#66923—28no—
94#66922—22no—
95#66922—22no—
96#66922D414yescomplemented
97#66922D414yescomplemented
98C4×Dic22—28no—
99C4×Dic22—24no—
100C4×Dic22—24no—
101C11×C4⋊C8—22no—
102C11×C4⋊C8—22no—
103C11×C22⋊C8—42no—
104C11⋊#95—21no—
105C11⋊#95—21no—
106C11⋊#95—21no—
107#66885—24no—
108#66881—24no—
109#66880—24no—
110#66880—24no—
111#66880—42no—
112#24151—111nomaximal, Sylow
113C24×C11C2×D411yescharacteristic
114C23×D11—21no—
115C22×C4×C11—22no—
116C22×C4×C11—22no—
117C22×C4×C11—22no—
118C22×C4×C11—21no—
119C22×C4×C11—21no—
120C22×C4×C11—21no—
121C22×C4×C11—21no—
122C22×C4×C11—21no—
123C22×C4×C11—21no—
124C22×C4×C11—41no—
125C22×C4×C11—41no—
126C22×C4×C11SD1614yescomplemented
127C22×C4×C11C2×D411yescharacteristic
128C22×C4×C11C2×D411yescharacteristic
129C22×C4×C11C2×D411yescharacteristic
130C22×C4×C11C2×D411yescharacteristic
131C22×C4×C11C2×D411yescharacteristic
132C22×C4×C11C2411yescharacteristic
133C2×C22⋊D11—24no—
134C2×C22⋊D11—42no—
135C2×C22⋊D11—42no—
136C22×Dic11—21no—
137C22×Dic11—22no—
138C22×Dic11—22no—
139C22×Dic11—21no—
140C22×Dic11—21no—
141C2×D44—21no—
142C2×D44—21no—
143C2×D44—21no—
144C2×D44—21no—
145C2×D44—21no—
146C2×D44#4112yes—
147C2×C4×D11—42no—
148C2×C4×D11—42no—
149C2×C4×D11—42no—
150C2×Dic22—24no—
151C2×Dic22—22no—
152C2×Dic22—22no—
153C2×Dic22—22no—
154C2×Dic22—22no—
155C2×Dic22—22no—
156C2×Dic22—42no—
157C2×Dic22—42no—
158C2×Dic22C2×D414yes—
159C2×C8×C11—22no—
160C2×C8×C11—22no—
161C2×C8×C11—22no—
162C2×C8×C11—22no—
163C2×C8×C11—22no—
164C2×C8×C11—22no—
165C42×C11—22no—
166C42×C11—22no—
167C42×C11—21no—
168C42×C11—21no—
169C42×C11C2×D412yes—
170C42×C11C2×D412yes—
171C22⋊Dic11—28no—
172C22⋊Dic11—42no—
173(C2×D11)⋊C4—24no—
174(C2×D11)⋊C4—42no—
175(C2×D11)⋊C4—42no—
176(C2×D11)⋊C4—42no—
177C4⋊Dic11—24no—
178C4⋊Dic11—24no—
179C4⋊Dic11—24no—
180C4⋊Dic11—22no—
181C4⋊Dic11—21no—
182C4⋊Dic11—21no—
183C4⋊Dic11—21no—
184C4⋊Dic11—21no—
185C4⋊Dic11—21no—
186C4⋊Dic11C2×D414yes—
187C4⋊Dic11#4112yes—
188Dic11⋊C4—28no—
189Dic11⋊C4—28no—
190Dic11⋊C4—42no—
191C4×Dic11—28no—
192C4×Dic11—42no—
193#4314—48no—
194C2×#533—111no—
195C2×#476—111no—
196C2×#461—111no—
197C4×#121—111no—
198#2035—112no—
199#2000—112no—
200#1987—111no—
201#1981—111no—
202#1880—112no—
203#1865—112no—
204#1797D1111yescharacteristic, complemented
205C23×C11—21no—
206C23×C11—21no—
207C23×C11—21no—
208C23×C11—21no—
209C23×C11—21no—
210C23×C11—21no—
211C23×C11C2×SD1612yes—
212C23×C11C22×D411yescharacteristic
213C22×D11—22no—
214C22×D11—41no—
215C22×D11—41no—
216C22×D11—41no—
217C22×D11—41no—
218C22×D11—41no—
219C22×D11—41no—
220C2×C4×C11—21no—
221C2×C4×C11—21no—
222C2×C4×C11—21no—
223C2×C4×C11—21no—
224C2×C4×C11—28no—
225C2×C4×C11—21no—
226C2×C4×C11—21no—
227C2×C4×C11—21no—
228C2×C4×C11—21no—
229C2×C4×C11—21no—
230C2×C4×C11—21no—
231C2×C4×C11—44no—
232C2×C4×C11—41no—
233C2×C4×C11—41no—
234C2×C4×C11—41no—
235C2×C4×C11—41no—
236C2×C4×C11#12012yes—
237C2×C4×C11#12012yes—
238C2×C4×C11C2×SD1612yes—
239C2×C4×C11C2×SD1612yes—
240C2×C4×C11#13712yes—
241C2×C4×C11C22×D411yescharacteristic
242C2×C4×C11C22×D411yescharacteristic
243C2×C4×C11C2×#4111yescharacteristic
244C2×C4×C11C2×#4111yescharacteristic
245C2×C4×C11H(5,𝔽2)11yescharacteristic
246C2×C4×C112−1+411yescharacteristic
247C22⋊D11—48no—
248C2×Dic11—24no—
249C2×Dic11—24no—
250C2×Dic11—22no—
251C2×Dic11—42no—
252C2×Dic11—42no—
253C2×Dic11—41no—
254C2×Dic11—42no—
255C2×Dic11—42no—
256C2×Dic11—41no—
257C2×Dic11—41no—
258C2×Dic11—41no—
259C2×Dic11—41no—
260C2×Dic11—41no—
261C2×Dic11—41no—
262C2×Dic11—41no—
263D44—22no—
264D44—22no—
265D44—41no—
266D44—41no—
267C4×D11—84no—
268Dic22—24no—
269Dic22—24no—
270Dic22—48no—
271Dic22—42no—
272Dic22—42no—
273C8×C11—42no—
274C8×C11—42no—
275C22×SD16—112no—
276#533—224no—
277#533—224no—
278C2×C4⋊Q8—111no—
279C2×C4⋊Q8—111no—
280C2×#125—221no—
281C2×C22⋊Q8—112no—
282C2×C22⋊Q8—221no—
283C2×#121—111no—
284C2×C4×Q8—112no—
285C2×C4×Q8—221no—
286C2×C4×D4—111no—
287C2×C4×D4—221no—
288C2×#117—221no—
289C22×C42C2×D1111yescharacteristic
290#476—224no—
291#461—224no—
292C2×#106—112no—
293C2×C4⋊C8C2×D1111yescharacteristic
294C2×C4⋊C8C2×D1111yescharacteristic
295C2×#103—112no—
296C2×#103—112no—
297C2×#103—112no—
298C2×#102—112no—
299C2×C22⋊C8C2×D1112yes—
300#390—111no—
301#389—111no—
302#388—221no—
303#386—221no—
304#380—111no—
305C4×C4⋊C4—111no—
306C4×C22⋊C4—221no—
307#340—112no—
308#340—112no—
309#336C2×D1112yes—
310C22×C11—22no—
311C22×C11—22no—
312C22×C11—22no—
313C22×C11—41no—
314C22×C11—41no—
315C22×C11—41no—
316C22×C11—41no—
317C22×C11C2×#12011yescharacteristic
318C22×C11#52511yescharacteristic
319C22×C11#53711yescharacteristic
320C22×C11C22×SD1611yescharacteristic
321C22×C11C2×#13711yescharacteristic
322C22×C11#57511yescharacteristic
323C22×C11#57511yescharacteristic
324C2×D11—42no—
325C2×D11—42no—
326C2×D11—42no—
327C2×D11—81no—
328C2×D11—81no—
329C2×D11—81no—
330C2×D11—81no—
331C4×C11—22no—
332C4×C11—22no—
333C4×C11—41no—
334C4×C11—41no—
335C4×C11#44814yes—
336C4×C11#54611yescharacteristic
337C4×C11#54711yescharacteristic
338C4×C11#54811yescharacteristic
339C4×C11#55011yescharacteristic
340Dic11—42no—
341Dic11—44no—
342Dic11—44no—
343Dic11—81no—
344Dic11—81no—
345C22×Q8—112no—
346C22×Q8—221no—
347C22×D4—111no—
348C22×D4—221no—
349C23×C4—221no—
350C23×C4D4411yescharacteristic
351C23×C4D4411yescharacteristic
352C23×C4C22×D1111yescharacteristic
353C2×SD16—224no—
354C2×SD16—224no—
355C2×SD16—442no—
356C22×C8C22×D1112yes—
357C22×C8C22×D1112yes—
358C4⋊Q8—114no—
359C4⋊Q8—114no—
360C4⋊Q8—222no—
361C4⋊Q8—222no—
362#125—228no—
363C22⋊Q8—224no—
364C22⋊Q8—224no—
365C22⋊Q8—228no—
366#121—222no—
367#121—222no—
368C4×Q8—224no—
369C4×Q8—224no—
370C4×Q8—228no—
371C4×D4—222no—
372C4×D4—222no—
373C4×D4—444no—
374#117—228no—
375C2×C4⋊C4—112no—
376C2×C4⋊C4—111no—
377C2×C4⋊C4—111no—
378C2×C4⋊C4—111no—
379C2×C4⋊C4—221no—
380C2×C4⋊C4—221no—
381C2×C4⋊C4—222no—
382C2×C4⋊C4—222no—
383C2×C4⋊C4—221no—
384C2×C4⋊C4—221no—
385C2×C22⋊C4—222no—
386C2×C22⋊C4—221no—
387C2×C22⋊C4—221no—
388C2×C22⋊C4—221no—
389C2×C22⋊C4—221no—
390C2×C22⋊C4—221no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—222no—
394C2×C42—221no—
395C2×C42D4411yescharacteristic
396C2×C42D4411yescharacteristic
397C2×C42C22×D1111yescharacteristic
398C2×C42C22×D1111yescharacteristic
399#106—224no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402#103—224no—
403#103—224no—
404#103—442no—
405#102—224no—
406C22⋊C8—42no—
407#95—221no—
408#95—221no—
409#95—221no—
410C2×C11—42no—
411C2×C11#240711yescharacteristic
412C2×C11C2×#44811yescharacteristic
413C2×C11#300511yescharacteristic
414C2×C11#310111yescharacteristic
415C2×C11#311611yescharacteristic
416C2×C11#324611yescharacteristic
417C2×C11#325211yescharacteristic
418D11—82no—
419C24—221no—
420C24C2×D4411yescharacteristic
421C2×Q8—114no—
422C2×Q8—224no—
423C2×Q8—222no—
424C2×Q8—222no—
425C2×Q8—222no—
426C2×Q8—222no—
427C2×Q8—222no—
428C2×Q8—442no—
429C2×Q8—442no—
430C2×D4—112no—
431C2×D4—224no—
432C2×D4—221no—
433C2×D4—221no—
434C2×D4—221no—
435C2×D4—221no—
436C2×D4—221no—
437C2×D4—442no—
438C2×D4—442no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—221no—
451C22×C4—222no—
452C22×C4—222no—
453C22×C4—221no—
454C22×C4—221no—
455C22×C4—442no—
456C22×C4—442no—
457C22×C4—442no—
458C22×C4#431414yescomplemented
459C22×C4C2×D4411yescharacteristic
460C22×C4C2×D4411yescharacteristic
461C22×C4C2×D4411yescharacteristic
462C22×C4C2×D4411yescharacteristic
463C22×C4C2×D4411yescharacteristic
464C22×C4C23×D1111yescharacteristic
465SD16—448no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—114no—
473C4⋊C4—112no—
474C4⋊C4—228no—
475C4⋊C4—228no—
476C4⋊C4—224no—
477C4⋊C4—224no—
478C4⋊C4—224no—
479C4⋊C4—222no—
480C4⋊C4—221no—
481C4⋊C4—221no—
482C4⋊C4—221no—
483C4⋊C4—221no—
484C4⋊C4—221no—
485C4⋊C4—442no—
486C22⋊C4—224no—
487C22⋊C4—228no—
488C22⋊C4—442no—
489C22⋊C4—442no—
490C22⋊C4—442no—
491C22⋊C4—442no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—21no—
496C42—228no—
497C42—442no—
498C42D4×D1112yes—
499C42D4×D1112yes—
500C11#2415111yescharacteristic, complemented, Sylow
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—222no—
508C23—441no—
509C23—441no—
510C23—441no—
511C23—441no—
512C23—441no—
513C23—441no—
514C23C2×#431412yes—
515C23C22×D4411yescharacteristic
516Q8—224no—
517Q8—224no—
518Q8—448no—
519Q8—442no—
520Q8—442no—
521D4—222no—
522D4—222no—
523D4—448no—
524D4—441no—
525D4—441no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—224no—
543C2×C4—224no—
544C2×C4—222no—
545C2×C4—441no—
546C2×C4—441no—
547C2×C4—442no—
548C2×C4—442no—
549C2×C4—442no—
550C2×C4—442no—
551C2×C4—441no—
552C2×C4—441no—
553C2×C4—441no—
554C2×C4—441no—
555C2×C4—441no—
556C2×C4—441no—
557C2×C4—884no—
558C2×C4#6693512yes—
559C2×C4#6693512yes—
560C2×C4C2×#431412yes—
561C2×C4C2×#431412yes—
562C2×C4#6696212yes—
563C2×C4C2×D4×D1111yescharacteristic
564C2×C4C2×D4×D1111yescharacteristic
565C2×C4#6703711yescharacteristic
566C2×C4#6704011yescharacteristic
567C2×C4#41×D1111yescharacteristic
568C2×C4#41×D1111yescharacteristic
569C8—42no—
570C8—42no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—442no—
579C22—442no—
580C22—442no—
581C22—881no—
582C22—881no—
583C22—881no—
584C22—881no—
585C22C2×#6693511yescharacteristic
586C22#1062304111yescharacteristic
587C22#1062305511yescharacteristic
588C22C22×#431411yescharacteristic
589C22C2×#6696211yescharacteristic
590C22#1062327911yescharacteristic
591C22#1062327911yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—444no—
597C4—442no—
598C4—444no—
599C4—881no—
600C4—881no—
601C4#1062235114yes—
602C4#1062313011yescharacteristic
603C4#1062314311yescharacteristic
604C4#1062316211yescharacteristic
605C4#1062316411yescharacteristic
606C2—42no—
607C2—882no—
608C2#1307196211yescharacteristic
609C2C2×#1062235111yescharacteristic
610C2#1307786111yescharacteristic
611C2#1307862511yescharacteristic
612C2#1307877911yescharacteristic
613C2#1307993311yescharacteristic
614C2#1307999011yescharacteristic
subgroupG/H
centerC23C22×D44
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1797C2
socleC23×C11C22×D4
prime radicalC2×#66983C22
hypercenter#1797D11
hypocenterC11#24151
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C2×#10623001C2semidirect
2. C2×#10622455C2semidirect
3. C2×#10622454C2semidirect
5. #13069813C2semidirect
7. #13069563C2semidirect
8. #13069538C2semidirect
9. #13068970C2semidirect
10. #13068887C2semidirect
11. C11×#1797C2semidirect
96. #66922D4semidirect
97. #66922D4semidirect
126. C22×C4×C11SD16semidirect
204. #1797D11semidirect
458. C22×C4#4314semidirect
500. C11#24151semidirect

Maximal subgroups

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

subgroupindex
C2×#106230012
C2×#106224552
C2×#106224542
C4×#669832
#130698132
#130696212
#130695632
#130695382
#130689702
#130688872
C11×#17972
#2415111