Finite Group Databasebeta

← #15769243

Subgroups — #15769243

Order 3328.

1066 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718868C211yesmaximal, characteristic
2#13123254C211yesmaximal, characteristic, complemented
3#13117219C211yesmaximal, characteristic
4C2×#11718151C211yesmaximal, characteristic, complemented
5C2×#11718148C211yesmaximal, characteristic, complemented
6#13114574C211yesmaximal, characteristic, complemented
7C2×C13×#407C211yesmaximal, characteristic
8C24×C4×C13C2211yescharacteristic
9C22×C13×M16C2211yescharacteristic
10C22×C13×M16C2211yescharacteristic
11C2×#88180C2211yescharacteristic
12C2×#88180C2211yescharacteristic
13C22×#6099C2211yescharacteristic
14C22×#6099C2211yescharacteristic
15C13×#407—22no—
16C13×#407C414yescomplemented
17C2×C13×C22⋊C8—22no—
18#11718908C412yes—
19#11718873—22no—
20#11718872C412yes—
21C4×C22⋊Dic13—22no—
22#11718868—22no—
23#11718868C414yescomplemented
24C2×C22⋊#1055—22no—
25#11718783—22no—
26M16×Dic13—22no—
27#11718762—22no—
28#11718151—22no—
29#11718148—22no—
30#11718146—22no—
31C25×C13C2×C411yescharacteristic
32C23×C4×C13—21no—
33C23×C4×C13—21no—
34C23×C4×C13—21no—
35C23×C4×C13—21no—
36C23×C4×C13—21no—
37C23×C4×C13—21no—
38C23×C4×C13—21no—
39C23×C4×C13—21no—
40C23×C4×C13—21no—
41C23×C4×C13—21no—
42C23×C4×C13—21no—
43C23×C4×C13—21no—
44C23×C4×C13C2×C411yescharacteristic
45C23×C4×C13C2×C412yes—
46C23×C4×C13C2×C412yes—
47C23×C4×C13C2311yescharacteristic
48C2×C13×M16—21no—
49C2×C13×M16—21no—
50C2×C13×M16—21no—
51C2×C13×M16—21no—
52C2×C13×M16—21no—
53C2×C13×M16—21no—
54C2×C13×M16—21no—
55C2×C13×M16—21no—
56C2×C13×M16—21no—
57C2×C13×M16C2×C412yes—
58C2×C13×M16C2×C412yes—
59C2×C13×M16D412yes—
60C22×C8×C13—21no—
61C22×C8×C13D412yes—
62C2×C22⋊Dic13—24no—
63C2×C22⋊Dic13C2×C412yes—
64C2×C22⋊Dic13C2×C412yes—
65#88180—21no—
66#88180—21no—
67#88180—21no—
68#88180—21no—
69#88180—21no—
70#88180—21no—
71#88180—21no—
72#88180—21no—
73C2×C4⋊Dic13—21no—
74C2×C4⋊Dic13D412yes—
75C2×C4×Dic13—21no—
76C2×C4×Dic13D412yes—
77C2×#6099—21no—
78C2×#6099—21no—
79C2×#6099—21no—
80C2×#6099—21no—
81C2×#6099—21no—
82C2×#6099—21no—
83C2×#6099—21no—
84C2×#6099—21no—
85C2×#6099—21no—
86C2×#6099C2×C412yes—
87C2×#6099C2×C412yes—
88C2×#6099D412yes—
89C22×#1055—21no—
90C22×#1055D412yes—
91C13×C22⋊C8—28no—
92#88071—24no—
93C22⋊#1055—28no—
94#88055—24no—
95C8×Dic13—24no—
96#10937—131nomaximal, Sylow
97C24×C13—21no—
98C24×C13—21no—
99C24×C13—21no—
100C24×C13—21no—
101C24×C13—21no—
102C24×C13—21no—
103C24×C13—44no—
104C24×C13C4212yes—
105C24×C13C22×C411yescharacteristic
106C22×C4×C13—21no—
107C22×C4×C13—21no—
108C22×C4×C13—21no—
109C22×C4×C13—21no—
110C22×C4×C13—21no—
111C22×C4×C13—21no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—41no—
125C22×C4×C13—41no—
126C22×C4×C13—41no—
127C22×C4×C13—41no—
128C22×C4×C13—41no—
129C22×C4×C13—41no—
130C22×C4×C13—41no—
131C22×C4×C13—41no—
132C22×C4×C13—41no—
133C22×C4×C13—41no—
134C22×C4×C13—41no—
135C22×C4×C13—41no—
136C22×C4×C13—41no—
137C22×C4×C13—41no—
138C22×C4×C13—41no—
139C22×C4×C13—41no—
140C22×C4×C13C4212yes—
141C22×C4×C13C22×C411yescharacteristic
142C22×C4×C13C22×C411yescharacteristic
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13#4111yescharacteristic
150C22×C4×C13#4111yescharacteristic
151C22×C4×C13#4111yescharacteristic
152C22×C4×C13#4111yescharacteristic
153C22×C4×C13#4111yescharacteristic
154C22×C4×C13#4111yescharacteristic
155C22×Dic13—22no—
156C22×Dic13C22⋊C414yes—
157C13×M16—22no—
158C13×M16—22no—
159C13×M16—22no—
160C13×M16—22no—
161C13×M16—22no—
162C13×M16—22no—
163C13×M16—41no—
164C13×M16—41no—
165C2×C8×C13—22no—
166C2×C8×C13—22no—
167C2×C8×C13—22no—
168C2×C8×C13—41no—
169C2×C8×C13—41no—
170C2×C8×C13C22⋊C414yescomplemented
171C22⋊Dic13—22no—
172C22⋊Dic13—22no—
173C22⋊Dic13—22no—
174C22⋊Dic13—22no—
175C22⋊Dic13—48no—
176C4⋊Dic13—22no—
177C4⋊Dic13—22no—
178C4⋊Dic13—41no—
179C4⋊Dic13—41no—
180C4×Dic13—22no—
181C4×Dic13—22no—
182C4×Dic13—41no—
183C4×Dic13—41no—
184#6099—22no—
185#6099—22no—
186#6099—22no—
187#6099—22no—
188#6099—22no—
189#6099—22no—
190#6099—41no—
191#6099—41no—
192C2×#1055—22no—
193C2×#1055—22no—
194C2×#1055—22no—
195C2×#1055—41no—
196C2×#1055—41no—
197C2×#1055C22⋊C414yescomplemented
198C2×#407—131no—
199C2×#407D1311yescharacteristic
200#2873—131no—
201#2280—131no—
202C2×#344—131no—
203C2×#344—131no—
204#1742—131no—
205C23×C13—24no—
206C23×C13—24no—
207C23×C13—24no—
208C23×C13—22no—
209C23×C13—22no—
210C23×C13—22no—
211C23×C13—44no—
212C23×C13—42no—
213C23×C13—42no—
214C23×C13—42no—
215C23×C13—42no—
216C23×C13—44no—
217C23×C13—44no—
218C23×C13—44no—
219C23×C13—41no—
220C23×C13—41no—
221C23×C13—41no—
222C23×C13—41no—
223C23×C13C2×C4211yescharacteristic
224C23×C13C2×C22⋊C411yescharacteristic
225C23×C13#11711yescharacteristic
226C23×C13C4×D411yescharacteristic
227C23×C13C4×D411yescharacteristic
228C23×C13C4×D411yescharacteristic
229C23×C13C4×D411yescharacteristic
230C2×C4×C13—22no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13—41no—
237C2×C4×C13—42no—
238C2×C4×C13—42no—
239C2×C4×C13—42no—
240C2×C4×C13—42no—
241C2×C4×C13—42no—
242C2×C4×C13—42no—
243C2×C4×C13—42no—
244C2×C4×C13—42no—
245C2×C4×C13C2×C22⋊C411yescharacteristic
246C2×C4×C13C2×C22⋊C411yescharacteristic
247C2×C4×C13#11711yescharacteristic
248C2×C4×C13#11711yescharacteristic
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×C13#12011yescharacteristic
258C2×C4×C13#12111yescharacteristic
259C2×C4×C13#12111yescharacteristic
260C2×C4×C13#12111yescharacteristic
261C2×C4×C13#12111yescharacteristic
262C2×C4×C13#12111yescharacteristic
263C2×C4×C13#12111yescharacteristic
264C2×C4×C13#12311yescharacteristic
265C2×C4×C13#12311yescharacteristic
266C2×C4×C13#12311yescharacteristic
267C2×C4×C13#12411yescharacteristic
268C2×C4×C13#12411yescharacteristic
269C2×C4×C13#12411yescharacteristic
270C2×C4×C13#12611yescharacteristic
271C2×C4×C13#12611yescharacteristic
272C2×C4×C13#12711yescharacteristic
273C2×Dic13—24no—
274C2×Dic13—44no—
275C2×Dic13—44no—
276C2×Dic13—42no—
277C8×C13—24no—
278C8×C13—42no—
279#1055—24no—
280#1055—42no—
281C24×C4C2×D1311yescharacteristic
282C22×M16—131no—
283C22×M16—131no—
284C22×M16C2×D1311yescharacteristic
285C22×M16C2×D1311yescharacteristic
286C2×#117—131no—
287C2×#117—131no—
288#407—22no—
289#407—134no—
290#407—262no—
291#407Dic1314yescomplemented
292C2×C22⋊C8—22no—
293C2×C22⋊C8—262no—
294#405—262no—
295#405—262no—
296C4×M16—262no—
297#381—132no—
298#380—262no—
299#379—132no—
300C4×C22⋊C4—262no—
301#344—262no—
302#344—262no—
303#343—262no—
304C22×C13—22no—
305C22×C13—22no—
306C22×C13—22no—
307C22×C13—24no—
308C22×C13—24no—
309C22×C13—24no—
310C22×C13—41no—
311C22×C13—41no—
312C22×C13—41no—
313C22×C13—41no—
314C22×C13—44no—
315C22×C13—42no—
316C22×C13—42no—
317C22×C13—42no—
318C22×C13—44no—
319C22×C13—44no—
320C22×C13—44no—
321C22×C13—42no—
322C22×C13C4×C22⋊C411yescharacteristic
323C22×C13C4×C22⋊C411yescharacteristic
324C22×C13C4×C22⋊C411yescharacteristic
325C22×C13#38611yescharacteristic
326C22×C13#38811yescharacteristic
327C22×C13#38811yescharacteristic
328C22×C13#39011yescharacteristic
329C4×C13—42no—
330C4×C13#38611yescharacteristic
331C4×C13#38611yescharacteristic
332C4×C13#38811yescharacteristic
333C4×C13#38811yescharacteristic
334C4×C13#38811yescharacteristic
335C4×C13#38811yescharacteristic
336C4×C13#39011yescharacteristic
337C4×C13#39011yescharacteristic
338Dic13—44no—
339Dic13—82no—
340C25C4×D1311yescharacteristic
341C23×C4—21no—
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—21no—
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—21no—
351C23×C4—21no—
352C23×C4—21no—
353C23×C4C4×D1311yescharacteristic
354C23×C4C4×D1312yes—
355C23×C4C2×Dic1312yes—
356C23×C4C22×D1311yescharacteristic
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—132no—
367C2×M16—132no—
368C2×M16—132no—
369C2×M16—261no—
370C2×M16—261no—
371C2×M16—261no—
372C2×M16—261no—
373C2×M16—261no—
374C2×M16—261no—
375C2×M16—261no—
376C2×M16—261no—
377C2×M16—261no—
378C2×M16C2×Dic1312yes—
379C2×M16C2×Dic1312yes—
380C2×M16C22⋊D1312yes—
381C22×C8—21no—
382C22×C8—132no—
383C22×C8—261no—
384C22×C8C22⋊D1312yes—
385#117—261no—
386#117—261no—
387#117—261no—
388#117—261no—
389#117—261no—
390#117—261no—
391#117—261no—
392#117—261no—
393C2×C4⋊C4—132no—
394C2×C4⋊C4—261no—
395C2×C22⋊C4—132no—
396C2×C22⋊C4—132no—
397C2×C22⋊C4—264no—
398C2×C42—132no—
399C2×C42—261no—
400C22⋊C8—28no—
401C22⋊C8—268no—
402#97—264no—
403C4×C8—264no—
404#95—264no—
405C2×C13—21no—
406C2×C13—21no—
407C2×C13—21no—
408C2×C13—21no—
409C2×C13—21no—
410C2×C13—21no—
411C2×C13—44no—
412C2×C13#143711yescharacteristic
413C2×C13#189112yes—
414C24—21no—
415C24—21no—
416C24—21no—
417C24—21no—
418C24—21no—
419C24—21no—
420C24—44no—
421C24C4×Dic1312yes—
422C24C2×C4×D1311yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—134no—
458C22×C4—262no—
459C22×C4C4×Dic1312yes—
460C22×C4C2×C4×D1311yescharacteristic
461C22×C4#612711yescharacteristic
462C22×C4#612711yescharacteristic
463C22×C4D4×D1311yescharacteristic
464C22×C4D4×D1311yescharacteristic
465C22×C4D4×D1311yescharacteristic
466C22×C4D4×D1311yescharacteristic
467C22×C4#612911yescharacteristic
468C22×C4#612911yescharacteristic
469C22×C4#612911yescharacteristic
470C22×C4#612911yescharacteristic
471C22×C4C22×Dic1311yescharacteristic
472C22×C4C2×C22⋊D1311yescharacteristic
473C22×C4C2×C22⋊D1311yescharacteristic
474M16—22no—
475M16—22no—
476M16—22no—
477M16—22no—
478M16—22no—
479M16—22no—
480M16—41no—
481M16—41no—
482M16—262no—
483M16—262no—
484M16—262no—
485M16—262no—
486M16—262no—
487M16—262no—
488M16—521no—
489M16—521no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—134no—
496C2×C8—262no—
497C2×C8—262no—
498C2×C8—262no—
499C2×C8—521no—
500C2×C8—521no—
501C2×C8C22⋊Dic1314yescomplemented
502C4⋊C4—262no—
503C4⋊C4—262no—
504C4⋊C4—521no—
505C4⋊C4—521no—
506C22⋊C4—262no—
507C22⋊C4—262no—
508C22⋊C4—262no—
509C22⋊C4—262no—
510C22⋊C4—528no—
511C42—262no—
512C42—262no—
513C42—521no—
514C42—521no—
515C13#1093711yescharacteristic, complemented, Sylow
516C23—24no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23#8813111yescharacteristic
535C23C22⋊C4×D1311yescharacteristic
536C23#8813511yescharacteristic
537C23#8813511yescharacteristic
538C23C2×C4×Dic1311yescharacteristic
539C23C4×C22⋊D1311yescharacteristic
540C23C4×C22⋊D1311yescharacteristic
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—264no—
557C2×C4—524no—
558C2×C4—524no—
559C2×C4—522no—
560C2×C4#8813111yescharacteristic
561C2×C4#8813311yescharacteristic
562C2×C4#8813311yescharacteristic
563C2×C4C22⋊C4×D1311yescharacteristic
564C2×C4#8813511yescharacteristic
565C2×C4#8813511yescharacteristic
566C2×C4#8813711yescharacteristic
567C2×C4#8813711yescharacteristic
568C2×C4#8813811yescharacteristic
569C2×C4#8813811yescharacteristic
570C2×C4#8813911yescharacteristic
571C2×C4#8813911yescharacteristic
572C2×C4#8818011yescharacteristic
573C2×C4C4×C22⋊D1311yescharacteristic
574C2×C4C4×C22⋊D1311yescharacteristic
575C2×C4D4×Dic1311yescharacteristic
576C2×C4D4×Dic1311yescharacteristic
577C2×C4D4×Dic1311yescharacteristic
578C2×C4D4×Dic1311yescharacteristic
579C2×C4#8818911yescharacteristic
580C2×C4#8819011yescharacteristic
581C2×C4#8819111yescharacteristic
582C2×C4#8819211yescharacteristic
583C2×C4#8819211yescharacteristic
584C2×C4#8819311yescharacteristic
585C2×C4#8819311yescharacteristic
586C2×C4#8819411yescharacteristic
587C2×C4C2×C22⋊Dic1311yescharacteristic
588C8—24no—
589C8—42no—
590C8—264no—
591C8—522no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—44no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22C22⋊C4×Dic1311yescharacteristic
611C22C22⋊C4×Dic1311yescharacteristic
612C22#1171860511yescharacteristic
613C22#1171860611yescharacteristic
614C22#1171861711yescharacteristic
615C22#1171861811yescharacteristic
616C22C4×C22⋊Dic1311yescharacteristic
617C4—42no—
618C4—524no—
619C4—1042no—
620C4#1171860511yescharacteristic
621C4#1171860611yescharacteristic
622C4#1171861211yescharacteristic
623C4#1171861211yescharacteristic
624C4#1171861711yescharacteristic
625C4#1171861811yescharacteristic
626C4#1171888111yescharacteristic
627C4#1171888211yescharacteristic
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—44no—
635C2#1311307311yescharacteristic
636C2#1311525112yes—
subgroupG/H
centerC2×C4#88191
commutatorC23×C13C2×C42
FrattiniC23×C4C22×D13
FittingC2×C13×#407C2
socleC22×C13#386
prime radicalC25×C13C2×C4
cosocleC23×C4×C13C23
hypercenterC2×#407D13
hypocenterC13#10937
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. #13123254C2semidirect
4. C2×#11718151C2semidirect
5. C2×#11718148C2semidirect
6. #13114574C2semidirect
16. C13×#407C4semidirect
23. #11718868C4semidirect
170. C2×C8×C13C22⋊C4semidirect
197. C2×#1055C22⋊C4semidirect
291. #407Dic13semidirect
501. C2×C8C22⋊Dic13semidirect
515. C13#10937semidirect

Maximal subgroups

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

subgroupindex
C2×#117188682
#131232542
#131172192
C2×#117181512
C2×#117181482
#131145742
C2×C13×#4072
#1093713