Finite Group Databasebeta

← #14759348

Subgroups — #14759348

Order 2816.

1880 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic11C211yesmaximal, characteristic
2C2×#10622831C211yesmaximal, characteristic
3C2×#10622371C212yesmaximal, complemented
4#13071144C211yesmaximal, characteristic, complemented
5#13068948C212yesmaximal, complemented
6#13068921C211yesmaximal, characteristic, complemented
7#13068875C212yesmaximal, complemented
8#13068808C211yesmaximal, characteristic, complemented
9#102×Dic11C212yesmaximal, complemented
10C22⋊C8×Dic11C211yesmaximal, characteristic, complemented
11C11×#1786C211yesmaximal, characteristic
12C23×D4×C11C2211yescharacteristic
13C24×Dic11—21no—
14C22×C22⋊Dic11—21no—
15C2×D4×Dic11—28no—
16C2×D4×Dic11—21no—
17C2×D4×Dic11—21no—
18C2×D4×Dic11C2212yes—
19C2×D4×Dic11C2212yes—
20C22×#4324C2212yes—
21C22×C4⋊Dic11C2211yescharacteristic
22C2×#66977C2211yescharacteristic
23C22×C4×Dic11C2211yescharacteristic
24C2×#66940C2211yescharacteristic, complemented
25C2×C11×#102C2212yes—
26C2×C11×#102C2212yes—
27C2×C11×C22⋊C8C2211yescharacteristic
28C11×#380C2211yescharacteristic
29#10622831C418yescomplemented
30C2×#66896C2212yes—
31C2×C22⋊(C11⋊C8)C2211yescharacteristic
32C2×#66880C2212yescomplemented
33C2×C8×Dic11C2212yescomplemented
34C4⋊C4×Dic11C2211yescharacteristic, complemented
35#10622587C2211yescharacteristic, complemented
36#10622586C2211yescharacteristic, complemented
37C2×#66872C2212yescomplemented
38#10622559—21no—
39C22⋊C4×Dic11—21no—
40#10622371—28no—
41#10622265—21no—
42#10622262—21no—
43#10622189C2212yescomplemented
44#10622188C2212yescomplemented
45#10622167C2212yescomplemented
46C25×C11—21no—
47C22×D4×C11—24no—
48C22×D4×C11—24no—
49C22×D4×C11—21no—
50C22×D4×C11—22no—
51C22×D4×C11C2×C414yes—
52C22×D4×C11C2312yes—
53C23×C4×C11C2311yescharacteristic
54C23×Dic11—28no—
55C23×Dic11—21no—
56C23×Dic11—22no—
57C23×Dic11—22no—
58C23×Dic11—44no—
59C23×Dic11—44no—
60C23×Dic11D411yescharacteristic
61C23×Dic11D411yescharacteristic
62C22×Dic22C2311yescharacteristic
63C22×C8×C11C2312yes—
64C2×C11×C4⋊C4C2311yescharacteristic
65C2×C11×C4⋊C4C2311yescharacteristic
66C2×C11×C22⋊C4—21no—
67C2×C22⋊Dic11—28no—
68C2×C22⋊Dic11—21no—
69C2×C22⋊Dic11—22no—
70C2×C22⋊Dic11—22no—
71D4×Dic11—28no—
72D4×Dic11—28no—
73D4×Dic11—24no—
74D4×Dic11—24no—
75D4×Dic11—24no—
76D4×Dic11D418yes—
77C2×#4324—24no—
78C2×#4324—24no—
79C2×#4324C2×C418yescomplemented
80C2×C4⋊Dic11—21no—
81C2×C4⋊Dic11—24no—
82C2×C4⋊Dic11C2311yescharacteristic
83C2×C4⋊Dic11C2312yes—
84#66977—22no—
85#66977C2×C414yescomplemented
86C2×Dic11⋊C4—21no—
87C2×Dic11⋊C4—21no—
88C2×Dic11⋊C4—21no—
89C2×C4×Dic11—24no—
90C2×C4×Dic11—21no—
91C2×C4×Dic11—21no—
92C2×C4×Dic11D411yescharacteristic
93C2×C4×Dic11D411yescharacteristic
94C2×C4×Dic11C2311yescharacteristic
95C2×C4×Dic11C2311yescharacteristic
96C22×C11⋊C8C2312yes—
97#66940—24no—
98C11×#102—24no—
99C11×#102—24no—
100C11×C22⋊C8—22no—
101C11×#95—21no—
102#66896C2×C418yescomplemented
103C11⋊#95—21no—
104C11⋊#95—21no—
105C11⋊#95—21no—
106C11⋊#95—21no—
107C22⋊(C11⋊C8)C2×C414yescomplemented
108#66880—24no—
109C8×Dic11—24no—
110#66872—24no—
111C4×#450—111nomaximal, Sylow
112C24×C11—24no—
113C24×C11—24no—
114C24×C11—24no—
115C24×C11—22no—
116C24×C11—42no—
117C24×C11—42no—
118C24×C11—42no—
119C24×C11—42no—
120C24×C11C2×D411yescharacteristic
121C2×D4×C11—24no—
122C2×D4×C11—24no—
123C2×D4×C11—216no—
124C2×D4×C11—216no—
125C2×D4×C11—24no—
126C2×D4×C11—22no—
127C2×D4×C11—22no—
128C2×D4×C11—22no—
129C2×D4×C11C22×C414yes—
130C2×D4×C11C2×D414yes—
131C2×D4×C11#4114yes—
132C22×C4×C11—22no—
133C22×C4×C11—22no—
134C22×C4×C11—21no—
135C22×C4×C11—21no—
136C22×C4×C11C22×C412yes—
137C22×C4×C11C2×D411yescharacteristic
138C22×C4×C11#4112yes—
139C22×C4×C11C2411yescharacteristic
140C22×Dic11—22no—
141C22×Dic11—22no—
142C22×Dic11—24no—
143C22×Dic11—28no—
144C22×Dic11—22no—
145C22×Dic11—28no—
146C22×Dic11—22no—
147C22×Dic11—21no—
148C22×Dic11—21no—
149C22×Dic11—22no—
150C22×Dic11—41no—
151C22×Dic11—42no—
152C22×Dic11—432no—
153C22×Dic11—44no—
154C22×Dic11—44no—
155C22×Dic11—44no—
156C22×Dic11—44no—
157C22×Dic11—42no—
158C22×Dic11SD1614yescomplemented
159C22×Dic11C2×D411yescharacteristic
160C22×Dic11C2×D411yescharacteristic
161C22×Dic11C2×D411yescharacteristic
162C22×Dic11C2×D411yescharacteristic
163C2×Dic22—22no—
164C2×Dic22—22no—
165C2×Dic22—21no—
166C2×Dic22C22×C412yescomplemented
167C2×C8×C11—22no—
168C2×C8×C11—22no—
169C2×C8×C11—22no—
170C11×C4⋊C4—22no—
171C11×C4⋊C4—22no—
172C11×C22⋊C4—42no—
173C22⋊Dic11—28no—
174C22⋊Dic11—28no—
175C22⋊Dic11—24no—
176#4324—216no—
177C4⋊Dic11—24no—
178C4⋊Dic11—22no—
179C4⋊Dic11—21no—
180C4⋊Dic11C22×C412yescomplemented
181C4⋊Dic11C2×D414yes—
182Dic11⋊C4—24no—
183Dic11⋊C4—42no—
184Dic11⋊C4—42no—
185C4×Dic11—22no—
186C4×Dic11—22no—
187C4×Dic11—22no—
188C4×Dic11—42no—
189C4×Dic11—42no—
190C4×Dic11C2×D414yes—
191C2×C11⋊C8—22no—
192C2×C11⋊C8—22no—
193C2×C11⋊C8C22×C414yescomplemented
194C22×C4×D4—111no—
195C2×#450—111no—
196C2×C4×SD16—112no—
197C4×C22⋊Q8—111no—
198#1877—112no—
199#1872—111no—
200#1864—112no—
201#1853—111no—
202#1786D1111yescharacteristic
203C4×#102—112no—
204C4×C22⋊C8—111no—
205C23×C11—21no—
206C23×C11—21no—
207C23×C11—21no—
208C23×C11—21no—
209C23×C11—216no—
210C23×C11—24no—
211C23×C11—24no—
212C23×C11—24no—
213C23×C11—48no—
214C23×C11—48no—
215C23×C11—48no—
216C23×C11—48no—
217C23×C11—48no—
218C23×C11—42no—
219C23×C11—42no—
220C23×C11—42no—
221C23×C11—42no—
222C23×C11—48no—
223C23×C11C4×D412yes—
224C23×C11C2×SD1612yes—
225C23×C11#13512yes—
226C23×C11C22×D411yescharacteristic
227D4×C11—216no—
228D4×C11—24no—
229D4×C11—24no—
230D4×C11—24no—
231D4×C11C4×D418yes—
232C2×C4×C11—21no—
233C2×C4×C11—21no—
234C2×C4×C11—24no—
235C2×C4×C11—24no—
236C2×C4×C11—41no—
237C2×C4×C11—41no—
238C2×C4×C11—41no—
239C2×C4×C11—41no—
240C2×C4×C11—41no—
241C2×C4×C11—41no—
242C2×C4×C11C4×D412yes—
243C2×C4×C11C23×C411yescharacteristic
244C2×C4×C11C22×D412yes—
245C2×C4×C11C2×#4112yes—
246C2×C4×C11C2×#4111yescharacteristic
247C2×Dic11—21no—
248C2×Dic11—21no—
249C2×Dic11—21no—
250C2×Dic11—21no—
251C2×Dic11—28no—
252C2×Dic11—22no—
253C2×Dic11—22no—
254C2×Dic11—21no—
255C2×Dic11—24no—
256C2×Dic11—21no—
257C2×Dic11—21no—
258C2×Dic11—44no—
259C2×Dic11—416no—
260C2×Dic11—48no—
261C2×Dic11—41no—
262C2×Dic11—41no—
263C2×Dic11—42no—
264C2×Dic11—41no—
265C2×Dic11—42no—
266C2×Dic11—41no—
267C2×Dic11—416no—
268C2×Dic11#12012yes—
269C2×Dic11#12012yes—
270C2×Dic11C2×SD1612yes—
271C2×Dic11C2×SD1612yes—
272C2×Dic11AGL(1,ℤ/8)12yes—
273Dic22—24no—
274Dic22—42no—
275C8×C11—42no—
276C11⋊C8—24no—
277C23×D4C2×D1111yescharacteristic
278C24×C4—221no—
279C22×SD16—112no—
280C2×C22⋊Q8—111no—
281C2×C4×Q8—111no—
282C2×C4×D4—112no—
283C2×C4×D4—112no—
284C2×C4×D4—221no—
285C2×C4×D4—221no—
286C2×C4×D4—228no—
287C22×C4⋊C4—111no—
288C22×C22⋊C4—221no—
289C22×C42—111no—
290#450—118no—
291C4×SD16—228no—
292C2×#106—112no—
293C2×#103—112no—
294C2×#102—112no—
295C2×#102C2×D1112yes—
296C2×#102C2×D1112yes—
297C2×C22⋊C8—111no—
298C2×C22⋊C8C2×D1111yescharacteristic
299C2×C4×C8—112no—
300#391—111no—
301#389—111no—
302#387—221no—
303#385—221no—
304#380C2×D1111yescharacteristic
305C4×C4⋊C4—111no—
306C4×C4⋊C4—221no—
307C4×C22⋊C4—221no—
308#340—112no—
309#340—112no—
310#336—112no—
311C22×C11—22no—
312C22×C11—22no—
313C22×C11—22no—
314C22×C11—22no—
315C22×C11—22no—
316C22×C11—22no—
317C22×C11—28no—
318C22×C11—432no—
319C22×C11—44no—
320C22×C11—44no—
321C22×C11—44no—
322C22×C11—44no—
323C22×C11—44no—
324C22×C11—44no—
325C22×C11C4×SD1614yescomplemented
326C22×C11C2×C4×D411yescharacteristic
327C22×C11C2×#12011yescharacteristic
328C22×C11#52511yescharacteristic
329C22×C11C22×SD1611yescharacteristic
330C22×C11C2×#13511yescharacteristic
331C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
332C22×C11#57511yescharacteristic
333C4×C11—22no—
334C4×C11—81no—
335C4×C11—81no—
336C4×C11C2×C4×D412yes—
337C4×C11#52512yes—
338Dic11—22no—
339Dic11—22no—
340Dic11—42no—
341Dic11—44no—
342Dic11—41no—
343Dic11—42no—
344Dic11#45014yescomplemented
345C25—21no—
346C22×Q8—111no—
347C22×D4—24no—
348C22×D4—24no—
349C22×D4—21no—
350C22×D4—22no—
351C22×D4C4×D1114yes—
352C22×D4C22×D1112yes—
353C23×C4—111no—
354C23×C4—111no—
355C23×C4—222no—
356C23×C4—222no—
357C23×C4—228no—
358C23×C4—221no—
359C23×C4—444no—
360C23×C4—444no—
361C23×C4C22×D1111yescharacteristic
362C2×SD16—118no—
363C2×SD16—224no—
364C2×SD16—224no—
365C22×C8—112no—
366C22×C8C22×D1112yes—
367C22⋊Q8—114no—
368C22⋊Q8—222no—
369C4×Q8—224no—
370C4×D4—118no—
371C4×D4—224no—
372C4×D4—224no—
373C4×D4—228no—
374C4×D4—228no—
375C4×D4—224no—
376C2×C4⋊C4—112no—
377C2×C4⋊C4—111no—
378C2×C4⋊C4—224no—
379C2×C4⋊C4—221no—
380C2×C4⋊C4—221no—
381C2×C4⋊C4—221no—
382C2×C4⋊C4—221no—
383C2×C4⋊C4C22×D1111yescharacteristic
384C2×C4⋊C4C22×D1111yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—222no—
387C2×C22⋊C4—222no—
388C2×C22⋊C4—228no—
389C2×C22⋊C4—221no—
390C2×C42—111no—
391C2×C42—111no—
392C2×C42—111no—
393C2×C42—111no—
394C2×C42—224no—
395C2×C42—221no—
396C2×C42—221no—
397#106—224no—
398#103—224no—
399#102—24no—
400#102—24no—
401#102—118no—
402C22⋊C8—22no—
403C22⋊C8—114no—
404C4×C8—224no—
405#95—21no—
406#95—221no—
407#95—221no—
408#95—221no—
409#95—221no—
410C2×C11—24no—
411C2×C11—48no—
412C2×C11C4×#12011yescharacteristic
413C2×C11C2×C4×SD1611yescharacteristic
414C2×C11C4×AGL(1,ℤ/8)11yescharacteristic
415C2×C11C2×#45011yescharacteristic
416C2×C11#300211yescharacteristic
417C2×C11#310111yescharacteristic
418C2×C11#310511yescharacteristic
419C24—24no—
420C24—24no—
421C24—24no—
422C24—22no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24D4×D1111yescharacteristic
428C2×Q8—112no—
429C2×Q8—221no—
430C2×Q8—222no—
431C2×Q8—222no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—216no—
435C2×D4—216no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4C2×C4×D1114yes—
441C2×D4D4×D1114yes—
442C2×D4#434114yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—111no—
448C22×C4—114no—
449C22×C4—111no—
450C22×C4—111no—
451C22×C4—111no—
452C22×C4—222no—
453C22×C4—222no—
454C22×C4—222no—
455C22×C4—228no—
456C22×C4—224no—
457C22×C4—228no—
458C22×C4—222no—
459C22×C4—221no—
460C22×C4—221no—
461C22×C4—222no—
462C22×C4—444no—
463C22×C4—444no—
464C22×C4—444no—
465C22×C4—441no—
466C22×C4—4432no—
467C22×C4—444no—
468C22×C4—442no—
469C22×C4—442no—
470C22×C4C2×C4×D1112yes—
471C22×C4D4×D1111yescharacteristic
472C22×C4#434112yes—
473C22×C4C23×D1111yescharacteristic
474SD16—2216no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—114no—
479C2×C8—222no—
480C2×C8—222no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—112no—
484C4⋊C4—114no—
485C4⋊C4—224no—
486C4⋊C4—224no—
487C4⋊C4—222no—
488C4⋊C4—221no—
489C4⋊C4—442no—
490C4⋊C4—442no—
491C22⋊C4—42no—
492C22⋊C4—228no—
493C22⋊C4—228no—
494C22⋊C4—224no—
495C42—114no—
496C42—222no—
497C42—222no—
498C42—222no—
499C42—442no—
500C42—442no—
501C11C4×#45011yescharacteristic, complemented, Sylow
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—216no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23(C22⋊D11)⋊C412yes—
521C23#6696512yes—
522C23SD16×D1112yes—
523C23C2×D4×D1111yescharacteristic
524Q8—224no—
525Q8—442no—
526D4—216no—
527D4—24no—
528D4—24no—
529D4—24no—
530D4(C22⋊D11)⋊C418yes—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—112no—
542C2×C4—112no—
543C2×C4—112no—
544C2×C4—112no—
545C2×C4—112no—
546C2×C4—228no—
547C2×C4—221no—
548C2×C4—221no—
549C2×C4—221no—
550C2×C4—221no—
551C2×C4—222no—
552C2×C4—222no—
553C2×C4—221no—
554C2×C4—221no—
555C2×C4—221no—
556C2×C4—224no—
557C2×C4—4416no—
558C2×C4—444no—
559C2×C4—4416no—
560C2×C4—448no—
561C2×C4—442no—
562C2×C4—441no—
563C2×C4—442no—
564C2×C4—441no—
565C2×C4—441no—
566C2×C4—441no—
567C2×C4(C22⋊D11)⋊C412yes—
568C2×C4C22×C4×D1111yescharacteristic
569C2×C4C2×D4×D1112yes—
570C2×C4C2×#434112yes—
571C2×C4C2×#434111yescharacteristic
572C8—42no—
573C8—224no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—28no—
581C22—432no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22#1062237114yescomplemented
589C22C2×(C22⋊D11)⋊C411yescharacteristic
590C22#1062311711yescharacteristic
591C22D11×#12011yescharacteristic
592C22C2×#6696511yescharacteristic
593C22C2×SD16×D1111yescharacteristic
594C22D11×AGL(1,ℤ/8)11yescharacteristic
595C22#1062330411yescharacteristic
596C4—22no—
597C4—81no—
598C4—81no—
599C4—114no—
600C4—222no—
601C4—222no—
602C4—442no—
603C4—444no—
604C4—441no—
605C4—442no—
606C4C2×(C22⋊D11)⋊C412yes—
607C4#1062311712yes—
608C2—24no—
609C2—48no—
610C2#1307111511yescharacteristic
611C2C2×#1062237111yescharacteristic
612C2#1307742011yescharacteristic
613C2D11×#45011yescharacteristic
614C2#1307782911yescharacteristic
615C2#1307861911yescharacteristic
616C2#1307865811yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1786C2
socleC23×C11C22×D4
prime radicalC23×D4×C11C22
cosocleC22×C4×C11C24
hypercenter#1786D11
hypocenterC11C4×#450
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
3. C2×#10622371C2semidirect
4. #13071144C2semidirect
5. #13068948C2semidirect
6. #13068921C2semidirect
7. #13068875C2semidirect
8. #13068808C2semidirect
9. #102×Dic11C2semidirect
10. C22⋊C8×Dic11C2semidirect
24. C2×#66940C22semidirect
29. #10622831C4semidirect
32. C2×#66880C22semidirect
33. C2×C8×Dic11C22semidirect
34. C4⋊C4×Dic11C22semidirect
35. #10622587C22semidirect
36. #10622586C22semidirect
37. C2×#66872C22semidirect
43. #10622189C22semidirect
44. #10622188C22semidirect
45. #10622167C22semidirect
79. C2×#4324C2×C4semidirect
85. #66977C2×C4semidirect
102. #66896C2×C4semidirect
107. C22⋊(C11⋊C8)C2×C4semidirect
158. C22×Dic11SD16semidirect
166. C2×Dic22C22×C4semidirect
180. C4⋊Dic11C22×C4semidirect
193. C2×C11⋊C8C22×C4semidirect
325. C22×C11C4×SD16semidirect
344. Dic11#450semidirect
501. C11C4×#450semidirect
588. C22#10622371semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic112
C2×#106228312
C2×#106223712
#130711442
#130689482
#130689212
#130688752
#130688082
#102×Dic112
C22⋊C8×Dic112
C11×#17862
C4×#45011