Finite Group Databasebeta

← #14755195

Subgroups — #14755195

Order 2816.

1926 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66961C211yesmaximal, characteristic
2C2×#10623054C211yesmaximal, characteristic, complemented
3C2×#10622726C211yesmaximal, characteristic, complemented
4C22×#66885C211yesmaximal, characteristic, complemented
5#13071196C211yesmaximal, characteristic, complemented
6#13068970C214yesmaximal, complemented
7#13068963C214yesmaximal, complemented
8#13068336C211yesmaximal, characteristic, complemented
9C11×#1729C211yesmaximal, characteristic, complemented
10C22×#4339C2211yescharacteristic
11C23×D44C2211yescharacteristic
12C22×C11×M16C2211yescharacteristic
13C23×C8×C11C2211yescharacteristic
14C2×C11×#117C2211yescharacteristic
15C2×#66979C2211yescharacteristic
16C22×C4⋊Dic11C2211yescharacteristic
17C2×#66961—24no—
18C2×#66961—24no—
19C2×#66961C418yescomplemented
20C22×D88C2212yes—
21C22×#4314C2212yes—
22#10623054—24no—
23#10623054—24no—
24C2×#66945—21no—
25C2×C4⋊C4×D11—21no—
26C2×(C22⋊D11)⋊C4—22no—
27C2×C4×D44C2212yescomplemented
28C2×C4×Dic22C2212yescomplemented
29#10622726—24no—
30C2×#66885—24no—
31C2×#66885C2214yes—
32C2×#66885C2212yescomplemented
33C2×#66880C2212yescomplemented
34#10622601—21no—
35#10622601—21no—
36#10622568—22no—
37#10622552C2214yes—
38C11×#336C2214yescomplemented
39#10622189C2214yescomplemented
40C24×D11—21no—
41C23×C4×C11C2311yescharacteristic
42C22×C22⋊D11—21no—
43C23×Dic11—21no—
44C2×#4339—22no—
45C2×#4339—22no—
46C2×#4339—22no—
47C2×#4339C2×C414yes—
48C2×#4339D414yes—
49C2×#4339D414yes—
50C22×D44—22no—
51C22×D44—22no—
52C22×D44—22no—
53C22×D44—22no—
54C22×D44—21no—
55C22×D44—22no—
56C22×D44C2×C414yescomplemented
57C22×D44C2312yes—
58C22×D44C2311yescharacteristic
59C22×C4×D11—21no—
60C22×C4×D11—22no—
61C22×Dic22C2311yescharacteristic
62C2×C11×M16—22no—
63C2×C11×M16—22no—
64C2×C11×M16C2×C414yescomplemented
65C22×C8×C11—22no—
66C22×C8×C11—22no—
67C22×C8×C11—22no—
68C22×C8×C11C2312yes—
69C22×C8×C11C2312yes—
70C11×#117—22no—
71C11×#117—22no—
72C2×C11×C4⋊C4D411yescharacteristic
73C2×C11×C4⋊C4D411yescharacteristic
74C2×C11×C22⋊C4D412yes—
75C2×C42×C11C2312yes—
76C2×C22⋊Dic11—21no—
77C2×(C2×D11)⋊C4—22no—
78C2×(C2×D11)⋊C4—24no—
79C2×(C2×D11)⋊C4—24no—
80#66979—22no—
81#66979—22no—
82C2×C4⋊Dic11—21no—
83C2×C4⋊Dic11—22no—
84C2×C4⋊Dic11D414yes—
85C2×C4⋊Dic11C2312yes—
86C2×C4⋊Dic11C2312yes—
87C2×Dic11⋊C4—22no—
88C2×C4×Dic11—22no—
89C2×C4×Dic11—21no—
90#66961—232no—
91C2×D88—24no—
92C2×D88—24no—
93C2×D88C2×C418yescomplemented
94C2×#4314—24no—
95C2×#4314—24no—
96C2×#4314C2×C418yescomplemented
97#66945—28no—
98C4⋊C4×D11—28no—
99(C22⋊D11)⋊C4—216no—
100C4×D44—24no—
101C4×D44—24no—
102C4×Dic22—24no—
103C4×Dic22—24no—
104#66885—24no—
105#66885—28no—
106#66880—24no—
107#20932—111nomaximal, Sylow
108C24×C11C2×D411yescharacteristic
109C23×D11—21no—
110C23×D11—22no—
111C23×D11—24no—
112C23×D11—42no—
113C23×D11—42no—
114C23×D11—82no—
115C23×D11—82no—
116C22×C4×C11C22×C412yes—
117C22×C4×C11C2×D412yes—
118C22×C4×C11C2×D412yes—
119C22×C4×C11C2×D412yes—
120C22×C4×C11C2×D412yes—
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2×D412yes—
123C22×C4×C11#4112yes—
124C22×C4×C11#4112yes—
125C22×C4×C11C2411yescharacteristic
126C2×C22⋊D11—24no—
127C2×C22⋊D11—24no—
128C2×C22⋊D11—24no—
129C22×Dic11—24no—
130C22×Dic11—21no—
131C22×Dic11—22no—
132C22×Dic11—22no—
133C22×Dic11—82no—
134#4339—28no—
135#4339—28no—
136#4339—28no—
137#4339C22⋊C4116yescomplemented
138C2×D44—28no—
139C2×D44—24no—
140C2×D44—28no—
141C2×D44—21no—
142C2×D44—22no—
143C2×D44—22no—
144C2×D44—22no—
145C2×D44—21no—
146C2×D44—21no—
147C2×D44—22no—
148C2×D44—22no—
149C2×D44—42no—
150C2×D44—44no—
151C2×D44—42no—
152C2×D44—44no—
153C2×D44C22×C414yes—
154C2×D44C22×C412yes—
155C2×D44C2×D412yes—
156C2×D44C2×D412yes—
157C2×C4×D11—28no—
158C2×C4×D11—24no—
159C2×C4×D11—24no—
160C2×C4×D11—24no—
161C2×C4×D11—84no—
162C2×Dic22—21no—
163C2×Dic22—21no—
164C2×Dic22—21no—
165C2×Dic22C22×C412yes—
166C2×Dic22C2×D412yes—
167C2×Dic22C2×D412yes—
168C11×M16—28no—
169C2×C8×C11—22no—
170C2×C8×C11—22no—
171C2×C8×C11—22no—
172C2×C8×C11—22no—
173C2×C8×C11—22no—
174C2×C8×C11—44no—
175C2×C8×C11C22×C414yes—
176C11×C4⋊C4—22no—
177C11×C4⋊C4—22no—
178C11×C22⋊C4—24no—
179C42×C11—22no—
180C42×C11—22no—
181C22⋊Dic11—42no—
182(C2×D11)⋊C4—28no—
183(C2×D11)⋊C4—84no—
184(C2×D11)⋊C4—84no—
185C4⋊Dic11—24no—
186C4⋊Dic11—22no—
187C4⋊Dic11—22no—
188C4⋊Dic11—22no—
189C4⋊Dic11C2×D414yes—
190Dic11⋊C4—28no—
191C4×Dic11—28no—
192C4×Dic11—42no—
193D88—216no—
194#4314—216no—
195C22×AGL(1,ℤ/8)—111no—
196C2×#520—111no—
197C2×#416—111no—
198C22×#102—111no—
199#2309—111no—
200#1880—114no—
201#1879—114no—
202#1736—111no—
203#1729D1111yescharacteristic, complemented
204C23×C11—22no—
205C23×C11—22no—
206C23×C11C4×D412yes—
207C23×C11#12012yes—
208C23×C11#12112yes—
209C23×C11C22×D411yescharacteristic
210C22×D11—22no—
211C22×D11—22no—
212C22×D11—22no—
213C22×D11—24no—
214C22×D11—42no—
215C22×D11—42no—
216C22×D11—42no—
217C22×D11—42no—
218C22×D11—44no—
219C22×D11—44no—
220C22×D11—48no—
221C22×D11—88no—
222C22×D11—84no—
223C22×D11—84no—
224C22×D11—88no—
225C2×C4×C11—22no—
226C2×C4×C11—22no—
227C2×C4×C11—24no—
228C2×C4×C11—24no—
229C2×C4×C11C2×C22⋊C414yes—
230C2×C4×C11C4×D412yes—
231C2×C4×C11C4×D414yes—
232C2×C4×C11#12012yes—
233C2×C4×C11#12114yes—
234C2×C4×C11#12112yes—
235C2×C4×C11#12314yes—
236C2×C4×C11C23×C411yescharacteristic
237C2×C4×C11C22×D411yescharacteristic
238C2×C4×C11C22×D411yescharacteristic
239C2×C4×C11C22×D411yescharacteristic
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—44no—
248C2×Dic11—42no—
249C2×Dic11—42no—
250C2×Dic11—42no—
251C2×Dic11—84no—
252D44—24no—
253D44—22no—
254D44—24no—
255D44—22no—
256D44—22no—
257D44—42no—
258D44—42no—
259D44—48no—
260D44C2×C22⋊C414yes—
261C4×D11—216no—
262C4×D11—88no—
263Dic22—22no—
264Dic22—22no—
265Dic22—22no—
266Dic22C2×C22⋊C414yes—
267C8×C11—24no—
268C8×C11—42no—
269C22×#41—111no—
270C23×D4—111no—
271C2×AGL(1,ℤ/8)—118no—
272C2×AGL(1,ℤ/8)—224no—
273C2×AGL(1,ℤ/8)—224no—
274C22×SD16—112no—
275C22×D8—112no—
276C22×M16C2×D1111yescharacteristic
277C23×C8C2×D1111yescharacteristic
278#520—224no—
279#520—224no—
280C2×C4×Q8—112no—
281C2×C4×D4—112no—
282C2×C4×D4—222no—
283C2×#117—111no—
284C2×#117—221no—
285C2×#117C2×D1111yescharacteristic
286C22×C4⋊C4—111no—
287C22×C4⋊C4—221no—
288#416—224no—
289C2×#103—112no—
290C2×#102—114no—
291C2×#102—112no—
292C2×#102—224no—
293#390—114no—
294#390—221no—
295#390—221no—
296#379—222no—
297#340—114no—
298#336C2×D1114yescomplemented
299C22×C11—24no—
300C22×C11—22no—
301C22×C11—22no—
302C22×C11—24no—
303C22×C11#38614yes—
304C22×C11C2×C4×D411yescharacteristic
305C22×C11C2×#12011yescharacteristic
306C22×C11C2×#12111yescharacteristic
307C22×C11#57612yes—
308C22×C11#57712yes—
309C2×D11—24no—
310C2×D11—44no—
311C2×D11—42no—
312C2×D11—42no—
313C2×D11—44no—
314C2×D11—48no—
315C2×D11—44no—
316C2×D11—816no—
317C2×D11—82no—
318C2×D11—82no—
319C2×D11—84no—
320C4×C11—42no—
321C4×C11—42no—
322C4×C11#38614yes—
323C4×C11C22×C22⋊C411yescharacteristic
324C4×C11C2×C4×D411yescharacteristic
325C4×C11C2×#12111yescharacteristic
326C4×C11C2×#12311yescharacteristic
327Dic11—24no—
328Dic11—44no—
329Dic11—82no—
330C25—221no—
331C2×#41—114no—
332C2×#41—114no—
333C2×#41—114no—
334C2×#41—222no—
335C2×#41—222no—
336C2×#41—222no—
337C22×Q8—111no—
338C22×D4—114no—
339C22×D4—111no—
340C22×D4—112no—
341C22×D4—221no—
342C22×D4—221no—
343C22×D4—222no—
344C22×D4—222no—
345C22×D4—222no—
346C22×D4—222no—
347C22×D4—222no—
348C23×C4—221no—
349C23×C4—221no—
350C23×C4—222no—
351C23×C4C22×D1111yescharacteristic
352AGL(1,ℤ/8)—2232no—
353C2×SD16—118no—
354C2×SD16—224no—
355C2×SD16—224no—
356C2×D8—118no—
357C2×D8—224no—
358C2×D8—224no—
359C2×M16—22no—
360C2×M16—22no—
361C2×M16C4×D1114yescomplemented
362C22×C8—22no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8C22×D1112yes—
366C22×C8C22×D1112yes—
367C4×Q8—224no—
368C4×Q8—224no—
369C4×D4—224no—
370C4×D4—224no—
371C4×D4—2216no—
372#117—22no—
373#117—22no—
374#117—228no—
375#117—222no—
376#117—222no—
377C2×C4⋊C4—114no—
378C2×C4⋊C4—112no—
379C2×C4⋊C4—112no—
380C2×C4⋊C4—228no—
381C2×C4⋊C4—221no—
382C2×C4⋊C4—222no—
383C2×C4⋊C4—222no—
384C2×C4⋊C4D4411yescharacteristic
385C2×C4⋊C4D4411yescharacteristic
386C2×C22⋊C4—222no—
387C2×C22⋊C4—221no—
388C2×C22⋊C4—224no—
389C2×C22⋊C4—224no—
390C2×C22⋊C4D4412yes—
391C2×C42—222no—
392C2×C42—221no—
393C2×C42C22×D1112yes—
394#103—224no—
395#102—228no—
396#102—224no—
397C2×C11—24no—
398C2×C11C2×#38611yescharacteristic
399C2×C11#294612yes—
400C2×C11#300811yescharacteristic
401C2×C11#301011yescharacteristic
402C2×C11#304512yes—
403D11—44no—
404D11—84no—
405C24—221no—
406C24—222no—
407C24—224no—
408C24—442no—
409C24—442no—
410C24—882no—
411C24—882no—
412C24C2×D4411yescharacteristic
413#41—1116no—
414#41—228no—
415#41—228no—
416#41—228no—
417C2×Q8—112no—
418C2×Q8—112no—
419C2×Q8—112no—
420C2×Q8—221no—
421C2×Q8—221no—
422C2×Q8—221no—
423C2×D4—112no—
424C2×D4—114no—
425C2×D4—112no—
426C2×D4—112no—
427C2×D4—224no—
428C2×D4—224no—
429C2×D4—224no—
430C2×D4—224no—
431C2×D4—228no—
432C2×D4—228no—
433C2×D4—221no—
434C2×D4—222no—
435C2×D4—221no—
436C2×D4—222no—
437C2×D4—222no—
438C2×D4—222no—
439C2×D4—221no—
440C2×D4—222no—
441C2×D4—442no—
442C2×D4—444no—
443C2×D4—444no—
444C2×D4—442no—
445C22×C4—228no—
446C22×C4—221no—
447C22×C4—224no—
448C22×C4—224no—
449C22×C4—224no—
450C22×C4—222no—
451C22×C4—224no—
452C22×C4—222no—
453C22×C4—884no—
454C22×C4—882no—
455C22×C4C2×C4×D1112yes—
456C22×C4C2×D4412yes—
457C22×C4C2×D4412yes—
458C22×C4C2×D4411yescharacteristic
459C22×C4#433912yes—
460C22×C4D4×D1112yes—
461C22×C4D4×D1112yes—
462C22×C4D4×D1112yes—
463C22×C4#434312yes—
464C22×C4C23×D1111yescharacteristic
465SD16—2216no—
466D8—2216no—
467M16—28no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—44no—
474C2×C8C2×C4×D1114yes—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—114no—
478C4⋊C4—228no—
479C4⋊C4—224no—
480C4⋊C4—222no—
481C4⋊C4—222no—
482C4⋊C4—222no—
483C22⋊C4—24no—
484C22⋊C4—228no—
485C22⋊C4—442no—
486C22⋊C4—884no—
487C22⋊C4—884no—
488C42—22no—
489C42—22no—
490C42—228no—
491C42—442no—
492C11#2093211yescharacteristic, complemented, Sylow
493C23—22no—
494C23—22no—
495C23—222no—
496C23—222no—
497C23—222no—
498C23—224no—
499C23—442no—
500C23—442no—
501C23—442no—
502C23—444no—
503C23—444no—
504C23—442no—
505C23—448no—
506C23—888no—
507C23—884no—
508C23—888no—
509C23—884no—
510C23C4×D4412yes—
511C23#6693512yes—
512C23#6694812yes—
513C23C22×D4411yescharacteristic
514Q8—114no—
515Q8—222no—
516Q8—222no—
517Q8—222no—
518D4—114no—
519D4—2216no—
520D4—224no—
521D4—222no—
522D4—222no—
523D4—224no—
524D4—222no—
525D4—442no—
526D4—442no—
527D4—448no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—222no—
533C2×C4—222no—
534C2×C4—222no—
535C2×C4—2216no—
536C2×C4—224no—
537C2×C4—444no—
538C2×C4—442no—
539C2×C4—442no—
540C2×C4—442no—
541C2×C4—888no—
542C2×C4—884no—
543C2×C4C4×D4412yes—
544C2×C4C22⋊C4×D1114yes—
545C2×C4#6693512yes—
546C2×C4#6693714yes—
547C2×C4#6694614yes—
548C2×C4#6694714yes—
549C2×C4#6694812yes—
550C2×C4C22×C4×D1111yescharacteristic
551C2×C4C2×#433911yescharacteristic
552C2×C4C2×D4×D1111yescharacteristic
553C2×C4C2×D4×D1111yescharacteristic
554C2×C4C2×D4×D1111yescharacteristic
555C2×C4C2×#434311yescharacteristic
556C8—24no—
557C8—42no—
558C22—22no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—224no—
563C22—442no—
564C22—444no—
565C22—442no—
566C22—444no—
567C22—444no—
568C22—448no—
569C22—8816no—
570C22—882no—
571C22—884no—
572C22—882no—
573C22#1062227814yes—
574C22C2×C4×D4411yescharacteristic
575C22C2×#6693511yescharacteristic
576C22C2×#6694811yescharacteristic
577C22#1062328212yes—
578C22#1062328312yes—
579C4—42no—
580C4—42no—
581C4—224no—
582C4—444no—
583C4—882no—
584C4#1062227814yes—
585C4C2×C22⋊C4×D1111yescharacteristic
586C4C2×#6693711yescharacteristic
587C4C2×#6694611yescharacteristic
588C4C2×#6694711yescharacteristic
589C2—24no—
590C2—444no—
591C2—884no—
592C2C2×#1062227811yescharacteristic
593C2#1307745012yes—
594C2#1307788611yescharacteristic
595C2#1307790411yescharacteristic
596C2#1307815712yes—
subgroupG/H
centerC23C22×D44
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1729C2
socleC23×C11C22×D4
prime radicalC22×#66961C2
cosocleC22×C4×C11C24
hypercenter#1729D11
hypocenterC11#20932
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1729

Complements

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

NH ≅ G/Ntype
2. C2×#10623054C2semidirect
3. C2×#10622726C2semidirect
4. C22×#66885C2semidirect
5. #13071196C2semidirect
6. #13068970C2semidirect
7. #13068963C2semidirect
8. #13068336C2semidirect
9. C11×#1729C2semidirect
19. C2×#66961C4semidirect
27. C2×C4×D44C22semidirect
28. C2×C4×Dic22C22semidirect
32. C2×#66885C22semidirect
33. C2×#66880C22semidirect
38. C11×#336C22semidirect
39. #10622189C22semidirect
56. C22×D44C2×C4semidirect
64. C2×C11×M16C2×C4semidirect
93. C2×D88C2×C4semidirect
96. C2×#4314C2×C4semidirect
137. #4339C22⋊C4semidirect
203. #1729D11semidirect
298. #336C2×D11semidirect
361. C2×M16C4×D11semidirect
492. C11#20932semidirect

Maximal subgroups

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

subgroupindex
C22×#669612
C2×#106230542
C2×#106227262
C22×#668852
#130711962
#130689702
#130689632
#130683362
C11×#17292
#2093211