Finite Group Databasebeta

← #14755039

Subgroups — #14755039

Order 2816.

1970 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623056C211yesmaximal, characteristic
2C2×#10623035C211yesmaximal, characteristic, complemented
3C2×#10622744C211yesmaximal, characteristic, complemented
4C22×#66873C211yesmaximal, characteristic, complemented
5C2×#10622742C211yesmaximal, characteristic, complemented
6#13068937C214yesmaximal, complemented
7#13068932C214yesmaximal, complemented
8#13068332C211yesmaximal, characteristic, complemented
9C11×#1736C211yesmaximal, characteristic, complemented
10C22×#4339C2211yescharacteristic
11C23×D44C2211yescharacteristic
12C22×C11×M16C2211yescharacteristic
13C2×C11×#117C2211yescharacteristic
14C22×C11×C4⋊C4C2211yescharacteristic
15C2×C4×C22⋊D11—22no—
16C2×#66979C2211yescharacteristic
17C23×C11⋊C8C2211yescharacteristic
18#10623056—24no—
19#10623056C418yescomplemented
20C2×#66948C2214yes—
21#10623035—24no—
22#10623035—24no—
23C2×#66946—21no—
24C2×#66945C2212yescomplemented
25C2×C4⋊C4×D11C2212yescomplemented
26C2×#66940—21no—
27C2×#66935—22no—
28C2×#66928—21no—
29C2×#66927—21no—
30#10622744—24no—
31#10622742—24no—
32C2×#66885C2212yes—
33C2×#66885C2212yes—
34C2×(C2×D11)⋊C8C2214yescomplemented
35C2×#66873—24no—
36C2×#66873C2214yes—
37C11×#340C2214yescomplemented
38#10622167C2212yescomplemented
39#10622167C2212yescomplemented
40C24×D11—21no—
41C23×C4×C11—21no—
42C23×C4×C11C2311yescharacteristic
43C22×C22⋊D11D412yes—
44C2×#4339—22no—
45C2×#4339—22no—
46C2×#4339—22no—
47C2×#4339—22no—
48C2×#4339C2×C414yes—
49C2×#4339D414yes—
50C22×D44—22no—
51C22×D44—22no—
52C22×D44—22no—
53C22×D44—22no—
54C22×D44—21no—
55C22×D44—22no—
56C22×D44—24no—
57C22×D44C2×C414yescomplemented
58C22×D44D411yescharacteristic
59C22×D44C2312yes—
60C22×C4×D11—22no—
61C22×C4×D11C2312yes—
62C22×Dic22D411yescharacteristic
63C2×C11×M16—22no—
64C2×C11×M16—22no—
65C2×C11×M16—22no—
66C22×C8×C11C2312yes—
67C11×#117—22no—
68C11×#117C2×C414yescomplemented
69C2×C11×C4⋊C4—21no—
70C2×C11×C4⋊C4—22no—
71C2×C11×C4⋊C4D414yescomplemented
72C2×C11×C4⋊C4C2312yes—
73C2×C11×C4⋊C4C2312yes—
74C2×C11×C22⋊C4—21no—
75C2×C42×C11—21no—
76C2×C22⋊Dic11—21no—
77C4×C22⋊D11—216no—
78C2×(C2×D11)⋊C4—22no—
79C2×(C2×D11)⋊C4—24no—
80#66979—22no—
81#66979D414yescomplemented
82C2×C4⋊Dic11C2311yescharacteristic
83C2×C4⋊Dic11C2311yescharacteristic
84C2×Dic11⋊C4—22no—
85C2×C4×Dic11—21no—
86C2×C4×Dic11—22no—
87C22×C11⋊C8—22no—
88C22×C11⋊C8—22no—
89C22×C11⋊C8—22no—
90C22×C11⋊C8C2312yes—
91#66948—28no—
92#66948C2×C4116yescomplemented
93#66946—28no—
94#66945—24no—
95#66945—24no—
96C4⋊C4×D11—24no—
97C4⋊C4×D11—24no—
98#66940—28no—
99#66935—216no—
100#66928—28no—
101#66927—28no—
102#66885—24no—
103#66885—24no—
104(C2×D11)⋊C8—28no—
105#66873—28no—
106#20914—111nomaximal, Sylow
107C24×C11C2×D411yescharacteristic
108C23×D11—22no—
109C23×D11—24no—
110C23×D11—42no—
111C23×D11—42no—
112C23×D11—82no—
113C23×D11—82no—
114C23×D11C2×D412yes—
115C22×C4×C11—22no—
116C22×C4×C11—22no—
117C22×C4×C11—24no—
118C22×C4×C11—82no—
119C22×C4×C11C22×C412yes—
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2×D412yes—
122C22×C4×C11C2×D412yes—
123C22×C4×C11C2×D412yes—
124C22×C4×C11#4112yes—
125C22×C4×C11#4112yes—
126C22×C4×C11C2411yescharacteristic
127C2×C22⋊D11—24no—
128C2×C22⋊D11—24no—
129C2×C22⋊D11C22⋊C418yes—
130C22×Dic11—21no—
131C22×Dic11—21no—
132C22×Dic11C2×D412yes—
133#4339—28no—
134#4339—28no—
135#4339—28no—
136#4339—28no—
137C2×D44—216no—
138C2×D44—24no—
139C2×D44—28no—
140C2×D44—28no—
141C2×D44—22no—
142C2×D44—22no—
143C2×D44—22no—
144C2×D44—22no—
145C2×D44—22no—
146C2×D44—22no—
147C2×D44—22no—
148C2×D44—42no—
149C2×D44—44no—
150C2×D44—44no—
151C2×D44—42no—
152C2×D44—84no—
153C2×D44—84no—
154C2×D44C22⋊C414yes—
155C2×D44C22×C414yes—
156C2×C4×D11—28no—
157C2×C4×D11—22no—
158C2×C4×D11—22no—
159C2×C4×D11—22no—
160C2×C4×D11—22no—
161C2×C4×D11—84no—
162C2×C4×D11C22×C414yes—
163C2×C4×D11C2×D414yes—
164C2×Dic22—22no—
165C2×Dic22—22no—
166C2×Dic22C22⋊C414yes—
167C11×M16—44no—
168C2×C8×C11—22no—
169C2×C8×C11—22no—
170C2×C8×C11—22no—
171C11×C4⋊C4—24no—
172C11×C4⋊C4—22no—
173C11×C4⋊C4—22no—
174C11×C4⋊C4C22×C414yes—
175C11×C4⋊C4C2×D414yes—
176C11×C22⋊C4—24no—
177C42×C11—24no—
178C22⋊Dic11—24no—
179(C2×D11)⋊C4—28no—
180(C2×D11)⋊C4—216no—
181C4⋊Dic11—21no—
182C4⋊Dic11—21no—
183C4⋊Dic11C2×D412yes—
184C4⋊Dic11C2×D412yes—
185Dic11⋊C4—28no—
186C4×Dic11—28no—
187C4×Dic11—24no—
188C2×C11⋊C8—22no—
189C2×C11⋊C8—22no—
190C2×C11⋊C8—22no—
191C2×C11⋊C8—44no—
192C2×#535—111no—
193C2×#520—111no—
194C2×#418—111no—
195C22×#102—111no—
196C2×#408—111no—
197#1875—114no—
198#1874—114no—
199#1736D1111yescharacteristic, complemented
200#1735—111no—
201C23×C11—22no—
202C23×C11—22no—
203C23×C11C2×C22⋊C412yes—
204C23×C11#12012yes—
205C23×C11#12312yes—
206C23×C11C22×D411yescharacteristic
207C22×D11—22no—
208C22×D11—22no—
209C22×D11—24no—
210C22×D11—24no—
211C22×D11—24no—
212C22×D11—42no—
213C22×D11—42no—
214C22×D11—44no—
215C22×D11—44no—
216C22×D11—48no—
217C22×D11—88no—
218C22×D11—88no—
219C22×D11—84no—
220C22×D11—84no—
221C22×D11C2×C22⋊C414yes—
222C2×C4×C11—24no—
223C2×C4×C11—24no—
224C2×C4×C11—44no—
225C2×C4×C11—42no—
226C2×C4×C11—42no—
227C2×C4×C11—84no—
228C2×C4×C11C2×C22⋊C412yes—
229C2×C4×C11C4×D414yes—
230C2×C4×C11C4×D414yes—
231C2×C4×C11#12012yes—
232C2×C4×C11#12114yes—
233C2×C4×C11#12114yes—
234C2×C4×C11#12312yes—
235C2×C4×C11C23×C411yescharacteristic
236C2×C4×C11C22×D411yescharacteristic
237C2×C4×C11C22×D411yescharacteristic
238C2×C4×C11C22×D411yescharacteristic
239C2×C4×C11C2×#4111yescharacteristic
240C2×C4×C11C2×#4111yescharacteristic
241C22⋊D11—216no—
242C2×Dic11—22no—
243C2×Dic11—22no—
244C2×Dic11—22no—
245C2×Dic11—22no—
246C2×Dic11—42no—
247C2×Dic11—42no—
248C2×Dic11C2×C22⋊C414yes—
249D44—24no—
250D44—24no—
251D44—28no—
252D44—42no—
253D44—42no—
254D44—48no—
255D44—816no—
256C4×D11—24no—
257C4×D11—24no—
258C4×D11—24no—
259C4×D11—24no—
260C4×D11—88no—
261Dic22—28no—
262C8×C11—42no—
263C11⋊C8—42no—
264C22×#41—111no—
265C23×D4—111no—
266C22×M16C2×D1111yescharacteristic
267C23×C8—111no—
268#535—118no—
269#535—224no—
270C2×#127—221no—
271C2×#124—221no—
272C2×#121—114no—
273C2×#120—222no—
274#520—224no—
275#520—224no—
276C2×C4×Q8—221no—
277C2×C4×D4—221no—
278C2×C4×D4—222no—
279C2×#117—111no—
280C2×#117—112no—
281C2×#117C2×D1111yescharacteristic
282C22×C4⋊C4—112no—
283C22×C4⋊C4C2×D1111yescharacteristic
284#418—224no—
285C2×#102—112no—
286C2×#102—112no—
287C2×#102—114no—
288C2×#102—224no—
289#408—224no—
290C2×C22⋊C8—114no—
291#340—112no—
292#340—112no—
293#340C2×D1114yescomplemented
294C22×C11—24no—
295C22×C11—24no—
296C22×C11—22no—
297C22×C11—22no—
298C22×C11#38614yes—
299C22×C11C22×C22⋊C411yescharacteristic
300C22×C11C2×#12011yescharacteristic
301C22×C11C2×#12311yescharacteristic
302C22×C11#57612yes—
303C22×C11#57712yes—
304C2×D11—28no—
305C2×D11—24no—
306C2×D11—44no—
307C2×D11—42no—
308C2×D11—42no—
309C2×D11—48no—
310C2×D11—44no—
311C2×D11—816no—
312C2×D11—84no—
313C2×D11—82no—
314C2×D11—82no—
315C4×C11—44no—
316C4×C11—44no—
317C4×C11#38614yes—
318C4×C11C2×C4×D411yescharacteristic
319C4×C11C2×C4×D411yescharacteristic
320C4×C11C2×#12111yescharacteristic
321C4×C11C2×#12111yescharacteristic
322Dic11—24no—
323Dic11—42no—
324Dic11—42no—
325C25—221no—
326C2×#41—114no—
327C2×#41—114no—
328C2×#41—222no—
329C2×#41—222no—
330C2×#41—222no—
331C2×#41—222no—
332C22×Q8—111no—
333C22×D4—112no—
334C22×D4—114no—
335C22×D4—111no—
336C22×D4—112no—
337C22×D4—224no—
338C22×D4—222no—
339C22×D4—222no—
340C22×D4—222no—
341C22×D4—221no—
342C22×D4—222no—
343C22×D4—222no—
344C23×C4—21no—
345C23×C4—112no—
346C23×C4—222no—
347C23×C4C22×D1111yescharacteristic
348C2×M16—22no—
349C2×M16—22no—
350C2×M16—22no—
351C22×C8—112no—
352C22×C8—222no—
353C22×C8—222no—
354C22×C8—222no—
355C22×C8C22×D1112yes—
356#127—228no—
357#124—228no—
358#121—1116no—
359#121—228no—
360#120—2216no—
361C4×Q8—228no—
362C4×D4—228no—
363C4×D4—2216no—
364#117—22no—
365#117—114no—
366#117—224no—
367#117—222no—
368#117—224no—
369#117C4×D1114yescomplemented
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—111no—
373C2×C4⋊C4—111no—
374C2×C4⋊C4—224no—
375C2×C4⋊C4—222no—
376C2×C4⋊C4—224no—
377C2×C4⋊C4D4414yescomplemented
378C2×C4⋊C4C22×D1112yes—
379C2×C4⋊C4C22×D1112yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—224no—
382C2×C22⋊C4—222no—
383C2×C22⋊C4—221no—
384C2×C42—21no—
385C2×C42—222no—
386C2×C42—221no—
387#102—224no—
388#102—224no—
389#102—228no—
390C22⋊C8—228no—
391C2×C11—24no—
392C2×C11C2×#38611yescharacteristic
393C2×C11#289712yes—
394C2×C11#300811yescharacteristic
395C2×C11#301011yescharacteristic
396C2×C11#309612yes—
397D11—44no—
398D11—84no—
399C24—112no—
400C24—222no—
401C24—224no—
402C24—442no—
403C24—442no—
404C24—882no—
405C24—882no—
406C24D4×D1111yescharacteristic
407#41—228no—
408#41—228no—
409#41—228no—
410#41—228no—
411C2×Q8—114no—
412C2×Q8—222no—
413C2×Q8—222no—
414C2×D4—118no—
415C2×D4—114no—
416C2×D4—114no—
417C2×D4—2216no—
418C2×D4—224no—
419C2×D4—224no—
420C2×D4—228no—
421C2×D4—224no—
422C2×D4—222no—
423C2×D4—228no—
424C2×D4—222no—
425C2×D4—222no—
426C2×D4—222no—
427C2×D4—222no—
428C2×D4—222no—
429C2×D4—222no—
430C2×D4—442no—
431C2×D4—444no—
432C2×D4—442no—
433C2×D4—444no—
434C2×D4—884no—
435C2×D4—884no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—82no—
440C22×C4—112no—
441C22×C4—114no—
442C22×C4—114no—
443C22×C4—228no—
444C22×C4—222no—
445C22×C4—222no—
446C22×C4—222no—
447C22×C4—222no—
448C22×C4—221no—
449C22×C4—221no—
450C22×C4—884no—
451C22×C4C2×C4×D1112yes—
452C22×C4C2×D4412yes—
453C22×C4#433912yes—
454C22×C4D4×D1111yescharacteristic
455C22×C4D4×D1112yes—
456C22×C4D4×D1112yes—
457C22×C4#434312yes—
458C22×C4C23×D1111yescharacteristic
459M16—44no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—222no—
464C2×C8—222no—
465C2×C8—222no—
466C2×C8—444no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—112no—
471C4⋊C4—112no—
472C4⋊C4—228no—
473C4⋊C4—221no—
474C4⋊C4—221no—
475C4⋊C4C2×C4×D1114yes—
476C4⋊C4C2×D4414yes—
477C22⋊C4—24no—
478C22⋊C4—2216no—
479C22⋊C4—228no—
480C22⋊C4—224no—
481C42—24no—
482C42—228no—
483C42—224no—
484C11#2091411yescharacteristic, complemented, Sylow
485C23—22no—
486C23—22no—
487C23—114no—
488C23—222no—
489C23—222no—
490C23—224no—
491C23—224no—
492C23—224no—
493C23—442no—
494C23—442no—
495C23—448no—
496C23—444no—
497C23—444no—
498C23—888no—
499C23—888no—
500C23—884no—
501C23—884no—
502C23C22⋊C4×D1112yes—
503C23#6693512yes—
504C23#6694712yes—
505C23C2×D4×D1111yescharacteristic
506Q8—228no—
507D4—2216no—
508D4—224no—
509D4—224no—
510D4—228no—
511D4—442no—
512D4—442no—
513D4—448no—
514D4—8816no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—114no—
522C2×C4—222no—
523C2×C4—222no—
524C2×C4—224no—
525C2×C4—224no—
526C2×C4—224no—
527C2×C4—224no—
528C2×C4—222no—
529C2×C4—222no—
530C2×C4—442no—
531C2×C4—442no—
532C2×C4—888no—
533C2×C4C4×D4414yes—
534C2×C4C22⋊C4×D1112yes—
535C2×C4#6693512yes—
536C2×C4#6693714yes—
537C2×C4#6694614yes—
538C2×C4#6694712yes—
539C2×C4#6694814yes—
540C2×C4C22×C4×D1111yescharacteristic
541C2×C4C22×D4411yescharacteristic
542C2×C4C2×#433911yescharacteristic
543C2×C4C2×D4×D1111yescharacteristic
544C2×C4C2×D4×D1111yescharacteristic
545C2×C4C2×#434311yescharacteristic
546C8—42no—
547C8—442no—
548C22—24no—
549C22—22no—
550C22—22no—
551C22—24no—
552C22—228no—
553C22—224no—
554C22—444no—
555C22—442no—
556C22—442no—
557C22—448no—
558C22—444no—
559C22—8816no—
560C22—884no—
561C22—882no—
562C22—882no—
563C22#1062227814yes—
564C22C2×C22⋊C4×D1111yescharacteristic
565C22C2×#6693511yescharacteristic
566C22C2×#6694711yescharacteristic
567C22#1062330512yes—
568C22#1062330912yes—
569C4—44no—
570C4—44no—
571C4—224no—
572C4—442no—
573C4—442no—
574C4#1062227814yes—
575C4C2×C4×D4411yescharacteristic
576C4C2×#6693711yescharacteristic
577C4C2×#6694611yescharacteristic
578C4C2×#6694811yescharacteristic
579C2—24no—
580C2—444no—
581C2—884no—
582C2C2×#1062227811yescharacteristic
583C2#1307714012yes—
584C2#1307788511yescharacteristic
585C2#1307790311yescharacteristic
586C2#1307857612yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1736C2
socleC23×C11C22×D4
prime radicalC2×#10623056C2
cosocleC22×C4×C11C24
hypercenter#1736D11
hypocenterC11#20914
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×#10623035C2semidirect
3. C2×#10622744C2semidirect
4. C22×#66873C2semidirect
5. C2×#10622742C2semidirect
6. #13068937C2semidirect
7. #13068932C2semidirect
8. #13068332C2semidirect
9. C11×#1736C2semidirect
19. #10623056C4semidirect
24. C2×#66945C22semidirect
25. C2×C4⋊C4×D11C22semidirect
34. C2×(C2×D11)⋊C8C22semidirect
37. C11×#340C22semidirect
38. #10622167C22semidirect
39. #10622167C22semidirect
57. C22×D44C2×C4semidirect
68. C11×#117C2×C4semidirect
71. C2×C11×C4⋊C4D4semidirect
81. #66979D4semidirect
92. #66948C2×C4semidirect
199. #1736D11semidirect
293. #340C2×D11semidirect
369. #117C4×D11semidirect
377. C2×C4⋊C4D44semidirect
484. C11#20914semidirect

Maximal subgroups

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

subgroupindex
C2×#106230562
C2×#106230352
C2×#106227442
C22×#668732
C2×#106227422
#130689372
#130689322
#130683322
C11×#17362
#2091411