Finite Group Databasebeta

← #15870284

Subgroups — #15870284

Order 3328.

844 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×#11718706C211yesmaximal, characteristic
2#13117520C211yesmaximal, characteristic
3#13117193C211yesmaximal, characteristic
4#13115473C211yesmaximal, characteristic, complemented
5#13115473C211yesmaximal, characteristic, complemented
6#13115431C211yesmaximal, characteristic, complemented
7#13115417C211yesmaximal, characteristic, complemented
8#13115029C211yesmaximal, characteristic
9#13114932C211yesmaximal, characteristic
10#13114732C211yesmaximal, characteristic, complemented
11#13114724C211yesmaximal, characteristic, complemented
12#13114668C211yesmaximal, characteristic, complemented
13#103×Dic13C211yesmaximal, characteristic, complemented
14#13114604C211yesmaximal, characteristic
15C13×#1872C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C22×C13×C4⋊C4C2211yescharacteristic
18C2×Q8×Dic13C2211yescharacteristic
19C2×#88180C2211yescharacteristic
20C2×#88180C2211yescharacteristic
21C2×C13×#103C2211yescharacteristic
22C2×C13×#103C2211yescharacteristic
23C2×C13×C22⋊C8C2211yescharacteristic
24#11718891C2211yescharacteristic
25C2×#88075C2211yescharacteristic
26C2×#88075C2211yescharacteristic
27#11718873C2211yescharacteristic
28C4×C22⋊Dic13—21no—
29C4×C22⋊Dic13—21no—
30C2×C22⋊#1055C2211yescharacteristic
31C2×#88055C2211yescharacteristic
32C2×C8×Dic13C2211yescharacteristic
33#11718706C418yescomplemented
34#11718646—21no—
35#11718646C2211yescharacteristic
36#11718642—21no—
37C4⋊C4×Dic13—21no—
38C4⋊C4×Dic13—21no—
39C4⋊C4×Dic13C2211yescharacteristic
40C4⋊C4×Dic13C2211yescharacteristic
41C4⋊C4×Dic13C2211yescharacteristic
42C4⋊C4×Dic13C2211yescharacteristic
43C4⋊C4×Dic13C2211yescharacteristic
44C2×#88048C2211yescharacteristic
45C2×#88048C2211yescharacteristic
46C2×#88048C2211yescharacteristic
47C2×#88047C2211yescharacteristic
48#11718612—21no—
49#11718612—21no—
50C13×#340C2211yescharacteristic
51C13×#340C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×Q8×C13C2311yescharacteristic
61C23×C4×C13—21no—
62C23×C4×C13C2311yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C13×C22⋊Q8—22no—
66C13×C22⋊Q8C2×C414yes—
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—22no—
70C2×C13×C4⋊C4—22no—
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C22⋊C4—21no—
75C2×C22⋊Dic13—21no—
76C2×C22⋊Dic13—21no—
77C2×C22⋊Dic13D411yescharacteristic
78C2×C22⋊Dic13D411yescharacteristic
79Q8×Dic13—22no—
80Q8×Dic13—22no—
81#88180—22no—
82#88180—22no—
83#88180—22no—
84#88180C2×C414yes—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13D411yescharacteristic
88C2×C4⋊Dic13D411yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13C2311yescharacteristic
101C2×C4×Dic13C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C13×#103—22no—
105C13×#103—22no—
106C13×C22⋊C8—22no—
107#88075C2×C414yescomplemented
108#88075C2×C414yescomplemented
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118C22⋊#1055C2×C414yes—
119#88055—22no—
120C8×Dic13—22no—
121#88048—22no—
122#88048—22no—
123#88048C2×C414yescomplemented
124#88047C2×C414yescomplemented
125#21591—131nomaximal, Sylow
126C24×C13C2×D411yescharacteristic
127C2×Q8×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13C22×C412yes—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—24no—
142C22×C4×C13—81no—
143C22×C4×C13—81no—
144C22×C4×C13C22×C412yes—
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13#4112yes—
147C22×C4×C13C2411yescharacteristic
148C22×Dic13—21no—
149C22×Dic13—21no—
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13C2×D411yescharacteristic
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C22×Dic13C2×D411yescharacteristic
156C2×C8×C13—21no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C13×C4⋊C4—24no—
163C13×C4⋊C4—24no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4C22×C412yes—
168C13×C4⋊C4#4112yes—
169C13×C4⋊C4#4112yes—
170C13×C22⋊C4—24no—
171C22⋊Dic13—24no—
172C22⋊Dic13—22no—
173C22⋊Dic13—22no—
174C22⋊Dic13—81no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13—22no—
178C4⋊Dic13—22no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—42no—
184C4⋊Dic13—81no—
185C4⋊Dic13C22×C412yes—
186C4⋊Dic13C22×C412yes—
187C4×Dic13—24no—
188C4×Dic13—21no—
189C4×Dic13—21no—
190C4×Dic13—42no—
191C4×Dic13—81no—
192C4×Dic13—81no—
193C4×Dic13—81no—
194C4×Dic13—81no—
195C4×Dic13—81no—
196C4×Dic13#4112yes—
197C4×Dic13#4112yes—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055C22×C412yes—
203C2×#1055C22×C412yes—
204C2×#485—131no—
205#2323—131no—
206#2274—131no—
207#1927—131no—
208#1927—131no—
209#1921—131no—
210#1919—131no—
211#1872D1311yescharacteristic
212#1853—131no—
213#1825—131no—
214#1776—131no—
215C4×#106—131no—
216#1763—131no—
217C4×#103—131no—
218#1750—131no—
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13C4×D412yes—
226C23×C13C22×D411yescharacteristic
227Q8×C13—22no—
228Q8×C13—22no—
229Q8×C13—41no—
230Q8×C13—41no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—84no—
250C2×C4×C13C4×D412yes—
251C2×C4×C13C23×C411yescharacteristic
252C2×C4×C13C2×#4111yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13C2×#4111yescharacteristic
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—22no—
262C2×Dic13—22no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13#12312yes—
280C2×Dic13#12312yes—
281C8×C13—41no—
282C8×C13—41no—
283#1055—22no—
284#1055—22no—
285C2×C22⋊Q8C2×D1311yescharacteristic
286C2×C4×Q8—131no—
287C2×#117—131no—
288C2×#117—131no—
289C22×C4⋊C4C2×D1311yescharacteristic
290#485—138no—
291C2×#107—131no—
292C2×#106—131no—
293C2×#106—131no—
294C2×#106—131no—
295C2×#103—131no—
296C2×#103—131no—
297C2×#103C2×D1311yescharacteristic
298C2×#103C2×D1311yescharacteristic
299C2×C22⋊C8—131no—
300C2×C22⋊C8C2×D1311yescharacteristic
301C2×#97—131no—
302C2×C4×C8—131no—
303#391—131no—
304#389—131no—
305#389—261no—
306#388—261no—
307#388—261no—
308#387—261no—
309#380—131no—
310C4×C4⋊C4—131no—
311C4×C4⋊C4—131no—
312C4×C4⋊C4—131no—
313C4×C4⋊C4—131no—
314C4×C4⋊C4—131no—
315C4×C4⋊C4—261no—
316C4×C4⋊C4—261no—
317C4×C22⋊C4—261no—
318C4×C22⋊C4—261no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340C2×D1311yescharacteristic
326#340C2×D1311yescharacteristic
327#336—131no—
328#336—131no—
329C22×C13—22no—
330C22×C13—22no—
331C22×C13—22no—
332C22×C13—41no—
333C22×C13—41no—
334C22×C13—41no—
335C22×C13—41no—
336C22×C13C2×C4×D411yescharacteristic
337C22×C13C2×#12311yescharacteristic
338C22×C13#52511yescharacteristic
339C22×C13#57711yescharacteristic
340C22×C13#57711yescharacteristic
341C22×C13#57711yescharacteristic
342C22×C13#57811yescharacteristic
343C4×C13—22no—
344C4×C13—42no—
345C4×C13—42no—
346C4×C13—42no—
347C4×C13—42no—
348C4×C13C4×#4111yescharacteristic
349C4×C13C4×#4111yescharacteristic
350C4×C13#52911yescharacteristic
351C4×C13#52911yescharacteristic
352Dic13—22no—
353Dic13—22no—
354Dic13—41no—
355Dic13—41no—
356Dic13—42no—
357Dic13—42no—
358Dic13—81no—
359Dic13—81no—
360C22×Q8C22×D1311yescharacteristic
361C23×C4—21no—
362C23×C4C22×D1311yescharacteristic
363C22×C8—131no—
364C22×C8—131no—
365C22×C8C22×D1311yescharacteristic
366C22×C8C22×D1311yescharacteristic
367C22⋊Q8—22no—
368C22⋊Q8C4×D1314yes—
369C4×Q8—262no—
370C4×Q8—262no—
371#117—134no—
372#117—262no—
373#117—262no—
374#117—262no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—131no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4C22×D1311yescharacteristic
388C2×C4⋊C4C22×D1311yescharacteristic
389C2×C4⋊C4C22×D1311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—131no—
392C2×C22⋊C4—131no—
393C2×C22⋊C4—261no—
394C2×C22⋊C4—261no—
395C2×C42—131no—
396C2×C42—131no—
397C2×C42—261no—
398C2×C42—261no—
399C2×C42—261no—
400C2×C42—261no—
401C2×C42—261no—
402C2×C42—261no—
403C2×C42—261no—
404#107—134no—
405#106—134no—
406#106—262no—
407#106—262no—
408#103—22no—
409#103—22no—
410#103—134no—
411#103—134no—
412C22⋊C8—22no—
413C22⋊C8—134no—
414#97—262no—
415C4×C8—262no—
416#95—261no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95—261no—
424#95—261no—
425C2×C13—42no—
426C2×C13C4×#12311yescharacteristic
427C2×C13#294711yescharacteristic
428C2×C13#295211yescharacteristic
429C2×C13#309711yescharacteristic
430C2×C13#309911yescharacteristic
431C2×C13#312111yescharacteristic
432C2×C13#312311yescharacteristic
433C24D4×D1311yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C2×C4×D1312yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—131no—
452C22×C4—131no—
453C22×C4—131no—
454C22×C4—131no—
455C22×C4—261no—
456C22×C4—261no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4C2×C4×D1312yes—
460C22×C4D4×D1311yescharacteristic
461C22×C4#612912yes—
462C22×C4C23×D1311yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—132no—
470C2×C8—132no—
471C2×C8—261no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8—261no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—132no—
481C4⋊C4—132no—
482C4⋊C4—262no—
483C4⋊C4—262no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C4⋊C4—261no—
488C4⋊C4—261no—
489C4⋊C4—261no—
490C4⋊C4—522no—
491C4⋊C4—1041no—
492C4⋊C4C2×C4×D1312yes—
493C4⋊C4#612912yes—
494C4⋊C4#612912yes—
495C22⋊C4—24no—
496C22⋊C4—262no—
497C22⋊C4—262no—
498C22⋊C4—264no—
499C22⋊C4—1041no—
500C42—132no—
501C42—132no—
502C42—264no—
503C42—261no—
504C42—261no—
505C42—522no—
506C42—1041no—
507C42—1041no—
508C42—1041no—
509C42—1041no—
510C42—1041no—
511C13#2159111yescharacteristic, complemented, Sylow
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23#8813512yes—
519C23C2×D4×D1311yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—132no—
544C2×C4—132no—
545C2×C4—261no—
546C2×C4—261no—
547C2×C4—261no—
548C2×C4—261no—
549C2×C4—262no—
550C2×C4—262no—
551C2×C4—261no—
552C2×C4—261no—
553C2×C4—261no—
554C2×C4—261no—
555C2×C4—261no—
556C2×C4—261no—
557C2×C4—521no—
558C2×C4—521no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—521no—
563C2×C4—521no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4#8813512yes—
568C2×C4C22×C4×D1311yescharacteristic
569C2×C4C2×#612911yescharacteristic
570C2×C4C2×#612911yescharacteristic
571C2×C4C2×#612911yescharacteristic
572C2×C4#41×D1311yescharacteristic
573C2×C4#41×D1311yescharacteristic
574C8—41no—
575C8—41no—
576C8—262no—
577C8—262no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×#8813511yescharacteristic
586C22#1171929111yescharacteristic
587C22D13×#12311yescharacteristic
588C22#1171946711yescharacteristic
589C22#1171947111yescharacteristic
590C22#1171948011yescharacteristic
591C22#1171948311yescharacteristic
592C4—22no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—262no—
598C4—262no—
599C4—522no—
600C4—522no—
601C4—521no—
602C4—521no—
603C4—1041no—
604C4—1041no—
605C4C4×#612911yescharacteristic
606C4C4×#612911yescharacteristic
607C4#1171930511yescharacteristic
608C4#1171930511yescharacteristic
609C2—42no—
610C2#1311736111yescharacteristic
611C2#1312367311yescharacteristic
612C2#1312371511yescharacteristic
613C2#1312480411yescharacteristic
614C2#1312482011yescharacteristic
615C2#1312506611yescharacteristic
616C2#1312508211yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1872C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1872D13
hypocenterC13#21591
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #13115473C2semidirect
5. #13115473C2semidirect
6. #13115431C2semidirect
7. #13115417C2semidirect
10. #13114732C2semidirect
11. #13114724C2semidirect
12. #13114668C2semidirect
13. #103×Dic13C2semidirect
33. #11718706C4semidirect
107. #88075C2×C4semidirect
108. #88075C2×C4semidirect
123. #88048C2×C4semidirect
124. #88047C2×C4semidirect
511. C13#21591semidirect

Maximal subgroups

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

subgroupindex
C2×#117187062
#131175202
#131171932
#131154732
#131154312
#131154172
#131150292
#131149322
#131147322
#131147242
#131146682
#103×Dic132
#131146042
C13×#18722
#2159113