Finite Group Databasebeta

← #14755161

Subgroups — #14755161

Order 2816.

2046 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66961C211yesmaximal, characteristic
2C2×#10623053C211yesmaximal, characteristic, complemented
3C2×#10622579C211yesmaximal, characteristic, complemented
4C2×#10622614C211yesmaximal, characteristic, complemented
5C2×#10622601C211yesmaximal, characteristic, complemented
6#13068972C214yesmaximal, complemented
7#13068965C214yesmaximal, complemented
8#13068309C211yesmaximal, characteristic, complemented
9C11×#1736C211yesmaximal, characteristic, complemented
10C22×#4339C2211yescharacteristic
11C23×D44C2211yescharacteristic
12C22×C11×M16C2211yescharacteristic
13C2×C11×#117C2211yescharacteristic
14C22×C11×C4⋊C4C2211yescharacteristic
15C22×(C2×D11)⋊C4—22no—
16C22×C4×Dic11C2211yescharacteristic
17C22×#4318C2211yescharacteristic
18C2×#66961—24no—
19C2×#66961—24no—
20C2×#66961C418yescomplemented
21C22×D88C2212yes—
22C22×#4314C2212yes—
23#10623053—24no—
24#10623053—24no—
25C2×#66946C2212yescomplemented
26C2×#66940C2212yescomplemented
27C2×(C22⋊D11)⋊C4—22no—
28C2×#66925—21no—
29C2×C42×D11—21no—
30#10622614—24no—
31#10622601—21no—
32#10622601—21no—
33#10622601C2214yes—
34C2×#66874C2212yescomplemented
35#10622579—24no—
36C2×#66873C2214yes—
37C2×#66873C2212yescomplemented
38C11×#340C2214yescomplemented
39#10622188C2214yescomplemented
40C24×D11—21no—
41C23×C4×C11—21no—
42C23×C4×C11C2311yescharacteristic
43C22×C22⋊D11—21no—
44C23×Dic11D412yes—
45C2×#4339—22no—
46C2×#4339—22no—
47C2×#4339—22no—
48C2×#4339C2×C414yes—
49C2×#4339D414yes—
50C2×#4339D414yes—
51C22×D44—22no—
52C22×D44—22no—
53C22×D44—22no—
54C22×D44—22no—
55C22×D44—21no—
56C22×D44—22no—
57C22×D44C2×C414yescomplemented
58C22×D44C2312yes—
59C22×D44C2311yescharacteristic
60C22×C4×D11—21no—
61C22×C4×D11—22no—
62C22×Dic22C2311yescharacteristic
63C2×C11×M16—22no—
64C2×C11×M16—22no—
65C2×C11×M16C2×C414yescomplemented
66C22×C8×C11C2312yes—
67C11×#117—22no—
68C11×#117D414yes—
69C2×C11×C4⋊C4—22no—
70C2×C11×C4⋊C4—22no—
71C2×C11×C4⋊C4—21no—
72C2×C11×C4⋊C4C2312yes—
73C2×C11×C4⋊C4C2312yes—
74C2×C11×C22⋊C4—21no—
75C2×C42×C11—21no—
76C2×(C2×D11)⋊C4—22no—
77C2×(C2×D11)⋊C4—24no—
78C2×(C2×D11)⋊C4—24no—
79C2×(C2×D11)⋊C4—48no—
80C2×Dic11⋊C4—22no—
81C2×C4×Dic11—22no—
82C2×C4×Dic11—22no—
83C2×C4×Dic11—22no—
84C2×C4×Dic11D411yescharacteristic
85C2×C4×Dic11D411yescharacteristic
86C2×C4×Dic11C2312yes—
87C2×#4318—22no—
88C2×#4318—22no—
89C2×#4318—22no—
90C22×C11⋊C8C2312yes—
91#66961—232no—
92C2×D88—24no—
93C2×D88—24no—
94C2×D88C2×C418yescomplemented
95C2×#4314—24no—
96C2×#4314—24no—
97C2×#4314C2×C418yescomplemented
98#66946—24no—
99#66946—24no—
100#66940—24no—
101#66940—24no—
102(C22⋊D11)⋊C4—216no—
103#66925—28no—
104C42×D11—28no—
105#66874—24no—
106#66873—24no—
107#66873—28no—
108#20928—111nomaximal, Sylow
109C24×C11C2×D411yescharacteristic
110C23×D11—21no—
111C23×D11—24no—
112C23×D11—22no—
113C23×D11—44no—
114C23×D11—44no—
115C23×D11—42no—
116C23×D11—42no—
117C22×C4×C11—22no—
118C22×C4×C11—22no—
119C22×C4×C11—44no—
120C22×C4×C11—42no—
121C22×C4×C11C22×C412yes—
122C22×C4×C11C2×D412yes—
123C22×C4×C11C2×D412yes—
124C22×C4×C11C2×D411yescharacteristic
125C22×C4×C11C2×D412yes—
126C22×C4×C11#4112yes—
127C22×C4×C11#4112yes—
128C22×C4×C11C2411yescharacteristic
129C2×C22⋊D11—24no—
130C2×C22⋊D11—24no—
131C2×C22⋊D11—24no—
132C22×Dic11—24no—
133C22×Dic11—24no—
134C22×Dic11—21no—
135C22×Dic11—42no—
136C22×Dic11C2×D412yes—
137C22×Dic11C2×D412yes—
138#4339—28no—
139#4339—28no—
140#4339—28no—
141#4339C22⋊C4116yescomplemented
142C2×D44—28no—
143C2×D44—24no—
144C2×D44—28no—
145C2×D44—21no—
146C2×D44—22no—
147C2×D44—22no—
148C2×D44—22no—
149C2×D44—21no—
150C2×D44—21no—
151C2×D44—22no—
152C2×D44—22no—
153C2×D44—42no—
154C2×D44—42no—
155C2×D44—44no—
156C2×D44—44no—
157C2×D44C22×C414yes—
158C2×D44C22×C412yes—
159C2×D44C2×D412yes—
160C2×D44C2×D412yes—
161C2×C4×D11—28no—
162C2×C4×D11—24no—
163C2×C4×D11—24no—
164C2×C4×D11—24no—
165C2×C4×D11—44no—
166C2×C4×D11—44no—
167C2×Dic22—21no—
168C2×Dic22—21no—
169C2×Dic22—21no—
170C2×Dic22C22×C412yes—
171C2×Dic22C2×D412yes—
172C2×Dic22C2×D412yes—
173C11×M16—28no—
174C2×C8×C11—22no—
175C2×C8×C11—22no—
176C2×C8×C11C22×C414yes—
177C11×C4⋊C4—22no—
178C11×C4⋊C4—22no—
179C11×C4⋊C4—22no—
180C11×C4⋊C4—42no—
181C11×C4⋊C4C2×D414yes—
182C11×C22⋊C4—24no—
183C42×C11—24no—
184(C2×D11)⋊C4—28no—
185(C2×D11)⋊C4—48no—
186(C2×D11)⋊C4—48no—
187Dic11⋊C4—28no—
188C4×Dic11—28no—
189C4×Dic11—22no—
190C4×Dic11—22no—
191C4×Dic11—22no—
192C4×Dic11—22no—
193#4318—44no—
194C2×C11⋊C8—22no—
195C2×C11⋊C8—22no—
196C2×C11⋊C8—22no—
197D88—216no—
198#4314—216no—
199C22×AGL(1,ℤ/8)—111no—
200C2×C4×#41—111no—
201C2×#417—111no—
202C2×#418—111no—
203C2×#390—111no—
204#1880—114no—
205#1879—114no—
206#1736D1111yescharacteristic, complemented
207#1728—111no—
208C23×C11—22no—
209C23×C11—22no—
210C23×C11C4×D412yes—
211C23×C11#12012yes—
212C23×C11#12112yes—
213C23×C11C22×D411yescharacteristic
214C22×D11—22no—
215C22×D11—22no—
216C22×D11—22no—
217C22×D11—24no—
218C22×D11—42no—
219C22×D11—42no—
220C22×D11—416no—
221C22×D11—42no—
222C22×D11—42no—
223C22×D11—44no—
224C22×D11—44no—
225C22×D11—48no—
226C22×D11—88no—
227C22×D11—84no—
228C22×D11—84no—
229C2×C4×C11—24no—
230C2×C4×C11—42no—
231C2×C4×C11—42no—
232C2×C4×C11—42no—
233C2×C4×C11—44no—
234C2×C4×C11—84no—
235C2×C4×C11C2×C22⋊C414yes—
236C2×C4×C11C4×D414yes—
237C2×C4×C11C4×D412yes—
238C2×C4×C11#12012yes—
239C2×C4×C11#12112yes—
240C2×C4×C11#12114yes—
241C2×C4×C11#12314yes—
242C2×C4×C11C23×C411yescharacteristic
243C2×C4×C11C22×D411yescharacteristic
244C2×C4×C11C22×D411yescharacteristic
245C2×C4×C11C22×D411yescharacteristic
246C2×C4×C11C2×#4111yescharacteristic
247C2×C4×C11C2×#4111yescharacteristic
248C22⋊D11—216no—
249C2×Dic11—22no—
250C2×Dic11—22no—
251C2×Dic11—22no—
252C2×Dic11—22no—
253C2×Dic11—22no—
254C2×Dic11—22no—
255C2×Dic11—22no—
256C2×Dic11—22no—
257C2×Dic11—48no—
258C2×Dic11AGL(1,ℤ/8)14yescomplemented
259D44—24no—
260D44—22no—
261D44—24no—
262D44—22no—
263D44—22no—
264D44—42no—
265D44—42no—
266D44—48no—
267D44C2×C22⋊C414yescomplemented
268C4×D11—216no—
269C4×D11—416no—
270Dic22—22no—
271Dic22—22no—
272Dic22—22no—
273Dic22C2×C22⋊C414yescomplemented
274C8×C11—24no—
275C11⋊C8—42no—
276C22×#41—111no—
277C23×D4—111no—
278C2×AGL(1,ℤ/8)—118no—
279C2×AGL(1,ℤ/8)—224no—
280C2×AGL(1,ℤ/8)—224no—
281C22×SD16—112no—
282C22×D8—112no—
283C22×M16—111no—
284C22×M16C2×D1111yescharacteristic
285C4×#41—224no—
286C4×#41—224no—
287C2×C4×Q8—112no—
288C2×C4×D4—112no—
289C2×C4×D4—222no—
290C2×#117—221no—
291C2×#117C2×D1111yescharacteristic
292C22×C4⋊C4C2×D1111yescharacteristic
293C22×C22⋊C4—222no—
294C22×C42—111no—
295C22×C42—221no—
296#418—224no—
297#417—224no—
298C2×#103—112no—
299C2×#102—114no—
300C2×#102—112no—
301#390—114no—
302#390—221no—
303#390—221no—
304#340C2×D1114yescomplemented
305#336—114no—
306C22×C11—24no—
307C22×C11—24no—
308C22×C11—22no—
309C22×C11—22no—
310C22×C11#38614yes—
311C22×C11C2×C4×D411yescharacteristic
312C22×C11C2×#12011yescharacteristic
313C22×C11C2×#12111yescharacteristic
314C22×C11C2×AGL(1,ℤ/8)12yes—
315C22×C11#57512yes—
316C2×D11—24no—
317C2×D11—44no—
318C2×D11—44no—
319C2×D11—42no—
320C2×D11—42no—
321C2×D11—48no—
322C2×D11—44no—
323C2×D11—816no—
324C2×D11—84no—
325C2×D11—82no—
326C2×D11—82no—
327C4×C11—44no—
328C4×C11—82no—
329C4×C11#38614yes—
330C4×C11C22×C22⋊C411yescharacteristic
331C4×C11C2×C4×D411yescharacteristic
332C4×C11C2×#12111yescharacteristic
333C4×C11C2×#12311yescharacteristic
334Dic11—24no—
335Dic11—24no—
336Dic11—42no—
337C25—221no—
338C2×#41—114no—
339C2×#41—114no—
340C2×#41—114no—
341C2×#41—222no—
342C2×#41—222no—
343C2×#41—222no—
344C22×Q8—111no—
345C22×D4—114no—
346C22×D4—111no—
347C22×D4—112no—
348C22×D4—221no—
349C22×D4—222no—
350C22×D4—222no—
351C22×D4—222no—
352C22×D4—222no—
353C22×D4—221no—
354C22×D4—222no—
355C23×C4—21no—
356C23×C4—112no—
357C23×C4—222no—
358C23×C4—221no—
359C23×C4C22×D1111yescharacteristic
360AGL(1,ℤ/8)—2232no—
361C2×SD16—118no—
362C2×SD16—224no—
363C2×SD16—224no—
364C2×D8—118no—
365C2×D8—224no—
366C2×D8—224no—
367C2×M16—22no—
368C2×M16—22no—
369C2×M16—222no—
370C2×M16—222no—
371C2×M16—222no—
372C2×M16C4×D1114yescomplemented
373C22×C8—112no—
374C22×C8C22×D1112yes—
375C4×Q8—224no—
376C4×Q8—224no—
377C4×D4—2216no—
378C4×D4—224no—
379C4×D4—224no—
380#117—22no—
381#117—228no—
382#117D4414yes—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—222no—
387C2×C4⋊C4C22×D1112yes—
388C2×C4⋊C4C22×D1112yes—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—222no—
391C2×C22⋊C4—224no—
392C2×C22⋊C4—224no—
393C2×C22⋊C4—448no—
394C2×C42—21no—
395C2×C42—111no—
396C2×C42—111no—
397C2×C42—112no—
398C2×C42—228no—
399C2×C42—222no—
400C2×C42—222no—
401C2×C42—222no—
402#103—224no—
403#102—224no—
404#102—228no—
405C2×C11—24no—
406C2×C11C2×#38611yescharacteristic
407C2×C11C4×AGL(1,ℤ/8)12yes—
408C2×C11#300612yes—
409C2×C11#303811yescharacteristic
410C2×C11#304011yescharacteristic
411D11—44no—
412D11—84no—
413C24—221no—
414C24—224no—
415C24—222no—
416C24—444no—
417C24—444no—
418C24—442no—
419C24—442no—
420C24D4×D1111yescharacteristic
421#41—1116no—
422#41—228no—
423#41—228no—
424#41—228no—
425C2×Q8—112no—
426C2×Q8—112no—
427C2×Q8—112no—
428C2×Q8—221no—
429C2×Q8—221no—
430C2×Q8—221no—
431C2×D4—112no—
432C2×D4—114no—
433C2×D4—112no—
434C2×D4—112no—
435C2×D4—224no—
436C2×D4—224no—
437C2×D4—224no—
438C2×D4—228no—
439C2×D4—228no—
440C2×D4—224no—
441C2×D4—221no—
442C2×D4—222no—
443C2×D4—221no—
444C2×D4—222no—
445C2×D4—222no—
446C2×D4—222no—
447C2×D4—221no—
448C2×D4—222no—
449C2×D4—442no—
450C2×D4—442no—
451C2×D4—444no—
452C2×D4—444no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—112no—
458C22×C4—112no—
459C22×C4—228no—
460C22×C4—224no—
461C22×C4—221no—
462C22×C4—224no—
463C22×C4—224no—
464C22×C4—224no—
465C22×C4—224no—
466C22×C4—444no—
467C22×C4—444no—
468C22×C4—442no—
469C22×C4C2×C4×D1112yes—
470C22×C4C2×D4412yes—
471C22×C4#433912yes—
472C22×C4D4×D1112yes—
473C22×C4D4×D1112yes—
474C22×C4D4×D1111yescharacteristic
475C22×C4#434312yes—
476C22×C4C23×D1111yescharacteristic
477SD16—2216no—
478D8—2216no—
479M16—28no—
480M16—444no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—222no—
484C2×C8—222no—
485C2×C8—222no—
486C2×C8C2×C4×D1114yes—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—228no—
492C4⋊C4C2×D4414yes—
493C22⋊C4—24no—
494C22⋊C4—228no—
495C22⋊C4—448no—
496C22⋊C4—448no—
497C42—24no—
498C42—228no—
499C42—222no—
500C42—222no—
501C42—222no—
502C42—222no—
503C11#2092811yescharacteristic, complemented, Sylow
504C23—22no—
505C23—22no—
506C23—222no—
507C23—222no—
508C23—222no—
509C23—224no—
510C23—4416no—
511C23—442no—
512C23—442no—
513C23—442no—
514C23—442no—
515C23—448no—
516C23—444no—
517C23—444no—
518C23—884no—
519C23—884no—
520C23—888no—
521C23#6693512yes—
522C23#6693712yes—
523C23#6694612yes—
524C23C2×D4×D1111yescharacteristic
525Q8—114no—
526Q8—222no—
527Q8—222no—
528Q8—222no—
529D4—114no—
530D4—2216no—
531D4—222no—
532D4—224no—
533D4—224no—
534D4—222no—
535D4—222no—
536D4—442no—
537D4—442no—
538D4—448no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—84no—
545C2×C4—114no—
546C2×C4—222no—
547C2×C4—222no—
548C2×C4—222no—
549C2×C4—222no—
550C2×C4—222no—
551C2×C4—2216no—
552C2×C4—222no—
553C2×C4—222no—
554C2×C4—222no—
555C2×C4—4416no—
556C2×C4—448no—
557C2×C4C4×D4414yes—
558C2×C4C22⋊C4×D1114yes—
559C2×C4#6693512yes—
560C2×C4#6693712yes—
561C2×C4#6694612yes—
562C2×C4#6694714yes—
563C2×C4#6694814yes—
564C2×C4C22×C4×D1111yescharacteristic
565C2×C4C22×D4411yescharacteristic
566C2×C4C2×#433911yescharacteristic
567C2×C4C2×D4×D1111yescharacteristic
568C2×C4C2×D4×D1111yescharacteristic
569C2×C4C2×#434311yescharacteristic
570C8—24no—
571C8—442no—
572C22—22no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—224no—
577C22—442no—
578C22—442no—
579C22—444no—
580C22—444no—
581C22—448no—
582C22—444no—
583C22—8816no—
584C22—882no—
585C22—882no—
586C22—884no—
587C22#1062227814yes—
588C22C2×#6693511yescharacteristic
589C22C2×#6693711yescharacteristic
590C22C2×#6694611yescharacteristic
591C22D11×AGL(1,ℤ/8)12yes—
592C22#1062330812yes—
593C4—44no—
594C4—82no—
595C4—224no—
596C4—224no—
597C4—442no—
598C4#1062227814yes—
599C4C2×C4×D4411yescharacteristic
600C4C2×C22⋊C4×D1111yescharacteristic
601C4C2×#6694711yescharacteristic
602C4C2×#6694811yescharacteristic
603C2—24no—
604C2—444no—
605C2—884no—
606C2C2×#1062227811yescharacteristic
607C2#1307742312yes—
608C2#1307786912yes—
609C2#1307809511yescharacteristic
610C2#1307811311yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1736C2
socleC23×C11C22×D4
prime radicalC22×#66961C2
cosocleC22×C4×C11C24
hypercenter#1736D11
hypocenterC11#20928
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×#10623053C2semidirect
3. C2×#10622579C2semidirect
4. C2×#10622614C2semidirect
5. C2×#10622601C2semidirect
6. #13068972C2semidirect
7. #13068965C2semidirect
8. #13068309C2semidirect
9. C11×#1736C2semidirect
20. C2×#66961C4semidirect
25. C2×#66946C22semidirect
26. C2×#66940C22semidirect
34. C2×#66874C22semidirect
37. C2×#66873C22semidirect
38. C11×#340C22semidirect
39. #10622188C22semidirect
57. C22×D44C2×C4semidirect
65. C2×C11×M16C2×C4semidirect
94. C2×D88C2×C4semidirect
97. C2×#4314C2×C4semidirect
141. #4339C22⋊C4semidirect
206. #1736D11semidirect
258. C2×Dic11AGL(1,ℤ/8)semidirect
267. D44C2×C22⋊C4semidirect
273. Dic22C2×C22⋊C4semidirect
304. #340C2×D11semidirect
372. C2×M16C4×D11semidirect
503. C11#20928semidirect

Maximal subgroups

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

subgroupindex
C22×#669612
C2×#106230532
C2×#106225792
C2×#106226142
C2×#106226012
#130689722
#130689652
#130683092
C11×#17362
#2092811