Finite Group Databasebeta

← #15769224

Subgroups — #15769224

Order 3328.

1118 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88045C211yesmaximal, characteristic
2C22×#88065C211yesmaximal, characteristic
3C2×M16×Dic13C211yesmaximal, characteristic, complemented
4#13117324C211yesmaximal, characteristic
5C2×#11718124C211yesmaximal, characteristic, complemented
6C4×#88071C211yesmaximal, characteristic, complemented
7C2×C13×#339C211yesmaximal, characteristic, complemented
8C22×#6127C2211yescharacteristic
9C22×C13×M16C2211yescharacteristic
10C2×C13×#117C2211yescharacteristic
11C22×C42×C13C2211yescharacteristic
12C22×C4×Dic13C2211yescharacteristic
13C22×C4×Dic13C2211yescharacteristic
14C22×#6099C2211yescharacteristic
15C2×#88071—22no—
16C2×#88065—22no—
17C2×#88065—22no—
18C2×#88065C414yescomplemented
19M16×Dic13—22no—
20M16×Dic13—22no—
21C2×#88055—21no—
22C2×C8×Dic13—21no—
23#11718649C412yescomplemented
24#11718635C412yescomplemented
25#11718619—22no—
26C4×#6103—21no—
27C4×#6103—21no—
28C2×#88045—22no—
29C2×#88045—22no—
30C2×#88045C414yescomplemented
31C13×#339—22no—
32#11718124—22no—
33C23×C4×C13—21no—
34C23×C4×C13C2311yescharacteristic
35C22×C22⋊D13—21no—
36C23×Dic13—21no—
37C23×Dic13D412yes—
38C2×#6127—21no—
39C2×#6127—21no—
40C2×#6127—21no—
41C2×#6127—21no—
42C2×#6127—21no—
43C2×#6127—21no—
44C2×#6127—21no—
45C2×#6127—21no—
46C2×#6127—21no—
47C2×#6127—21no—
48C2×#6127C2×C412yes—
49C2×#6127C2×C412yes—
50C22×D52C2×C411yescharacteristic
51C22×C4×D13—21no—
52C22×Dic26C2×C411yescharacteristic
53C2×C13×M16—21no—
54C2×C13×M16—21no—
55C2×C13×M16—21no—
56C2×C13×M16C2×C412yes—
57C2×C13×M16D412yes—
58C2×C13×M16D412yes—
59C22×C8×C13—21no—
60C13×#117—21no—
61C13×#117—21no—
62C13×#117—21no—
63C13×#117—21no—
64C2×C13×C4⋊C4C2×C412yes—
65C2×C13×C22⋊C4—21no—
66C2×C42×C13—21no—
67C2×C42×C13—21no—
68C2×C42×C13—21no—
69C2×C42×C13—21no—
70C2×C42×C13—21no—
71C2×C42×C13—21no—
72C2×C42×C13C2×C412yes—
73C2×#6103—22no—
74C2×#6103—22no—
75C2×C4×Dic13—21no—
76C2×C4×Dic13—21no—
77C2×C4×Dic13—21no—
78C2×C4×Dic13—21no—
79C2×C4×Dic13—21no—
80C2×C4×Dic13—21no—
81C2×C4×Dic13—21no—
82C2×C4×Dic13—21no—
83C2×C4×Dic13—21no—
84C2×C4×Dic13C2×C412yes—
85C2×C4×Dic13C2×C412yes—
86C2×C4×Dic13D411yescharacteristic
87C2×C4×Dic13D411yescharacteristic
88C2×#6099—21no—
89C2×#6099—21no—
90C2×#6099—21no—
91C2×#6099C2×C412yes—
92C2×#6099D412yes—
93C2×#6099D412yes—
94C22×#1055—21no—
95#88071—28no—
96#88071—22no—
97#88071—22no—
98#88065—28no—
99#88055—24no—
100C8×Dic13—24no—
101#88045—28no—
102#10933—131nomaximal, Sylow
103C24×C13C2×D411yescharacteristic
104C23×D13—21no—
105C22×C4×C13—24no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—42no—
109C22×C4×C13—41no—
110C22×C4×C13—41no—
111C22×C4×C13C22×C411yescharacteristic
112C22×C4×C13C22×C411yescharacteristic
113C22×C4×C13C22×C411yescharacteristic
114C22×C4×C13C2×D411yescharacteristic
115C22×C4×C13C2×D411yescharacteristic
116C22×C4×C13C2×D411yescharacteristic
117C22×C4×C13C2×D411yescharacteristic
118C22×C4×C13C2×D411yescharacteristic
119C22×C4×C13#4111yescharacteristic
120C22×C4×C13#4111yescharacteristic
121C22×C4×C13#4111yescharacteristic
122C22×C4×C13#4111yescharacteristic
123C22×C4×C13#4111yescharacteristic
124C22×C4×C13#4111yescharacteristic
125C2×C22⋊D13—22no—
126C2×C22⋊D13—22no—
127C2×C22⋊D13—41no—
128C2×C22⋊D13—41no—
129C2×C22⋊D13—41no—
130C2×C22⋊D13—41no—
131C22×Dic13—24no—
132C22×Dic13—21no—
133C22×Dic13—22no—
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13—24no—
137C22×Dic13—42no—
138C22×Dic13—42no—
139C22×Dic13—41no—
140C22×Dic13—41no—
141C22×Dic13C22⋊C412yes—
142C22×Dic13C22⋊C412yes—
143#6127—22no—
144#6127—22no—
145#6127—22no—
146#6127—22no—
147#6127—22no—
148#6127—22no—
149#6127—22no—
150#6127—22no—
151#6127—41no—
152#6127—41no—
153#6127—41no—
154#6127—41no—
155#6127—41no—
156#6127—41no—
157#6127—41no—
158#6127—41no—
159C2×D52—21no—
160C2×D52—21no—
161C2×D52—21no—
162C2×D52—21no—
163C2×D52—21no—
164C2×D52C4212yes—
165C2×C4×D13—22no—
166C2×C4×D13—22no—
167C2×C4×D13—41no—
168C2×C4×D13—41no—
169C2×C4×D13—41no—
170C2×C4×D13—41no—
171C2×Dic26—21no—
172C2×Dic26—21no—
173C2×Dic26—21no—
174C2×Dic26—21no—
175C2×Dic26—21no—
176C2×Dic26C4212yes—
177C13×M16—22no—
178C13×M16—22no—
179C13×M16—22no—
180C13×M16C22⋊C414yescomplemented
181C2×C8×C13—22no—
182C2×C8×C13—22no—
183C2×C8×C13—22no—
184C13×C4⋊C4—22no—
185C13×C4⋊C4—22no—
186C13×C22⋊C4—41no—
187C13×C22⋊C4—41no—
188C42×C13—22no—
189C42×C13—22no—
190C42×C13—41no—
191C42×C13—41no—
192C42×C13—41no—
193C42×C13—41no—
194#6103—44no—
195#6103—44no—
196C4×Dic13—22no—
197C4×Dic13—22no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13—22no—
203C4×Dic13—22no—
204C4×Dic13—41no—
205C4×Dic13—41no—
206#6099—22no—
207#6099—22no—
208#6099—22no—
209#6099C22⋊C414yescomplemented
210C2×#1055—22no—
211C2×#1055—22no—
212C2×#1055—22no—
213C22×#104—131no—
214C22×#104—131no—
215C2×C4×M16—131no—
216#2297—131no—
217C2×#339—131no—
218C2×#339D1311yescharacteristic, complemented
219C4×#95—131no—
220C23×C13—22no—
221C23×C13—22no—
222C23×C13C2×C22⋊C411yescharacteristic
223C23×C13C4×D411yescharacteristic
224C23×C13C4×D411yescharacteristic
225C23×C13#12111yescharacteristic
226C23×C13#12111yescharacteristic
227C23×C13#12411yescharacteristic
228C23×C13#12711yescharacteristic
229C22×D13—22no—
230C22×D13—41no—
231C22×D13—41no—
232C22×D13—42no—
233C22×D13—42no—
234C2×C4×C13—24no—
235C2×C4×C13—44no—
236C2×C4×C13—44no—
237C2×C4×C13—42no—
238C2×C4×C13—42no—
239C2×C4×C13—42no—
240C2×C4×C13—42no—
241C2×C4×C13—42no—
242C2×C4×C13C2×C4211yescharacteristic
243C2×C4×C13C2×C22⋊C411yescharacteristic
244C2×C4×C13C2×C22⋊C411yescharacteristic
245C2×C4×C13#11711yescharacteristic
246C2×C4×C13#11711yescharacteristic
247C2×C4×C13#11711yescharacteristic
248C2×C4×C13C4×D411yescharacteristic
249C2×C4×C13C4×D411yescharacteristic
250C2×C4×C13C4×D411yescharacteristic
251C2×C4×C13C4×D411yescharacteristic
252C2×C4×C13C4×D411yescharacteristic
253C2×C4×C13C4×D411yescharacteristic
254C2×C4×C13C4×D411yescharacteristic
255C2×C4×C13C4×D411yescharacteristic
256C2×C4×C13C4×D411yescharacteristic
257C2×C4×C13C4×D411yescharacteristic
258C2×C4×C13#12011yescharacteristic
259C2×C4×C13#12111yescharacteristic
260C2×C4×C13#12111yescharacteristic
261C2×C4×C13#12111yescharacteristic
262C2×C4×C13#12111yescharacteristic
263C2×C4×C13#12311yescharacteristic
264C2×C4×C13#12311yescharacteristic
265C2×C4×C13#12311yescharacteristic
266C2×C4×C13#12411yescharacteristic
267C2×C4×C13#12411yescharacteristic
268C2×C4×C13#12611yescharacteristic
269C2×C4×C13#12611yescharacteristic
270C22⋊D13—42no—
271C22⋊D13—42no—
272C22⋊D13—42no—
273C22⋊D13—42no—
274C2×Dic13—22no—
275C2×Dic13—22no—
276C2×Dic13—24no—
277C2×Dic13—22no—
278C2×Dic13—22no—
279C2×Dic13—22no—
280C2×Dic13—22no—
281C2×Dic13—44no—
282C2×Dic13—44no—
283C2×Dic13—44no—
284C2×Dic13—44no—
285C2×Dic13—42no—
286C2×Dic13—41no—
287C2×Dic13—42no—
288C2×Dic13—41no—
289C2×Dic13#10414yescomplemented
290D52—22no—
291D52—22no—
292D52—41no—
293D52—41no—
294C4×D13—42no—
295C4×D13—42no—
296C4×D13—42no—
297C4×D13—42no—
298Dic26—22no—
299Dic26—22no—
300Dic26—41no—
301Dic26—41no—
302C8×C13—24no—
303#1055—24no—
304C22×#41—131no—
305C22×M16—131no—
306C22×M16C2×D1311yescharacteristic
307C2×#117C2×D1311yescharacteristic
308C22×C42—131no—
309C22×C42—131no—
310C22×C42C2×D1311yescharacteristic
311C2×#104—134no—
312C2×#104—134no—
313C2×#104—262no—
314C2×#104—262no—
315C2×#104—262no—
316C2×#104—262no—
317C4×M16—262no—
318C4×M16—262no—
319C2×#97—261no—
320C2×C4×C8—261no—
321#391—132no—
322#390—132no—
323#386—262no—
324C4×C22⋊C4—261no—
325C4×C22⋊C4—261no—
326C2×#95—262no—
327#339—22no—
328#339—262no—
329C22×C13—22no—
330C22×C13—22no—
331C22×C13—22no—
332C22×C13—22no—
333C22×C13—21no—
334C22×C13—21no—
335C22×C13—21no—
336C22×C13—21no—
337C22×C13C4×C22⋊C411yescharacteristic
338C22×C13#38611yescharacteristic
339C22×C13#38811yescharacteristic
340C22×C13#38811yescharacteristic
341C22×C13#39011yescharacteristic
342C22×C13#39011yescharacteristic
343C22×C13#39011yescharacteristic
344C22×C13C2×#10412yes—
345C22×C13#42112yes—
346C2×D13—42no—
347C2×D13—42no—
348C2×D13—42no—
349C2×D13—82no—
350C2×D13—82no—
351C4×C13—44no—
352C4×C13—82no—
353C4×C13C4×C22⋊C411yescharacteristic
354C4×C13C4×C22⋊C411yescharacteristic
355C4×C13#38611yescharacteristic
356C4×C13#38611yescharacteristic
357C4×C13#38811yescharacteristic
358C4×C13#38811yescharacteristic
359C4×C13#38811yescharacteristic
360C4×C13#38811yescharacteristic
361Dic13—24no—
362Dic13—44no—
363Dic13—42no—
364Dic13—42no—
365C2×#41—132no—
366C2×#41—132no—
367C2×#41—261no—
368C2×#41—261no—
369C2×#41—261no—
370C2×#41—261no—
371C2×#41—261no—
372C2×#41—261no—
373C2×#41—261no—
374C2×#41—261no—
375C2×#41—261no—
376C2×#41—261no—
377C22×Q8—131no—
378C22×D4—131no—
379C22×D4—261no—
380C23×C4—21no—
381C23×C4—132no—
382C23×C4—261no—
383C23×C4—261no—
384C23×C4C22×D1311yescharacteristic
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—132no—
389C2×M16—132no—
390C2×M16—132no—
391C2×M16—261no—
392C2×M16—261no—
393C2×M16—261no—
394C2×M16C4×D1312yes—
395C2×M16D5212yes—
396C2×M16C22⋊D1312yes—
397C22×C8—21no—
398C22×C8—261no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—21no—
403C2×C4⋊C4C4×D1312yes—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—262no—
406C2×C22⋊C4—262no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—132no—
414C2×C42—131no—
415C2×C42—131no—
416C2×C42—132no—
417C2×C42—261no—
418C2×C42—261no—
419C2×C42—261no—
420C2×C42—261no—
421C2×C42—261no—
422C2×C42—261no—
423C2×C42—261no—
424C2×C42—261no—
425C2×C42—261no—
426C2×C42C4×D1312yes—
427#104—268no—
428#104—268no—
429#97—264no—
430C4×C8—264no—
431#95—262no—
432#95—262no—
433#95—268no—
434C2×C13—21no—
435C2×C13—21no—
436C2×C13—21no—
437C2×C13—21no—
438C2×C13#143711yescharacteristic
439C2×C13C4×#10411yescharacteristic
440C2×C13#175911yescharacteristic
441C2×C13#187112yes—
442C2×C13#198012yes—
443D13—82no—
444C24—261no—
445C24D4×D1311yescharacteristic
446#41—262no—
447#41—262no—
448#41—262no—
449#41—262no—
450#41—262no—
451#41—262no—
452#41—262no—
453#41—262no—
454#41—521no—
455#41—521no—
456#41—521no—
457#41—521no—
458#41—521no—
459#41—521no—
460#41—521no—
461#41—521no—
462C2×Q8—132no—
463C2×Q8—261no—
464C2×Q8—261no—
465C2×Q8—261no—
466C2×Q8—261no—
467C2×Q8—261no—
468C2×D4—132no—
469C2×D4—262no—
470C2×D4—262no—
471C2×D4—261no—
472C2×D4—261no—
473C2×D4—261no—
474C2×D4—261no—
475C2×D4—261no—
476C2×D4—521no—
477C2×D4—521no—
478C2×D4—521no—
479C2×D4—521no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—42no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—132no—
487C22×C4—132no—
488C22×C4—264no—
489C22×C4—264no—
490C22×C4—261no—
491C22×C4—262no—
492C22×C4—262no—
493C22×C4—262no—
494C22×C4—262no—
495C22×C4—262no—
496C22×C4—522no—
497C22×C4—522no—
498C22×C4—521no—
499C22×C4—521no—
500C22×C4—521no—
501C22×C4—521no—
502C22×C4—521no—
503C22×C4—521no—
504C22×C4C2×C4×D1311yescharacteristic
505C22×C4C2×C4×D1311yescharacteristic
506C22×C4C2×C4×D1311yescharacteristic
507C22×C4C2×D5211yescharacteristic
508C22×C4#612711yescharacteristic
509C22×C4#612711yescharacteristic
510C22×C4D4×D1311yescharacteristic
511C22×C4D4×D1311yescharacteristic
512C22×C4D4×D1311yescharacteristic
513C22×C4#612911yescharacteristic
514C22×C4#612911yescharacteristic
515C22×C4#613111yescharacteristic
516C22×C4#613111yescharacteristic
517C22×C4C2×C22⋊D1311yescharacteristic
518M16—22no—
519M16—22no—
520M16—22no—
521M16—134no—
522M16—262no—
523M16—262no—
524M16—262no—
525M16#610314yescomplemented
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—262no—
530C2×C8—262no—
531C2×C8—262no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—524no—
537C22⋊C4—524no—
538C42—22no—
539C42—22no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—262no—
545C42—261no—
546C42—261no—
547C42—262no—
548C42—261no—
549C42—261no—
550C42—262no—
551C42—262no—
552C42—521no—
553C42—521no—
554C13#1093311yescharacteristic, complemented, Sylow
555C23—22no—
556C23—22no—
557C23—262no—
558C23—522no—
559C23—522no—
560C23—521no—
561C23—521no—
562C23C22⋊C4×D1311yescharacteristic
563C23#8813511yescharacteristic
564C23#8813811yescharacteristic
565C23#8813911yescharacteristic
566C23#8814711yescharacteristic
567C23#8814911yescharacteristic
568C23#8819411yescharacteristic
569Q8—262no—
570Q8—262no—
571Q8—521no—
572Q8—521no—
573D4—262no—
574D4—262no—
575D4—522no—
576D4—522no—
577D4—522no—
578D4—522no—
579D4—521no—
580D4—521no—
581C2×C4—24no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—134no—
590C2×C4—262no—
591C2×C4—262no—
592C2×C4—264no—
593C2×C4—262no—
594C2×C4—262no—
595C2×C4—262no—
596C2×C4—262no—
597C2×C4—524no—
598C2×C4—524no—
599C2×C4—524no—
600C2×C4—522no—
601C2×C4—524no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—522no—
605C2×C4—522no—
606C2×C4—522no—
607C2×C4—522no—
608C2×C4—522no—
609C2×C4C42×D1311yescharacteristic
610C2×C4#8812611yescharacteristic
611C2×C4C4×D5211yescharacteristic
612C2×C4C4×D5211yescharacteristic
613C2×C4C22⋊C4×D1311yescharacteristic
614C2×C4#8813511yescharacteristic
615C2×C4#8813511yescharacteristic
616C2×C4#8813511yescharacteristic
617C2×C4#8813611yescharacteristic
618C2×C4#8813811yescharacteristic
619C2×C4#8813911yescharacteristic
620C2×C4#8814011yescharacteristic
621C2×C4#8814611yescharacteristic
622C2×C4#8814611yescharacteristic
623C2×C4#8814711yescharacteristic
624C2×C4#8814711yescharacteristic
625C2×C4#8814711yescharacteristic
626C2×C4#8814811yescharacteristic
627C2×C4#8814811yescharacteristic
628C2×C4#8814911yescharacteristic
629C2×C4#8815211yescharacteristic
630C2×C4#8815211yescharacteristic
631C2×C4C2×#610311yescharacteristic
632C2×C4C4×C22⋊D1311yescharacteristic
633C2×C4C4×C22⋊D1311yescharacteristic
634C2×C4#8819311yescharacteristic
635C2×C4#8819311yescharacteristic
636C2×C4#8820111yescharacteristic
637C8—24no—
638C8—264no—
639C22—22no—
640C22—22no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—22no—
647C22—522no—
648C22—522no—
649C22—522no—
650C22—1042no—
651C22—1042no—
652C22#1171832611yescharacteristic
653C22#1171832711yescharacteristic
654C22#1171833011yescharacteristic
655C22#1171833111yescharacteristic
656C22D13×#10412yes—
657C22#1171848212yes—
658C22#1171861811yescharacteristic
659C22#1171861811yescharacteristic
660C22#1171864911yescharacteristic
661C4—44no—
662C4—82no—
663C4—264no—
664C4—524no—
665C4—522no—
666C4—522no—
667C4#1171832611yescharacteristic
668C4#1171832711yescharacteristic
669C4#1171833011yescharacteristic
670C4#1171833111yescharacteristic
671C4C4×#610311yescharacteristic
672C4#1171861911yescharacteristic
673C4#1171865111yescharacteristic
674C4#1171865111yescharacteristic
675C2—21no—
676C2—21no—
677C2—21no—
678C2—21no—
679C2—1042no—
680C2#1311307211yescharacteristic
681C2#1311464011yescharacteristic
682C2#1311464511yescharacteristic
683C2#1311513712yes—
684C2#1311575312yes—
subgroupG/H
centerC22×C4D4×D13
commutatorC2×C4×C13C2×C42
FrattiniC23×C4C22×D13
FittingC2×C13×#339C2
socleC23×C13C2×C22⋊C4
prime radicalC22×#6127C22
cosocleC23×C4×C13C23
hypercenterC2×#339D13
hypocenterC13#10933
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
3. C2×M16×Dic13C2semidirect
5. C2×#11718124C2semidirect
6. C4×#88071C2semidirect
7. C2×C13×#339C2semidirect
18. C2×#88065C4semidirect
23. #11718649C4semidirect
24. #11718635C4semidirect
30. C2×#88045C4semidirect
180. C13×M16C22⋊C4semidirect
209. #6099C22⋊C4semidirect
218. C2×#339D13semidirect
289. C2×Dic13#104semidirect
525. M16#6103semidirect
554. C13#10933semidirect

Maximal subgroups

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

subgroupindex
C22×#880452
C22×#880652
C2×M16×Dic132
#131173242
C2×#117181242
C4×#880712
C2×C13×#3392
#1093313