Finite Group Databasebeta

← #14754773

Subgroups — #14754773

Order 2816.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623035C211yesmaximal, characteristic
2C2×#10622746C211yesmaximal, characteristic
3C2×#10622615C211yesmaximal, characteristic
4#13071232C211yesmaximal, characteristic, complemented
5#13068881C212yesmaximal, complemented
6#13068874C212yesmaximal, complemented
7#13068867C212yesmaximal, complemented
8#13068860C212yesmaximal, complemented
9C2×#10622167C211yesmaximal, characteristic, complemented
10#13068314C211yesmaximal, characteristic, complemented
11C2×C11×#428C211yesmaximal, characteristic, complemented
12C22×#4339C2211yescharacteristic
13C22×C11×M16C2211yescharacteristic
14C2×C11×#117C2211yescharacteristic
15C22×C11×C4⋊C4C2211yescharacteristic
16C2×C4×C22⋊D11—22no—
17C2×#66979C2211yescharacteristic
18C22×C4⋊Dic11C2211yescharacteristic
19C23×C11⋊C8C2211yescharacteristic
20#10623035—24no—
21#10623035—24no—
22C2×#66946C2212yes—
23C2×#66945—21no—
24C2×C4⋊C4×D11—21no—
25C2×#66940C2212yes—
26C11×#428—24no—
27C2×C11×#107C2212yes—
28C2×C11×#106C2212yes—
29#10622746C418yescomplemented
30C2×#66885C2212yes—
31C2×#66880C2212yes—
32#10622615—24no—
33#10622586C2212yes—
34#10622586C2212yes—
35C2×#66874C2211yescharacteristic
36C2×#66874C2211yescharacteristic
37C2×#66873C2211yescharacteristic
38C2×#66873C2211yescharacteristic
39#10622555—22no—
40#10622541—21no—
41#10622541—21no—
42#10622188C2212yes—
43#10622188C2212yes—
44#10622167C2212yes—
45#10622167C2212yes—
46C23×C4×C11—21no—
47C23×C4×C11C2311yescharacteristic
48C22×C22⋊D11—21no—
49C23×Dic11—21no—
50C2×#4339—22no—
51C2×#4339—22no—
52C2×#4339—22no—
53C2×#4339C2×C414yes—
54C2×#4339D414yes—
55C2×#4339Q814yes—
56C22×D44C2311yescharacteristic
57C22×C4×D11—21no—
58C22×C4×D11—22no—
59C22×Dic22C2311yescharacteristic
60C2×C11×M16—22no—
61C2×C11×M16—22no—
62C2×C11×M16C2×C414yescomplemented
63C22×C8×C11C2312yes—
64C11×#117—22no—
65C11×#117—22no—
66C2×C11×C4⋊C4—21no—
67C2×C11×C4⋊C4—22no—
68C2×C11×C4⋊C4D414yes—
69C2×C11×C4⋊C4C2311yescharacteristic
70C2×C11×C4⋊C4C2311yescharacteristic
71C2×C11×C4⋊C4C2312yes—
72C2×C11×C22⋊C4—21no—
73C2×C42×C11—21no—
74C2×C22⋊Dic11D412yes—
75C4×C22⋊D11—216no—
76C2×(C2×D11)⋊C4—22no—
77#66979—22no—
78#66979—22no—
79C2×C4⋊Dic11—21no—
80C2×C4⋊Dic11—22no—
81C2×C4⋊Dic11C2×C414yescomplemented
82C2×C4⋊Dic11D411yescharacteristic
83C2×C4⋊Dic11D411yescharacteristic
84C2×C4⋊Dic11C2312yes—
85C2×Dic11⋊C4—22no—
86C2×Dic11⋊C4—22no—
87C2×Dic11⋊C4—22no—
88C2×C4×Dic11—22no—
89C2×C4×Dic11C2312yes—
90C22×C11⋊C8—22no—
91C22×C11⋊C8—22no—
92C22×C11⋊C8—22no—
93C22×C11⋊C8C2311yescharacteristic
94C22×C11⋊C8C2311yescharacteristic
95#66946—24no—
96#66946—24no—
97#66945—28no—
98C4⋊C4×D11—28no—
99#66940—24no—
100#66940—24no—
101C11×#107—24no—
102C11×#106—24no—
103C11⋊#95—22no—
104C11⋊#95—22no—
105#66885C2×C418yescomplemented
106#66880C2×C418yescomplemented
107#66874—22no—
108#66874—22no—
109#66873—22no—
110#66873—22no—
111#20878—111nomaximal, Sylow
112C24×C11C2×D411yescharacteristic
113C23×D11—21no—
114C22×C4×C11—22no—
115C22×C4×C11—24no—
116C22×C4×C11—21no—
117C22×C4×C11—21no—
118C22×C4×C11—81no—
119C22×C4×C11—81no—
120C22×C4×C11C22×C412yes—
121C22×C4×C11C2×D412yes—
122C22×C4×C11C2×D411yescharacteristic
123C22×C4×C11C2×D412yes—
124C22×C4×C11C2×Q812yes—
125C22×C4×C11#4112yes—
126C22×C4×C11#4112yes—
127C22×C4×C11C2411yescharacteristic
128C2×C22⋊D11—24no—
129C2×C22⋊D11—24no—
130C2×C22⋊D11—24no—
131C22×Dic11—24no—
132C22×Dic11—21no—
133C22×Dic11—22no—
134C22×Dic11—81no—
135C22×Dic11—81no—
136C22×Dic11C2×D412yes—
137C22×Dic11C2×D412yes—
138#4339—28no—
139#4339—28no—
140#4339—28no—
141#4339C4⋊C4116yescomplemented
142C2×D44—21no—
143C2×D44—21no—
144C2×D44—21no—
145C2×D44C22×C412yes—
146C2×D44C2×D412yes—
147C2×D44C2×Q812yes—
148C2×C4×D11—28no—
149C2×C4×D11—24no—
150C2×C4×D11—24no—
151C2×C4×D11—24no—
152C2×C4×D11—82no—
153C2×C4×D11—82no—
154C2×Dic22—21no—
155C2×Dic22—21no—
156C2×Dic22—21no—
157C2×Dic22C22×C412yes—
158C2×Dic22C2×D412yes—
159C2×Dic22C2×Q812yes—
160C11×M16—28no—
161C2×C8×C11—22no—
162C2×C8×C11—22no—
163C2×C8×C11C22×C414yes—
164C11×C4⋊C4—24no—
165C11×C4⋊C4—22no—
166C11×C4⋊C4—21no—
167C11×C4⋊C4—21no—
168C11×C4⋊C4—21no—
169C11×C4⋊C4—21no—
170C11×C4⋊C4C2×D414yes—
171C11×C22⋊C4—42no—
172C42×C11—42no—
173C22⋊Dic11—24no—
174(C2×D11)⋊C4—28no—
175C4⋊Dic11—24no—
176C4⋊Dic11—22no—
177C4⋊Dic11—22no—
178C4⋊Dic11—22no—
179C4⋊Dic11C22×C414yes—
180Dic11⋊C4—28no—
181Dic11⋊C4—82no—
182Dic11⋊C4—82no—
183C4×Dic11—28no—
184C4×Dic11—22no—
185C4×Dic11—22no—
186C2×C11⋊C8—21no—
187C2×C11⋊C8—21no—
188C2×C11⋊C8—21no—
189C2×C11⋊C8—21no—
190C2×C11⋊C8—21no—
191C2×C11⋊C8—21no—
192C2×C11⋊C8—44no—
193C2×#520—111no—
194C2×#428D1111yescharacteristic, complemented
195C2×#417—111no—
196C2×#416—111no—
197#2316—111no—
198#1865—112no—
199#1864—112no—
200#1863—112no—
201#1862—112no—
202C2×#340—111no—
203#1729—111no—
204C23×C11—21no—
205C23×C11—21no—
206C23×C11—21no—
207C23×C11—21no—
208C23×C11C4×D412yes—
209C23×C11#12012yes—
210C23×C11C22⋊Q812yes—
211C23×C11C22×D411yescharacteristic
212C22×D11—22no—
213C22×D11—22no—
214C22×D11—22no—
215C22×D11—41no—
216C22×D11—41no—
217C22×D11—41no—
218C22×D11—41no—
219C2×C4×C11—24no—
220C2×C4×C11—42no—
221C2×C4×C11—42no—
222C2×C4×C11—41no—
223C2×C4×C11—41no—
224C2×C4×C11—41no—
225C2×C4×C11—41no—
226C2×C4×C11—41no—
227C2×C4×C11—41no—
228C2×C4×C11—84no—
229C2×C4×C11C2×C4⋊C414yes—
230C2×C4×C11C4×D412yes—
231C2×C4×C11C4×D414yes—
232C2×C4×C11#12012yes—
233C2×C4×C11C22⋊Q812yes—
234C2×C4×C11C22⋊Q814yes—
235C2×C4×C11#12314yes—
236C2×C4×C11C23×C411yescharacteristic
237C2×C4×C11C22×D411yescharacteristic
238C2×C4×C11C22×D411yescharacteristic
239C2×C4×C11C22×Q811yescharacteristic
240C2×C4×C11C2×#4111yescharacteristic
241C2×C4×C11C2×#4111yescharacteristic
242C22⋊D11—216no—
243C2×Dic11—22no—
244C2×Dic11—22no—
245C2×Dic11—22no—
246C2×Dic11—24no—
247C2×Dic11—22no—
248C2×Dic11—22no—
249C2×Dic11—22no—
250C2×Dic11—22no—
251C2×Dic11—22no—
252C2×Dic11—22no—
253C2×Dic11—42no—
254C2×Dic11—42no—
255C2×Dic11—84no—
256D44—22no—
257D44—22no—
258D44—22no—
259D44C2×C4⋊C414yes—
260C4×D11—216no—
261C4×D11—88no—
262Dic22—22no—
263Dic22—22no—
264Dic22—22no—
265Dic22C2×C4⋊C414yes—
266C8×C11—24no—
267C11⋊C8—41no—
268C11⋊C8—41no—
269C22×#41—111no—
270C22×M16C2×D1111yescharacteristic
271C23×C8—111no—
272#520—224no—
273#520—224no—
274C2×C4×Q8—112no—
275C2×C4×D4—112no—
276C2×C4×D4—222no—
277C2×#117—111no—
278C2×#117—221no—
279C2×#117C2×D1111yescharacteristic
280C22×C4⋊C4—111no—
281C22×C4⋊C4—221no—
282C22×C4⋊C4C2×D1111yescharacteristic
283#428—24no—
284C2×#107C2×D1112yes—
285C2×#106C2×D1112yes—
286#417—118no—
287#416—224no—
288C2×#103—112no—
289C2×#103—111no—
290C2×#103—111no—
291C2×#102—112no—
292C2×#102—111no—
293C2×#102—111no—
294#389—112no—
295#389—112no—
296#389—221no—
297#389—221no—
298#380—222no—
299#340—112no—
300#340—112no—
301#336—112no—
302#336—112no—
303C22×C11—22no—
304C22×C11—22no—
305C22×C11—22no—
306C22×C11—22no—
307C22×C11—22no—
308C22×C11—22no—
309C22×C11#38514yes—
310C22×C11C2×C4×D411yescharacteristic
311C22×C11C2×#12011yescharacteristic
312C22×C11C2×C22⋊Q811yescharacteristic
313C22×C11#57611yescharacteristic
314C22×C11#57711yescharacteristic
315C22×C11#57711yescharacteristic
316C22×C11#57811yescharacteristic
317C2×D11—24no—
318C2×D11—42no—
319C2×D11—42no—
320C2×D11—42no—
321C2×D11—42no—
322C2×D11—42no—
323C2×D11—42no—
324C4×C11—44no—
325C4×C11—81no—
326C4×C11—81no—
327C4×C11#38514yes—
328C4×C11C22×C4⋊C411yescharacteristic
329C4×C11C2×C4×D411yescharacteristic
330C4×C11C2×C22⋊Q811yescharacteristic
331C4×C11C2×#12311yescharacteristic
332Dic11—24no—
333Dic11—44no—
334Dic11—42no—
335Dic11—42no—
336C2×#41—114no—
337C2×#41—114no—
338C2×#41—114no—
339C2×#41—222no—
340C2×#41—222no—
341C2×#41—222no—
342C22×Q8—111no—
343C22×D4—111no—
344C22×D4—221no—
345C23×C4—21no—
346C23×C4—222no—
347C23×C4—221no—
348C23×C4—221no—
349C23×C4C22×D1111yescharacteristic
350C2×M16—22no—
351C2×M16—22no—
352C2×M16C4×D1114yescomplemented
353C22×C8—111no—
354C22×C8—111no—
355C22×C8—222no—
356C22×C8—222no—
357C22×C8—222no—
358C22×C8C22×D1112yes—
359C4×Q8—224no—
360C4×Q8—224no—
361C4×D4—224no—
362C4×D4—2216no—
363C4×D4—224no—
364#117—22no—
365#117—22no—
366#117—228no—
367#117—222no—
368#117—222no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—114no—
372C2×C4⋊C4—112no—
373C2×C4⋊C4—111no—
374C2×C4⋊C4—111no—
375C2×C4⋊C4—228no—
376C2×C4⋊C4—222no—
377C2×C4⋊C4—221no—
378C2×C4⋊C4—222no—
379C2×C4⋊C4—222no—
380C2×C4⋊C4—222no—
381C2×C4⋊C4C22⋊D1114yes—
382C2×C4⋊C4C22×D1111yescharacteristic
383C2×C4⋊C4C22×D1111yescharacteristic
384C2×C4⋊C4C22×D1112yes—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—112no—
387C2×C22⋊C4—222no—
388C2×C42—21no—
389C2×C42—112no—
390C2×C42—222no—
391#107—24no—
392#106—24no—
393#103—118no—
394#103—222no—
395#103—222no—
396#102—118no—
397#102—222no—
398#102—222no—
399#95—222no—
400#95—222no—
401C2×C11—24no—
402C2×C11C2×#38511yescharacteristic
403C2×C11#294611yescharacteristic
404C2×C11#294711yescharacteristic
405C2×C11#301211yescharacteristic
406C2×C11#301311yescharacteristic
407C2×C11#308311yescharacteristic
408C2×C11#308411yescharacteristic
409D11—44no—
410C24—221no—
411C24D4×D1111yescharacteristic
412#41—1116no—
413#41—228no—
414#41—228no—
415#41—228no—
416C2×Q8—112no—
417C2×Q8—112no—
418C2×Q8—112no—
419C2×Q8—221no—
420C2×Q8—221no—
421C2×Q8—221no—
422C2×D4—112no—
423C2×D4—112no—
424C2×D4—112no—
425C2×D4—224no—
426C2×D4—224no—
427C2×D4—224no—
428C2×D4—221no—
429C2×D4—221no—
430C2×D4—221no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—112no—
438C22×C4—112no—
439C22×C4—228no—
440C22×C4—221no—
441C22×C4—224no—
442C22×C4—224no—
443C22×C4—224no—
444C22×C4—222no—
445C22×C4—224no—
446C22×C4—882no—
447C22×C4—882no—
448C22×C4—881no—
449C22×C4—881no—
450C22×C4C2×C4×D1112yes—
451C22×C4#433912yes—
452C22×C4D4×D1112yes—
453C22×C4D4×D1111yescharacteristic
454C22×C4Q8×D1112yes—
455C22×C4#434312yes—
456C22×C4C2×C22⋊D1112yes—
457C22×C4C23×D1111yescharacteristic
458M16—28no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—221no—
462C2×C8—221no—
463C2×C8—221no—
464C2×C8—221no—
465C2×C8—221no—
466C2×C8—221no—
467C2×C8—444no—
468C2×C8C2×C4×D1114yes—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—114no—
476C4⋊C4—228no—
477C4⋊C4—224no—
478C4⋊C4—222no—
479C4⋊C4—222no—
480C4⋊C4—222no—
481C4⋊C4—882no—
482C4⋊C4—882no—
483C4⋊C4C2×C22⋊D1114yes—
484C22⋊C4—42no—
485C22⋊C4—228no—
486C22⋊C4—224no—
487C42—42no—
488C42—228no—
489C42—222no—
490C42—222no—
491C11#2087811yescharacteristic, complemented, Sylow
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—222no—
497C23—222no—
498C23—222no—
499C23—441no—
500C23—441no—
501C23—441no—
502C23—441no—
503C23#6694612yes—
504C23#6694912yes—
505C23#6699012yes—
506C23C2×D4×D1111yescharacteristic
507Q8—114no—
508Q8—222no—
509Q8—222no—
510Q8—222no—
511D4—114no—
512D4—2216no—
513D4—222no—
514D4—222no—
515D4—222no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—222no—
527C2×C4—222no—
528C2×C4—222no—
529C2×C4—2216no—
530C2×C4—224no—
531C2×C4—222no—
532C2×C4—222no—
533C2×C4—222no—
534C2×C4—222no—
535C2×C4—222no—
536C2×C4—222no—
537C2×C4—442no—
538C2×C4—442no—
539C2×C4—888no—
540C2×C4—884no—
541C2×C4C4⋊C4×D1114yes—
542C2×C4#6694612yes—
543C2×C4#6694714yes—
544C2×C4#6694912yes—
545C2×C4C4×C22⋊D1114yes—
546C2×C4#6699012yes—
547C2×C4#6699914yes—
548C2×C4C22×C4×D1111yescharacteristic
549C2×C4C2×#433911yescharacteristic
550C2×C4C2×D4×D1111yescharacteristic
551C2×C4C2×Q8×D1111yescharacteristic
552C2×C4C2×#434311yescharacteristic
553C2×C4C22×C22⋊D1111yescharacteristic
554C8—24no—
555C8—441no—
556C8—441no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—224no—
564C22—442no—
565C22—442no—
566C22—442no—
567C22—442no—
568C22—442no—
569C22—442no—
570C22#1062260214yes—
571C22C2×#6694611yescharacteristic
572C22C2×#6694911yescharacteristic
573C22#1062330511yescharacteristic
574C22#1062330611yescharacteristic
575C22#1062330911yescharacteristic
576C22#1062331011yescharacteristic
577C22C2×#6699011yescharacteristic
578C4—44no—
579C4—81no—
580C4—81no—
581C4—224no—
582C4—442no—
583C4—442no—
584C4—444no—
585C4#1062260214yes—
586C4C2×C4⋊C4×D1111yescharacteristic
587C4C2×#6694711yescharacteristic
588C4C2×C4×C22⋊D1111yescharacteristic
589C4C2×#6699911yescharacteristic
590C2—24no—
591C2—444no—
592C2C2×#1062260211yescharacteristic
593C2#1307744611yescharacteristic
594C2#1307745711yescharacteristic
595C2#1307791911yescharacteristic
596C2#1307793011yescharacteristic
597C2#1307846611yescharacteristic
598C2#1307847711yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC2×C11×#428C2
socleC23×C11C22×D4
prime radicalC22×#4339C22
cosocleC22×C4×C11C24
hypercenterC2×#428D11
hypocenterC11#20878
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #13071232C2semidirect
5. #13068881C2semidirect
6. #13068874C2semidirect
7. #13068867C2semidirect
8. #13068860C2semidirect
9. C2×#10622167C2semidirect
10. #13068314C2semidirect
11. C2×C11×#428C2semidirect
29. #10622746C4semidirect
62. C2×C11×M16C2×C4semidirect
81. C2×C4⋊Dic11C2×C4semidirect
105. #66885C2×C4semidirect
106. #66880C2×C4semidirect
141. #4339C4⋊C4semidirect
194. C2×#428D11semidirect
352. C2×M16C4×D11semidirect
491. C11#20878semidirect

Maximal subgroups

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

subgroupindex
C2×#106230352
C2×#106227462
C2×#106226152
#130712322
#130688812
#130688742
#130688672
#130688602
C2×#106221672
#130683142
C2×C11×#4282
#2087811