Finite Group Databasebeta

← #15794208

Subgroups — #15794208

Order 3328.

2734 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88200C211yesmaximal, characteristic
2C22×C42×D13C211yesmaximal, characteristic
3C2×#11718891C211yesmaximal, characteristic, complemented
4C2×#11718648C211yesmaximal, characteristic
5#13113067C214yesmaximal, complemented
6#13113066C211yesmaximal, characteristic, complemented
7#13113065C211yesmaximal, characteristic, complemented
8#13113057C214yesmaximal
9C2×C13×#391C211yesmaximal, characteristic, complemented
10C23×Q8×C13C2211yescharacteristic
11C23×C4×D13C2212yes—
12C23×C4×D13C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C22×C42×C13C2211yescharacteristic
15C2×#88200—24no—
16C2×#88200—24no—
17C2×#88200C418yescomplemented
18C22×#6103C2212yes—
19C22×#6103C2212yes—
20C22×C4⋊Dic13C2211yescharacteristic
21C22×C4⋊Dic13C2211yescharacteristic
22C22×#6101C2212yes—
23C22×C4×Dic13C2212yes—
24C22×C4×Dic13C2211yescharacteristic
25C2×C42×D13—24no—
26C2×C42×D13—24no—
27C2×C42×D13—24no—
28C13×#391—24no—
29C2×C13×#95C2214yes—
30#11718891—24no—
31C2×#88071C2212yes—
32C2×#88071C2214yes—
33C2×#88071C2214yes—
34C2×#88071C2214yes—
35#11718648—24no—
36C24×D13C2311yescharacteristic
37C22×Q8×C13—22no—
38C22×Q8×C13—22no—
39C22×Q8×C13—22no—
40C22×Q8×C13—22no—
41C22×Q8×C13—22no—
42C22×Q8×C13—22no—
43C22×Q8×C13C2×C414yescomplemented
44C23×C4×C13C2312yes—
45C23×C4×C13C2312yes—
46C23×C4×C13C2312yes—
47C23×C4×C13C2311yescharacteristic
48C23×Dic13C2312yes—
49C23×Dic13C2312yes—
50C23×Dic13C2311yescharacteristic
51C23×Dic13C2312yes—
52C22×C4×D13—24no—
53C22×C4×D13—24no—
54C22×C4×D13—22no—
55C22×C4×D13—24no—
56C22×C4×D13—22no—
57C22×C4×D13—22no—
58C22×C4×D13—22no—
59C22×C4×D13—24no—
60C22×C4×D13—24no—
61C22×C4×D13—24no—
62C22×C4×D13C2×C414yes—
63C22×C4×D13D414yes—
64C22×C4×D13D414yes—
65C22×C4×D13Q818yescomplemented
66C2×C13×C4⋊C4—22no—
67C2×C13×C4⋊C4—22no—
68C2×C13×C4⋊C4—22no—
69C2×C42×C13—22no—
70C2×C42×C13—22no—
71C2×C42×C13D414yes—
72#88200—232no—
73C2×#6103—24no—
74C2×#6103—24no—
75C2×#6103—24no—
76C2×#6103—28no—
77C2×#6103C2×C418yes—
78C2×C4⋊Dic13—22no—
79C2×C4⋊Dic13—22no—
80C2×C4⋊Dic13—22no—
81C2×C4⋊Dic13—22no—
82C2×C4⋊Dic13—22no—
83C2×C4⋊Dic13C2×C414yescomplemented
84C2×#6101—24no—
85C2×#6101—24no—
86C2×#6101C2×C418yescomplemented
87C2×C4×Dic13—22no—
88C2×C4×Dic13—24no—
89C2×C4×Dic13—22no—
90C2×C4×Dic13—24no—
91C2×C4×Dic13—24no—
92C2×C4×Dic13D414yes—
93C42×D13—232no—
94C13×#95—28no—
95#88071—28no—
96#88071—28no—
97#88071—24no—
98#88071—28no—
99#14702—131nomaximal, Sylow
100C24×C13C2411yescharacteristic
101C23×D13—21no—
102C23×D13—21no—
103C23×D13—21no—
104C23×D13—21no—
105C23×D13—22no—
106C23×D13—22no—
107C23×D13C22×C412yes—
108C23×D13C2×D412yes—
109C23×D13C2×D412yes—
110C23×D13C2×Q814yes—
111C23×D13#4114yes—
112C2×Q8×C13—28no—
113C2×Q8×C13—28no—
114C2×Q8×C13—28no—
115C2×Q8×C13—44no—
116C2×Q8×C13—44no—
117C2×Q8×C13—44no—
118C2×Q8×C13—44no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13—22no—
129C22×C4×C13—22no—
130C22×C4×C13—22no—
131C22×C4×C13—24no—
132C22×C4×C13—24no—
133C22×C4×C13—22no—
134C22×C4×C13C22×C414yes—
135C22×C4×C13C22×C412yes—
136C22×C4×C13C2×D412yes—
137C22×C4×C13C2×D412yes—
138C22×C4×C13C2×D412yes—
139C22×C4×C13C2×D414yes—
140C22×C4×C13C2×D412yes—
141C22×C4×C13C2×Q814yes—
142C22×C4×C13#4112yes—
143C22×C4×C13#4112yes—
144C22×C4×C13#4114yes—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—21no—
150C22×Dic13—21no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—24no—
156C22×Dic13—24no—
157C22×Dic13—22no—
158C22×Dic13—22no—
159C22×Dic13—24no—
160C22×Dic13—22no—
161C22×Dic13—24no—
162C22×Dic13C22×C414yes—
163C22×Dic13C22×C412yes—
164C22×Dic13C2×D412yes—
165C22×Dic13C2×D412yes—
166C22×Dic13C2×D414yes—
167C22×Dic13C2×Q814yes—
168C22×Dic13#4114yes—
169C2×C4×D13—216no—
170C2×C4×D13—216no—
171C2×C4×D13—216no—
172C2×C4×D13—216no—
173C2×C4×D13—216no—
174C2×C4×D13—28no—
175C2×C4×D13—28no—
176C2×C4×D13—28no—
177C2×C4×D13—28no—
178C2×C4×D13—28no—
179C2×C4×D13—216no—
180C2×C4×D13—28no—
181C2×C4×D13—216no—
182C2×C4×D13C22⋊C4116yes—
183C13×C4⋊C4—44no—
184C42×C13—28no—
185#6103—216no—
186#6103—48no—
187C4⋊Dic13—28no—
188C4⋊Dic13—44no—
189#6101—216no—
190C4×Dic13—216no—
191C4×Dic13—28no—
192C22×C22⋊Q8—131no—
193C23×C42—131no—
194C2×#391—131no—
195C2×#391D1311yescharacteristic, complemented
196C2×#380—131no—
197#1436—134no—
198C4⋊#95—131no—
199C4⋊#95—131no—
200C22⋊#95—134no—
201C23×C13C23×C411yescharacteristic
202C23×C13C22×D411yescharacteristic
203C23×C13C22×D411yescharacteristic
204C23×C13C22×D411yescharacteristic
205C23×C13C22×D411yescharacteristic
206C23×C13C22×Q812yes—
207C23×C13C2×#4111yescharacteristic
208C23×C13C2×#4111yescharacteristic
209C23×C13C2×#4112yes—
210C23×C13C2×#4112yes—
211C23×C13C2×#4112yes—
212C22×D13—22no—
213C22×D13—22no—
214C22×D13—24no—
215C22×D13—22no—
216C22×D13—22no—
217C22×D13—22no—
218C22×D13—24no—
219C22×D13—24no—
220C22×D13—22no—
221C22×D13—24no—
222C22×D13—24no—
223C22×D13—24no—
224C22×D13—24no—
225C22×D13—24no—
226C22×D13—22no—
227C22×D13—22no—
228C22×D13—24no—
229C22×D13—24no—
230C22×D13C2×C22⋊C414yes—
231C22×D13C4×Q818yes—
232C22×D13C22⋊Q818yes—
233C22×D13#12414yes—
234C22×D13C4⋊Q814yes—
235Q8×C13—416no—
236C2×C4×C13—24no—
237C2×C4×C13—24no—
238C2×C4×C13—24no—
239C2×C4×C13—24no—
240C2×C4×C13—24no—
241C2×C4×C13—24no—
242C2×C4×C13—24no—
243C2×C4×C13—24no—
244C2×C4×C13—24no—
245C2×C4×C13—42no—
246C2×C4×C13—42no—
247C2×C4×C13—42no—
248C2×C4×C13—42no—
249C2×C4×C13—42no—
250C2×C4×C13—44no—
251C2×C4×C13—42no—
252C2×C4×C13—44no—
253C2×C4×C13—42no—
254C2×C4×C13C2×C22⋊C414yes—
255C2×C4×C13C4×D414yes—
256C2×C4×C13C4×D414yes—
257C2×C4×C13#12014yes—
258C2×C4×C13#12114yes—
259C2×C4×C13#12114yes—
260C2×C4×C13C22⋊Q818yescomplemented
261C2×C4×C13#12314yes—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—24no—
265C2×Dic13—22no—
266C2×Dic13—22no—
267C2×Dic13—22no—
268C2×Dic13—24no—
269C2×Dic13—24no—
270C2×Dic13—22no—
271C2×Dic13—24no—
272C2×Dic13—24no—
273C2×Dic13—24no—
274C2×Dic13—24no—
275C2×Dic13—28no—
276C2×Dic13—44no—
277C2×Dic13—42no—
278C2×Dic13—42no—
279C2×Dic13—42no—
280C2×Dic13—44no—
281C2×Dic13—42no—
282C2×Dic13—44no—
283C2×Dic13—42no—
284C2×Dic13C2×C22⋊C414yes—
285C2×Dic13C22⋊Q818yescomplemented
286C4×D13—264no—
287C4×D13—232no—
288C23×Q8C2×D1311yescharacteristic
289C24×C4—131no—
290C24×C4—132no—
291C2×C22⋊Q8—138no—
292C2×C22⋊Q8—264no—
293C2×C22⋊Q8—264no—
294C22×C4⋊C4—132no—
295C22×C4⋊C4—131no—
296C22×C4⋊C4—131no—
297C22×C4⋊C4C2×D1311yescharacteristic
298C22×C22⋊C4—132no—
299C22×C22⋊C4—132no—
300C22×C42—131no—
301C22×C42—132no—
302C22×C42—264no—
303C22×C42—264no—
304C22×C42—264no—
305C22×C42C2×D1311yescharacteristic
306#391—24no—
307#391—264no—
308#380—264no—
309C2×#95—134no—
310C2×#95—134no—
311C2×#95—134no—
312C2×#95—132no—
313C2×#95C2×D1314yes—
314C22×C13C22×C22⋊C411yescharacteristic
315C22×C13C2×C4×D411yescharacteristic
316C22×C13C2×C4×D411yescharacteristic
317C22×C13C2×C4×Q812yes—
318C22×C13C4×#4112yes—
319C22×C13C2×#12011yescharacteristic
320C22×C13C2×#12111yescharacteristic
321C22×C13C2×#12111yescharacteristic
322C22×C13C2×C22⋊Q812yes—
323C22×C13C2×C22⋊Q812yes—
324C22×C13C2×C22⋊Q812yes—
325C22×C13C2×#12311yescharacteristic
326C22×C13#52512yes—
327C22×C13#52512yes—
328C22×C13#52512yes—
329C22×C13C2×#12411yescharacteristic
330C22×C13#52912yes—
331C22×C13#52912yes—
332C22×C13C2×C4⋊Q811yescharacteristic
333C22×C13#53211yescharacteristic
334C22×C13#53211yescharacteristic
335C22×C13#53312yes—
336C22×C13#53312yes—
337C2×D13—28no—
338C2×D13—24no—
339C2×D13—24no—
340C2×D13—28no—
341C2×D13—24no—
342C2×D13—24no—
343C2×D13—24no—
344C2×D13—24no—
345C2×D13—28no—
346C2×D13—28no—
347C2×D13#39118yescomplemented
348C4×C13—28no—
349C4×C13—44no—
350C4×C13—82no—
351C4×C13#38618yes—
352Dic13—28no—
353Dic13—24no—
354Dic13—44no—
355Dic13—82no—
356C25—131no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—22no—
363C22×Q8C4×D1314yescomplemented
364C23×C4—131no—
365C23×C4—132no—
366C23×C4—134no—
367C23×C4—134no—
368C23×C4—134no—
369C23×C4—132no—
370C23×C4—138no—
371C23×C4—132no—
372C23×C4—264no—
373C23×C4—264no—
374C23×C4—264no—
375C23×C4—264no—
376C23×C4—262no—
377C23×C4—262no—
378C23×C4—262no—
379C23×C4—264no—
380C23×C4—264no—
381C23×C4—262no—
382C23×C4C22×D1312yes—
383C23×C4C22×D1312yes—
384C23×C4C22×D1312yes—
385C23×C4C22×D1311yescharacteristic
386C22⋊Q8—2632no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—138no—
391C2×C4⋊C4—134no—
392C2×C4⋊C4—264no—
393C2×C4⋊C4—264no—
394C2×C4⋊C4—262no—
395C2×C4⋊C4—262no—
396C2×C4⋊C4—262no—
397C2×C4⋊C4—262no—
398C2×C4⋊C4—262no—
399C2×C22⋊C4—138no—
400C2×C22⋊C4—264no—
401C2×C22⋊C4—264no—
402C2×C22⋊C4—268no—
403C2×C22⋊C4—264no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—134no—
407C2×C42—2632no—
408C2×C42—264no—
409C2×C42—264no—
410C2×C42—262no—
411C2×C42—264no—
412C2×C42—262no—
413C2×C42C22⋊D1314yes—
414#95—28no—
415#95—268no—
416#95—264no—
417#95—268no—
418#95—268no—
419C2×C13C2×#38611yescharacteristic
420C2×C13C2×#39111yescharacteristic
421C2×C13#229711yescharacteristic
422C2×C13C4×C22⋊Q812yes—
423C2×C13C4×C22⋊Q812yes—
424C2×C13#284812yes—
425C2×C13#285011yescharacteristic
426C2×C13#285011yescharacteristic
427C2×C13#285512yes—
428C2×C13#285711yescharacteristic
429C2×C13#285711yescharacteristic
430D13—28no—
431C24—132no—
432C24—132no—
433C24—132no—
434C24—134no—
435C24—134no—
436C24—261no—
437C24—261no—
438C24—261no—
439C24—262no—
440C24—262no—
441C24—261no—
442C24C23×D1311yescharacteristic
443C2×Q8—28no—
444C2×Q8—28no—
445C2×Q8—28no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—132no—
466C22×C4—134no—
467C22×C4—132no—
468C22×C4—132no—
469C22×C4—134no—
470C22×C4—134no—
471C22×C4—1316no—
472C22×C4—134no—
473C22×C4—2616no—
474C22×C4—2616no—
475C22×C4—2616no—
476C22×C4—2616no—
477C22×C4—2616no—
478C22×C4—2616no—
479C22×C4—261no—
480C22×C4—261no—
481C22×C4—262no—
482C22×C4—261no—
483C22×C4—262no—
484C22×C4—262no—
485C22×C4—261no—
486C22×C4—262no—
487C22×C4—268no—
488C22×C4—268no—
489C22×C4—268no—
490C22×C4—268no—
491C22×C4—268no—
492C22×C4—268no—
493C22×C4—2616no—
494C22×C4—262no—
495C22×C4—262no—
496C22×C4—264no—
497C22×C4—264no—
498C22×C4—262no—
499C22×C4—262no—
500C22×C4—262no—
501C22×C4—264no—
502C22×C4—264no—
503C22×C4C2×C4×D1314yes—
504C22×C4C2×C4×D1312yes—
505C22×C4#612712yes—
506C22×C4D4×D1312yes—
507C22×C4D4×D1312yes—
508C22×C4D4×D1312yes—
509C22×C4#612912yes—
510C22×C4Q8×D1314yes—
511C22×C4#613114yes—
512C22×C4C2×C22⋊D1314yes—
513C22×C4C2×C22⋊D1312yes—
514C4⋊C4—44no—
515C4⋊C4—2616no—
516C4⋊C4—268no—
517C4⋊C4—524no—
518C22⋊C4—2616no—
519C22⋊C4—528no—
520C42—28no—
521C42—2616no—
522C42—268no—
523C13#1470211yescharacteristic, complemented, Sylow
524C23—134no—
525C23—134no—
526C23—138no—
527C23—138no—
528C23—134no—
529C23—262no—
530C23—264no—
531C23—262no—
532C23—262no—
533C23—262no—
534C23—262no—
535C23—262no—
536C23—262no—
537C23—264no—
538C23—264no—
539C23—264no—
540C23—264no—
541C23—264no—
542C23—264no—
543C23—264no—
544C23—264no—
545C23—262no—
546C23—264no—
547C23C22×C4×D1311yescharacteristic
548C23C2×#612711yescharacteristic
549C23C2×D4×D1311yescharacteristic
550C23C2×D4×D1311yescharacteristic
551C23C2×D4×D1311yescharacteristic
552C23C2×#612911yescharacteristic
553C23C2×Q8×D1312yes—
554C23C2×#613112yes—
555C23#41×D1312yes—
556C23#41×D1312yes—
557C23C22×C22⋊D1311yescharacteristic
558Q8—416no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—134no—
578C2×C4—138no—
579C2×C4—2664no—
580C2×C4—262no—
581C2×C4—262no—
582C2×C4—262no—
583C2×C4—262no—
584C2×C4—264no—
585C2×C4—262no—
586C2×C4—262no—
587C2×C4—264no—
588C2×C4—264no—
589C2×C4—264no—
590C2×C4—264no—
591C2×C4—264no—
592C2×C4—2632no—
593C2×C4—264no—
594C2×C4—268no—
595C2×C4—522no—
596C2×C4—524no—
597C2×C4—522no—
598C2×C4—522no—
599C2×C4—522no—
600C2×C4—524no—
601C2×C4—524no—
602C2×C4—522no—
603C2×C4C22⋊C4×D1314yes—
604C2×C4#8813514yes—
605C2×C4#8813714yes—
606C2×C4#8813814yes—
607C2×C4C4×C22⋊D1314yes—
608C2×C4#8819114yes—
609C2×C4#8819214yes—
610C2×C4#8820018yescomplemented
611C22—138no—
612C22—268no—
613C22—264no—
614C22—264no—
615C22—264no—
616C22—268no—
617C22—264no—
618C22—264no—
619C22—264no—
620C22—268no—
621C22—268no—
622C22C2×C22⋊C4×D1311yescharacteristic
623C22C2×#8813511yescharacteristic
624C22C2×#8813711yescharacteristic
625C22C2×#8813811yescharacteristic
626C22C4×Q8×D1312yes—
627C22C4×#613112yes—
628C22#1171928312yes—
629C22#1171928512yes—
630C22D13×C22⋊Q812yes—
631C22D13×C22⋊Q812yes—
632C22#1171933212yes—
633C22#1171933212yes—
634C22#1171937311yescharacteristic
635C22D13×#12411yescharacteristic
636C22#1171938512yes—
637C22D13×C4⋊Q811yescharacteristic
638C22#1171943011yescharacteristic
639C22#1171943312yes—
640C22C2×C4×C22⋊D1311yescharacteristic
641C22C2×#8819111yescharacteristic
642C22C2×#8819211yescharacteristic
643C22C2×#8820012yes—
644C22#1171953312yes—
645C4—28no—
646C4—44no—
647C4—82no—
648C4—268no—
649C4—264no—
650C4—524no—
651C4—1042no—
652C4#1171861718yes—
653C2—268no—
654C2C2×#1171861711yescharacteristic
655C2D13×#39111yescharacteristic
656C2#1311732511yescharacteristic
657C2C4×#8820012yes—
658C2#1311737412yes—
659C2#1312312012yes—
660C2#1312313211yescharacteristic
661C2#1312314011yescharacteristic
662C2#1312317712yes—
663C2#1312318311yescharacteristic
664C2#1312318511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC2×C13×#391C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenterC2×#391D13
hypocenterC13#14702
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C2×#11718891C2semidirect
5. #13113067C2semidirect
6. #13113066C2semidirect
7. #13113065C2semidirect
9. C2×C13×#391C2semidirect
17. C2×#88200C4semidirect
43. C22×Q8×C13C2×C4semidirect
65. C22×C4×D13Q8semidirect
83. C2×C4⋊Dic13C2×C4semidirect
86. C2×#6101C2×C4semidirect
195. C2×#391D13semidirect
260. C2×C4×C13C22⋊Q8semidirect
285. C2×Dic13C22⋊Q8semidirect
347. C2×D13#391semidirect
363. C22×Q8C4×D13semidirect
523. C13#14702semidirect
610. C2×C4#88200semidirect

Maximal subgroups

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

subgroupindex
C22×#882002
C22×C42×D132
C2×#117188912
C2×#117186482
#131130672
#131130662
#131130652
#131130572
C2×C13×#3912
#1470213