Finite Group Databasebeta

← #14756477

Subgroups — #14756477

Order 2816.

1198 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
1C2×#10622746C211yesmaximal, characteristic
2C2×#10622579C211yesmaximal, characteristic
3C2×M16×Dic11C211yesmaximal, characteristic, complemented
4#13071104C211yesmaximal, characteristic
5C2×#10622586C211yesmaximal, characteristic, complemented
6#13069311C212yesmaximal, complemented
7#13069304C212yesmaximal, complemented
8#13069297C212yesmaximal, complemented
9#13069290C212yesmaximal, complemented
10#13068337C211yesmaximal, characteristic, complemented
11C11×#1736C211yesmaximal, characteristic, complemented
12C22×#4339C2211yescharacteristic
13C22×C11×M16C2211yescharacteristic
14C2×C11×#117C2211yescharacteristic
15C22×C11×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic11C2211yescharacteristic
17C22×Dic11⋊C4—21no—
18C22×C4×Dic11C2211yescharacteristic
19C22×#4318C2211yescharacteristic
20C2×C11⋊#95—21no—
21#10622746C418yescomplemented
22M16×Dic11—24no—
23C2×#66885C2212yes—
24C2×#66880C2212yes—
25C2×#66879C2212yes—
26C2×C8×Dic11C2212yes—
27#10622601C2211yescharacteristic
28#10622601C2211yescharacteristic
29#10622587C2211yescharacteristic
30#10622587C2211yescharacteristic
31#10622586—21no—
32#10622586—21no—
33#10622586C2212yes—
34#10622586C2212yes—
35C2×#66874C2212yes—
36#10622579—24no—
37C2×#66873C2212yes—
38#10622571—21no—
39#10622571—21no—
40C4×(C2×D11)⋊C4—21no—
41C4×(C2×D11)⋊C4—21no—
42C11×#340C2212yes—
43C11×#340C2212yes—
44#10622167C2212yes—
45#10622167C2212yes—
46C23×C4×C11—21no—
47C23×C4×C11C2311yescharacteristic
48C22×C22⋊D11—21no—
49C23×Dic11—21no—
50C23×Dic11D411yescharacteristic
51C23×Dic11D411yescharacteristic
52C2×#4339—22no—
53C2×#4339—22no—
54C2×#4339—22no—
55C2×#4339—22no—
56C2×#4339—22no—
57C2×#4339C2×C414yes—
58C22×D44C2311yescharacteristic
59C22×C4×D11—21no—
60C22×Dic22C2311yescharacteristic
61C2×C11×M16—22no—
62C2×C11×M16—22no—
63C2×C11×M16C2×C414yes—
64C22×C8×C11C2312yes—
65C11×#117—22no—
66C11×#117—22no—
67C2×C11×C4⋊C4—22no—
68C2×C11×C4⋊C4—22no—
69C2×C11×C4⋊C4—21no—
70C2×C11×C4⋊C4C2312yes—
71C2×C11×C4⋊C4C2311yescharacteristic
72C2×C11×C4⋊C4C2311yescharacteristic
73C2×C11×C22⋊C4—21no—
74C2×C42×C11—21no—
75C2×(C2×D11)⋊C4—21no—
76C2×(C2×D11)⋊C4—21no—
77C2×(C2×D11)⋊C4—21no—
78C2×(C2×D11)⋊C4—21no—
79C2×C4⋊Dic11—21no—
80C2×C4⋊Dic11—22no—
81C2×C4⋊Dic11C2×C414yes—
82C2×C4⋊Dic11C2312yes—
83C2×Dic11⋊C4—22no—
84C2×Dic11⋊C4—22no—
85C2×Dic11⋊C4—44no—
86C2×C4×Dic11—22no—
87C2×C4×Dic11—22no—
88C2×C4×Dic11D411yescharacteristic
89C2×C4×Dic11D411yescharacteristic
90C2×C4×Dic11C2311yescharacteristic
91C2×C4×Dic11C2311yescharacteristic
92C2×#4318—22no—
93C2×#4318—22no—
94C2×#4318D414yes—
95C22×C11⋊C8C2312yes—
96C11⋊#95—21no—
97C11⋊#95—21no—
98C11⋊#95—21no—
99C11⋊#95—21no—
100C11⋊#95—22no—
101C11⋊#95—22no—
102#66885C2×C418yescomplemented
103#66880C2×C418yescomplemented
104#66879—24no—
105C8×Dic11—24no—
106#66874—24no—
107#66873—24no—
108#21093—111nomaximal, Sylow
109C24×C11C2×D411yescharacteristic
110C23×D11—21no—
111C22×C4×C11—22no—
112C22×C4×C11—21no—
113C22×C4×C11—21no—
114C22×C4×C11—42no—
115C22×C4×C11—42no—
116C22×C4×C11—42no—
117C22×C4×C11C22×C412yes—
118C22×C4×C11C2×D411yescharacteristic
119C22×C4×C11C2×D412yes—
120C22×C4×C11#4112yes—
121C22×C4×C11#4112yes—
122C22×C4×C11#4112yes—
123C22×C4×C11#4112yes—
124C22×C4×C11C2411yescharacteristic
125C2×C22⋊D11—24no—
126C2×C22⋊D11—42no—
127C2×C22⋊D11—42no—
128C22×Dic11—22no—
129C22×Dic11—22no—
130C22×Dic11—24no—
131C22×Dic11—22no—
132C22×Dic11—22no—
133C22×Dic11—21no—
134C22×Dic11—22no—
135C22×Dic11—41no—
136C22×Dic11—41no—
137C22×Dic11—42no—
138C22×Dic11—42no—
139C22×Dic11C2×D411yescharacteristic
140C22×Dic11C2×D411yescharacteristic
141C22×Dic11C2×D411yescharacteristic
142C22×Dic11C2×D411yescharacteristic
143#4339—28no—
144#4339—28no—
145#4339—44no—
146#4339—44no—
147C2×D44—21no—
148C2×D44—21no—
149C2×D44—21no—
150C2×D44—21no—
151C2×D44—21no—
152C2×D44C22×C412yes—
153C2×C4×D11—24no—
154C2×C4×D11—42no—
155C2×C4×D11—42no—
156C2×Dic22—21no—
157C2×Dic22—21no—
158C2×Dic22—21no—
159C2×Dic22—21no—
160C2×Dic22—21no—
161C2×Dic22C22×C412yes—
162C11×M16—28no—
163C2×C8×C11—22no—
164C2×C8×C11—22no—
165C2×C8×C11C22×C414yes—
166C11×C4⋊C4—22no—
167C11×C4⋊C4—22no—
168C11×C4⋊C4—21no—
169C11×C4⋊C4—21no—
170C11×C4⋊C4—21no—
171C11×C4⋊C4—21no—
172C11×C4⋊C4—42no—
173C11×C22⋊C4—42no—
174C42×C11—42no—
175(C2×D11)⋊C4—42no—
176(C2×D11)⋊C4—42no—
177(C2×D11)⋊C4—42no—
178(C2×D11)⋊C4—42no—
179C4⋊Dic11—24no—
180C4⋊Dic11—22no—
181C4⋊Dic11C22×C414yes—
182Dic11⋊C4—44no—
183Dic11⋊C4—44no—
184C4×Dic11—22no—
185C4×Dic11—22no—
186C4×Dic11—21no—
187C4×Dic11—21no—
188C4×Dic11—21no—
189C4×Dic11—21no—
190#4318—28no—
191C2×C11⋊C8—22no—
192C2×C11⋊C8—22no—
193C2×C11⋊C8C2×D414yes—
194C2×#417—111no—
195C2×#417—111no—
196C2×C4×M16—111no—
197#2297—111no—
198C2×#389—111no—
199#1936—112no—
200#1935—112no—
201#1934—112no—
202#1933—112no—
203#1736—111no—
204#1736D1111yescharacteristic, complemented
205C23×C11—21no—
206C23×C11—21no—
207C23×C11—21no—
208C23×C11—21no—
209C23×C11C4×D412yes—
210C23×C11#12112yes—
211C23×C11#12412yes—
212C23×C11C22×D411yescharacteristic
213C22×D11—22no—
214C22×D11—41no—
215C22×D11—41no—
216C22×D11—41no—
217C22×D11—41no—
218C22×D11—41no—
219C22×D11—41no—
220C2×C4×C11—42no—
221C2×C4×C11—41no—
222C2×C4×C11—41no—
223C2×C4×C11—42no—
224C2×C4×C11—41no—
225C2×C4×C11—41no—
226C2×C4×C11—41no—
227C2×C4×C11—41no—
228C2×C4×C11—42no—
229C2×C4×C11—84no—
230C2×C4×C11#11714yes—
231C2×C4×C11C4×D414yes—
232C2×C4×C11C4×D412yes—
233C2×C4×C11#12112yes—
234C2×C4×C11#12314yes—
235C2×C4×C11#12412yes—
236C2×C4×C11#12614yes—
237C2×C4×C11C23×C411yescharacteristic
238C2×C4×C11C22×D411yescharacteristic
239C2×C4×C11C2×#4111yescharacteristic
240C2×C4×C11C2×#4111yescharacteristic
241C2×C4×C11C2×#4111yescharacteristic
242C2×C4×C11C2×#4111yescharacteristic
243C22⋊D11—48no—
244C2×Dic11—21no—
245C2×Dic11—21no—
246C2×Dic11—21no—
247C2×Dic11—21no—
248C2×Dic11—24no—
249C2×Dic11—22no—
250C2×Dic11—21no—
251C2×Dic11—21no—
252C2×Dic11—21no—
253C2×Dic11—21no—
254C2×Dic11—21no—
255C2×Dic11—21no—
256C2×Dic11—44no—
257C2×Dic11—44no—
258C2×Dic11—48no—
259C2×Dic11—41no—
260C2×Dic11—41no—
261C2×Dic11—42no—
262C2×Dic11—42no—
263C2×Dic11AGL(1,ℤ/8)12yes—
264C2×Dic11#13712yes—
265D44—22no—
266D44—22no—
267D44—41no—
268D44—41no—
269C4×D11—48no—
270Dic22—22no—
271Dic22—22no—
272Dic22—41no—
273Dic22—41no—
274C8×C11—24no—
275C11⋊C8—24no—
276C22×#41—111no—
277C22×M16—111no—
278C22×M16C2×D1111yescharacteristic
279C2×#117C2×D1111yescharacteristic
280C22×C4⋊C4—111no—
281C22×C4⋊C4—221no—
282C22×C4⋊C4C2×D1111yescharacteristic
283C22×C42—111no—
284#417—118no—
285#417—224no—
286C2×#103—112no—
287C2×#103—112no—
288C2×#102—112no—
289C2×#102—112no—
290C4×M16—224no—
291C2×#97—112no—
292C2×C4×C8—112no—
293#391—111no—
294#391—111no—
295#390—111no—
296#390—111no—
297#389—112no—
298#389—112no—
299#389—221no—
300#389—221no—
301#386—221no—
302#386—221no—
303C4×C22⋊C4—221no—
304C4×C22⋊C4—221no—
305C2×#95—221no—
306#340—112no—
307#340—112no—
308#340C2×D1112yes—
309#340C2×D1112yes—
310C22×C11—22no—
311C22×C11—22no—
312C22×C11—22no—
313C22×C11—22no—
314C22×C11—22no—
315C22×C11—22no—
316C22×C11#38814yes—
317C22×C11C2×C4×D411yescharacteristic
318C22×C11C2×#12111yescharacteristic
319C22×C11C2×#12411yescharacteristic
320C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
321C22×C11C2×#13711yescharacteristic
322C22×C11#57511yescharacteristic
323C22×C11#57511yescharacteristic
324C2×D11—42no—
325C2×D11—42no—
326C2×D11—42no—
327C2×D11—81no—
328C2×D11—81no—
329C2×D11—81no—
330C2×D11—81no—
331C4×C11—81no—
332C4×C11—81no—
333C4×C11—82no—
334C4×C11#38814yes—
335C4×C11C2×#11711yescharacteristic
336C4×C11C2×C4×D411yescharacteristic
337C4×C11C2×#12311yescharacteristic
338C4×C11C2×#12611yescharacteristic
339Dic11—22no—
340Dic11—22no—
341Dic11—44no—
342Dic11—41no—
343Dic11—41no—
344Dic11—42no—
345C2×#41—114no—
346C2×#41—222no—
347C2×#41—222no—
348C2×#41—222no—
349C2×#41—222no—
350C2×#41—222no—
351C22×Q8—111no—
352C22×D4—111no—
353C22×D4—221no—
354C23×C4—21no—
355C23×C4—111no—
356C23×C4—111no—
357C23×C4—221no—
358C23×C4—221no—
359C23×C4C22×D1111yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—114no—
363C2×M16—222no—
364C2×M16—222no—
365C2×M16C4×D1114yes—
366C22×C8—112no—
367C22×C8C22×D1112yes—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—114no—
374C2×C4⋊C4—112no—
375C2×C4⋊C4—221no—
376C2×C4⋊C4—222no—
377C2×C4⋊C4—222no—
378C2×C4⋊C4—222no—
379C2×C4⋊C4—444no—
380C2×C4⋊C4C22×D1112yes—
381C2×C4⋊C4C22×D1111yescharacteristic
382C2×C4⋊C4C22×D1111yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—221no—
385C2×C22⋊C4—221no—
386C2×C22⋊C4—221no—
387C2×C22⋊C4—221no—
388C2×C42—21no—
389C2×C42—111no—
390C2×C42—111no—
391C2×C42—111no—
392C2×C42—111no—
393C2×C42—222no—
394C2×C42—222no—
395#103—118no—
396#103—224no—
397#102—118no—
398#102—224no—
399#97—224no—
400C4×C8—224no—
401#95—222no—
402#95—222no—
403#95—221no—
404#95—221no—
405#95—221no—
406#95—221no—
407C2×C11—24no—
408C2×C11C2×#38811yescharacteristic
409C2×C11C4×AGL(1,ℤ/8)11yescharacteristic
410C2×C11C4×#13711yescharacteristic
411C2×C11#305511yescharacteristic
412C2×C11#305611yescharacteristic
413C2×C11#314111yescharacteristic
414C2×C11#314211yescharacteristic
415D11—82no—
416C24—221no—
417C24D4×D1111yescharacteristic
418#41—228no—
419#41—228no—
420#41—444no—
421#41—444no—
422C2×Q8—112no—
423C2×Q8—221no—
424C2×Q8—221no—
425C2×Q8—221no—
426C2×Q8—221no—
427C2×Q8—221no—
428C2×D4—112no—
429C2×D4—224no—
430C2×D4—221no—
431C2×D4—221no—
432C2×D4—221no—
433C2×D4—221no—
434C2×D4—221no—
435C2×D4—442no—
436C2×D4—442no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—111no—
444C22×C4—111no—
445C22×C4—111no—
446C22×C4—111no—
447C22×C4—222no—
448C22×C4—222no—
449C22×C4—222no—
450C22×C4—222no—
451C22×C4—224no—
452C22×C4—221no—
453C22×C4—222no—
454C22×C4—224no—
455C22×C4—441no—
456C22×C4—441no—
457C22×C4—442no—
458C22×C4—442no—
459C22×C4—442no—
460C22×C4—442no—
461C22×C4C2×C4×D1112yes—
462C22×C4#433912yes—
463C22×C4#433912yes—
464C22×C4D4×D1112yes—
465C22×C4D4×D1111yescharacteristic
466C22×C4#434112yes—
467C22×C4#434312yes—
468C22×C4C23×D1111yescharacteristic
469M16—28no—
470M16—228no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—114no—
474C2×C8—222no—
475C2×C8—222no—
476C2×C8C2×C4×D1114yes—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—114no—
485C4⋊C4—224no—
486C4⋊C4—222no—
487C4⋊C4—444no—
488C4⋊C4—444no—
489C22⋊C4—42no—
490C22⋊C4—442no—
491C22⋊C4—442no—
492C22⋊C4—442no—
493C22⋊C4—442no—
494C42—42no—
495C42—222no—
496C42—222no—
497C42—221no—
498C42—221no—
499C42—221no—
500C42—221no—
501C11#2109311yescharacteristic, complemented, Sylow
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—222no—
507C23—441no—
508C23—441no—
509C23—441no—
510C23—441no—
511C23—441no—
512C23—441no—
513C23#6693712yes—
514C23#6693812yes—
515C23#6694612yes—
516C23C2×D4×D1111yescharacteristic
517Q8—222no—
518Q8—222no—
519Q8—441no—
520Q8—441no—
521D4—222no—
522D4—222no—
523D4—448no—
524D4—441no—
525D4—441no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—112no—
537C2×C4—112no—
538C2×C4—221no—
539C2×C4—221no—
540C2×C4—221no—
541C2×C4—221no—
542C2×C4—221no—
543C2×C4—221no—
544C2×C4—224no—
545C2×C4—222no—
546C2×C4—221no—
547C2×C4—221no—
548C2×C4—221no—
549C2×C4—221no—
550C2×C4—444no—
551C2×C4—444no—
552C2×C4—448no—
553C2×C4—441no—
554C2×C4—441no—
555C2×C4—442no—
556C2×C4—442no—
557C2×C4—448no—
558C2×C4#6692514yes—
559C2×C4(C22⋊D11)⋊C414yes—
560C2×C4#6693712yes—
561C2×C4#6693812yes—
562C2×C4#6694612yes—
563C2×C4#6694714yes—
564C2×C4#6695114yes—
565C2×C4C22×C4×D1111yescharacteristic
566C2×C4C2×#433911yescharacteristic
567C2×C4C2×#433911yescharacteristic
568C2×C4C2×D4×D1111yescharacteristic
569C2×C4C2×#434111yescharacteristic
570C2×C4C2×#434311yescharacteristic
571C8—24no—
572C8—224no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—442no—
580C22—442no—
581C22—442no—
582C22—881no—
583C22—881no—
584C22—881no—
585C22—881no—
586C22#1062228214yes—
587C22C2×#6693711yescharacteristic
588C22C2×#6693811yescharacteristic
589C22C2×#6694611yescharacteristic
590C22D11×AGL(1,ℤ/8)11yescharacteristic
591C22#1062330411yescharacteristic
592C22D11×#13711yescharacteristic
593C22#1062330811yescharacteristic
594C4—82no—
595C4—81no—
596C4—81no—
597C4—222no—
598C4—222no—
599C4—444no—
600C4—442no—
601C4—441no—
602C4—441no—
603C4#1062228214yes—
604C4C2×#6692511yescharacteristic
605C4C2×(C22⋊D11)⋊C411yescharacteristic
606C4C2×#6694711yescharacteristic
607C4C2×#6695111yescharacteristic
608C2—24no—
609C2—882no—
610C2C2×#1062228211yescharacteristic
611C2#1307742311yescharacteristic
612C2#1307743411yescharacteristic
613C2#1307823311yescharacteristic
614C2#1307824411yescharacteristic
615C2#1307900711yescharacteristic
616C2#1307901411yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1736C2
socleC23×C11C22×D4
prime radicalC22×#4339C22
cosocleC22×C4×C11C24
hypercenter#1736D11
hypocenterC11#21093
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
3. C2×M16×Dic11C2semidirect
5. C2×#10622586C2semidirect
6. #13069311C2semidirect
7. #13069304C2semidirect
8. #13069297C2semidirect
9. #13069290C2semidirect
10. #13068337C2semidirect
11. C11×#1736C2semidirect
21. #10622746C4semidirect
102. #66885C2×C4semidirect
103. #66880C2×C4semidirect
204. #1736D11semidirect
501. C11#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#106227462
C2×#106225792
C2×M16×Dic112
#130711042
C2×#106225862
#130693112
#130693042
#130692972
#130692902
#130683372
C11×#17362
#2109311