Finite Group Databasebeta

← #15786001

Subgroups — #15786001

Order 3328.

1918 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718871C211yesmaximal, characteristic
2C2×#11718604C211yesmaximal, characteristic
3C2×#11718604C211yesmaximal, characteristic
4C2×C22⋊C4×Dic13C211yesmaximal, characteristic
5C22×#88071C211yesmaximal, characteristic
6#13113097C212yesmaximal, complemented
7#13113094C212yesmaximal, complemented
8#13113067C212yesmaximal, complemented
9#13113061C212yesmaximal, complemented
10#13113058C211yesmaximal, characteristic
11C13×C22⋊#95C211yesmaximal, characteristic
12C24×C4×C13C2211yescharacteristic
13C24×Dic13C2211yescharacteristic
14C24×Dic13C2211yescharacteristic
15C22×C13×C22⋊C4C2211yescharacteristic
16C22×C13×C22⋊C4C2211yescharacteristic
17C22×C22⋊Dic13C2211yescharacteristic
18C22×C22⋊Dic13C2211yescharacteristic
19C22×#6101C2212yes—
20C22×C4×Dic13C2211yescharacteristic
21C22×C4×Dic13C2211yescharacteristic
22C2×C13×#95C2212yes—
23C2×C13×#95C2212yes—
24#11718871—24no—
25C2×#88071—24no—
26C2×#88071C2211yescharacteristic
27C2×#88071C2212yes—
28C2×#88071C2212yes—
29C2×#88071C2211yescharacteristic
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36C2×#88071C2212yes—
37#11718604—24no—
38#11718604C418yescomplemented
39C22⋊C4×Dic13—24no—
40C25×C13C2311yescharacteristic
41C23×C4×C13—22no—
42C23×C4×C13—24no—
43C23×C4×C13—22no—
44C23×C4×C13—22no—
45C23×C4×C13—42no—
46C23×C4×C13C2312yes—
47C23×C4×C13C2311yescharacteristic
48C23×C4×C13C2311yescharacteristic
49C23×C4×C13C2312yes—
50C23×Dic13—22no—
51C23×Dic13—22no—
52C23×Dic13—22no—
53C23×Dic13—22no—
54C23×Dic13—22no—
55C23×Dic13—22no—
56C23×Dic13—22no—
57C23×Dic13—22no—
58C23×Dic13—24no—
59C23×Dic13—42no—
60C23×Dic13C2×C414yes—
61C23×Dic13D414yes—
62C23×Dic13C2312yes—
63C23×Dic13C2312yes—
64C23×Dic13C2311yescharacteristic
65C23×Dic13C2311yescharacteristic
66C23×Dic13C2311yescharacteristic
67C23×Dic13C2311yescharacteristic
68C2×C13×C22⋊C4—22no—
69C2×C13×C22⋊C4—22no—
70C2×C13×C22⋊C4—22no—
71C2×C13×C22⋊C4—22no—
72C2×C13×C22⋊C4—22no—
73C2×C13×C22⋊C4C2×C414yes—
74C2×C22⋊Dic13—22no—
75C2×C22⋊Dic13—22no—
76C2×C22⋊Dic13—22no—
77C2×C22⋊Dic13—22no—
78C2×C22⋊Dic13C2×C414yes—
79C2×C22⋊Dic13D414yes—
80C2×#6101—24no—
81C2×#6101—24no—
82C2×#6101—42no—
83C2×C4×Dic13—22no—
84C2×C4×Dic13—22no—
85C2×C4×Dic13—24no—
86C2×C4×Dic13—42no—
87C13×#95—24no—
88C13×#95—42no—
89#88071—22no—
90#88071—22no—
91#88071—24no—
92#88071—24no—
93#88071—24no—
94#88071—24no—
95#88071—24no—
96#88071—24no—
97#88071—42no—
98#88071C2×C418yescomplemented
99#88071C2×C418yescomplemented
100#13992—131nomaximal, Sylow
101C24×C13—22no—
102C24×C13—22no—
103C24×C13—22no—
104C24×C13—22no—
105C24×C13C22×C412yes—
106C24×C13C2×D412yes—
107C24×C13C2×D412yes—
108C24×C13#4112yes—
109C24×C13#4112yes—
110C24×C13#4112yes—
111C24×C13#4112yes—
112C24×C13C2411yescharacteristic
113C22×C4×C13—28no—
114C22×C4×C13—22no—
115C22×C4×C13—21no—
116C22×C4×C13—21no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—21no—
121C22×C4×C13—21no—
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—22no—
132C22×C4×C13—22no—
133C22×C4×C13—22no—
134C22×C4×C13—22no—
135C22×C4×C13—48no—
136C22×C4×C13—48no—
137C22×C4×C13—44no—
138C22×C4×C13—44no—
139C22×C4×C13C22×C412yes—
140C22×C4×C13C22×C412yes—
141C22×C4×C13#4114yes—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—22no—
145C22×Dic13—28no—
146C22×Dic13—28no—
147C22×Dic13—28no—
148C22×Dic13—28no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—22no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—22no—
159C22×Dic13—21no—
160C22×Dic13—21no—
161C22×Dic13—22no—
162C22×Dic13—22no—
163C22×Dic13—22no—
164C22×Dic13—22no—
165C22×Dic13—21no—
166C22×Dic13—21no—
167C22×Dic13—22no—
168C22×Dic13—22no—
169C22×Dic13—22no—
170C22×Dic13—22no—
171C22×Dic13—22no—
172C22×Dic13—22no—
173C22×Dic13—28no—
174C22×Dic13—44no—
175C22×Dic13—48no—
176C22×Dic13—48no—
177C22×Dic13—44no—
178C22×Dic13—44no—
179C22×Dic13—44no—
180C22×Dic13—44no—
181C22×Dic13C22×C414yes—
182C22×Dic13C22×C414yes—
183C22×Dic13C2×D412yes—
184C22×Dic13C2×D412yes—
185C22×Dic13C2×D412yes—
186C22×Dic13C2×D412yes—
187C22×Dic13#4114yes—
188C13×C22⋊C4—28no—
189C13×C22⋊C4—44no—
190C22⋊Dic13—28no—
191C22⋊Dic13—28no—
192#6101—48no—
193C4×Dic13—44no—
194C4×Dic13—44no—
195C2×#385—131no—
196C2×#381—131no—
197C2×#381—131no—
198C2×C4×C22⋊C4—131no—
199C22×#95—131no—
200#1443—132no—
201#1442—132no—
202#1436—132no—
203#1434—132no—
204C22⋊#95—131no—
205C22⋊#95D1311yescharacteristic
206C23×C13—24no—
207C23×C13—24no—
208C23×C13—22no—
209C23×C13—22no—
210C23×C13—24no—
211C23×C13—22no—
212C23×C13—22no—
213C23×C13—24no—
214C23×C13—24no—
215C23×C13—24no—
216C23×C13—24no—
217C23×C13—24no—
218C23×C13—24no—
219C23×C13—24no—
220C23×C13—24no—
221C23×C13—24no—
222C23×C13#11714yes—
223C23×C13C4×D414yes—
224C23×C13C4×D414yes—
225C23×C13#12114yes—
226C23×C13#12314yes—
227C23×C13#12414yes—
228C23×C13#12614yes—
229C23×C13C23×C411yescharacteristic
230C23×C13C22×D411yescharacteristic
231C23×C13C22×D411yescharacteristic
232C23×C13C2×#4111yescharacteristic
233C23×C13C2×#4111yescharacteristic
234C23×C13C2×#4111yescharacteristic
235C23×C13C2×#4111yescharacteristic
236C23×C13C2×#4112yes—
237C23×C13C2×#4112yes—
238C23×C13H(5,𝔽2)12yes—
239C23×C132−1+412yes—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—24no—
243C2×C4×C13—24no—
244C2×C4×C13—24no—
245C2×C4×C13—24no—
246C2×C4×C13—24no—
247C2×C4×C13—416no—
248C2×C4×C13—42no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—42no—
252C2×C4×C13—41no—
253C2×C4×C13—42no—
254C2×C4×C13—41no—
255C2×C4×C13—42no—
256C2×C4×C13—42no—
257C2×C4×C13—42no—
258C2×C4×C13—42no—
259C2×C4×C13—42no—
260C2×C4×C13—42no—
261C2×C4×C13—42no—
262C2×C4×C13—42no—
263C2×C4×C13—42no—
264C2×C4×C13—42no—
265C2×Dic13—22no—
266C2×Dic13—22no—
267C2×Dic13—24no—
268C2×Dic13—24no—
269C2×Dic13—24no—
270C2×Dic13—24no—
271C2×Dic13—24no—
272C2×Dic13—24no—
273C2×Dic13—24no—
274C2×Dic13—24no—
275C2×Dic13—22no—
276C2×Dic13—22no—
277C2×Dic13—22no—
278C2×Dic13—22no—
279C2×Dic13—22no—
280C2×Dic13—24no—
281C2×Dic13—24no—
282C2×Dic13—416no—
283C2×Dic13—42no—
284C2×Dic13—41no—
285C2×Dic13—42no—
286C2×Dic13—41no—
287C2×Dic13—42no—
288C2×Dic13—42no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—42no—
292C2×Dic13—42no—
293C2×Dic13—42no—
294C2×Dic13—42no—
295C2×Dic13—42no—
296C2×Dic13—42no—
297C2×Dic13—42no—
298C2×Dic13—42no—
299C2×Dic13—416no—
300C2×Dic13#12314yes—
301C24×C4—131no—
302C24×C4—131no—
303C24×C4C2×D1311yescharacteristic
304C22×C4⋊C4—132no—
305C22×C22⋊C4—131no—
306C22×C22⋊C4—131no—
307C22×C22⋊C4C2×D1311yescharacteristic
308C22×C22⋊C4C2×D1311yescharacteristic
309C22×C42—131no—
310C22×C42—131no—
311#385—264no—
312#381—138no—
313#381—264no—
314C4×C22⋊C4—264no—
315C2×#95—132no—
316C2×#95—132no—
317C2×#95—132no—
318C2×#95—132no—
319C2×#95—132no—
320C2×#95—132no—
321C2×#95—132no—
322C2×#95—131no—
323C2×#95—132no—
324C2×#95—131no—
325C2×#95—132no—
326C2×#95—264no—
327C2×#95C2×D1312yes—
328C2×#95C2×D1312yes—
329C22×C13—24no—
330C22×C13—28no—
331C22×C13—28no—
332C22×C13—24no—
333C22×C13—24no—
334C22×C13—24no—
335C22×C13—28no—
336C22×C13—28no—
337C22×C13—24no—
338C22×C13—24no—
339C22×C13#38818yes—
340C22×C13C2×#11711yescharacteristic
341C22×C13C2×C4×D411yescharacteristic
342C22×C13C2×C4×D411yescharacteristic
343C22×C13C4×#4112yes—
344C22×C13#52012yes—
345C22×C13C2×#12111yescharacteristic
346C22×C13C2×#12311yescharacteristic
347C22×C13C2×#12311yescharacteristic
348C22×C13#52511yescharacteristic
349C22×C13C2×#12411yescharacteristic
350C22×C13C2×#12611yescharacteristic
351C22×C13#52912yes—
352C22×C13#53411yescharacteristic
353C22×C13#53611yescharacteristic
354C22×C13#53912yes—
355C22×C13#54612yes—
356C22×C13#54712yes—
357C22×C13#55112yes—
358C22×C13#55112yes—
359C22×C13#55112yes—
360C22×C13#55212yes—
361C22×C13#55212yes—
362C22×C13#55212yes—
363C4×C13—42no—
364C4×C13—44no—
365C4×C13—42no—
366C4×C13—82no—
367Dic13—24no—
368Dic13—44no—
369Dic13—42no—
370Dic13—42no—
371Dic13—42no—
372Dic13—44no—
373C25C22×D1311yescharacteristic
374C23×C4—22no—
375C23×C4—24no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—42no—
379C23×C4—131no—
380C23×C4—134no—
381C23×C4—131no—
382C23×C4—132no—
383C23×C4—131no—
384C23×C4—132no—
385C23×C4—131no—
386C23×C4—134no—
387C23×C4—264no—
388C23×C4—262no—
389C23×C4—262no—
390C23×C4—262no—
391C23×C4—262no—
392C23×C4—262no—
393C23×C4—262no—
394C23×C4—262no—
395C23×C4—262no—
396C23×C4—522no—
397C23×C4C22×D1312yes—
398C23×C4C22×D1311yescharacteristic
399C23×C4C22×D1311yescharacteristic
400C23×C4C22×D1312yes—
401C2×C4⋊C4—264no—
402C2×C4⋊C4—264no—
403C2×C4⋊C4—522no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—134no—
410C2×C22⋊C4—134no—
411C2×C22⋊C4—262no—
412C2×C22⋊C4—262no—
413C2×C22⋊C4—262no—
414C2×C22⋊C4—262no—
415C2×C22⋊C4C4×D1314yes—
416C2×C42—262no—
417C2×C42—262no—
418C2×C42—264no—
419C2×C42—522no—
420#95—24no—
421#95—42no—
422#95—138no—
423#95—138no—
424#95—264no—
425#95—264no—
426#95—264no—
427#95—262no—
428#95—262no—
429#95—264no—
430#95—264no—
431#95—264no—
432#95—522no—
433C2×C13—28no—
434C2×C13C2×#38811yescharacteristic
435C2×C13C4×#12311yescharacteristic
436C2×C13#232211yescharacteristic
437C2×C13#234312yes—
438C2×C13#235912yes—
439C2×C13#248311yescharacteristic
440C2×C13#248411yescharacteristic
441C2×C13#248611yescharacteristic
442C2×C13#248812yes—
443C2×C13#249312yes—
444C2×C13#253411yescharacteristic
445C24—22no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24C2×C4×D1312yes—
450C24#612712yes—
451C24#612712yes—
452C24D4×D1312yes—
453C24D4×D1312yes—
454C24#612912yes—
455C24#613112yes—
456C24C23×D1311yescharacteristic
457C22×C4—28no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—132no—
484C22×C4—134no—
485C22×C4—134no—
486C22×C4—134no—
487C22×C4—132no—
488C22×C4—132no—
489C22×C4—132no—
490C22×C4—268no—
491C22×C4—261no—
492C22×C4—262no—
493C22×C4—261no—
494C22×C4—268no—
495C22×C4—268no—
496C22×C4—268no—
497C22×C4—268no—
498C22×C4—261no—
499C22×C4—261no—
500C22×C4—262no—
501C22×C4—262no—
502C22×C4—261no—
503C22×C4—262no—
504C22×C4—262no—
505C22×C4—261no—
506C22×C4—261no—
507C22×C4—262no—
508C22×C4—262no—
509C22×C4—262no—
510C22×C4—262no—
511C22×C4—262no—
512C22×C4—262no—
513C22×C4—262no—
514C22×C4—262no—
515C22×C4—262no—
516C22×C4—261no—
517C22×C4—262no—
518C22×C4—262no—
519C22×C4—262no—
520C22×C4—262no—
521C22×C4—262no—
522C22×C4—524no—
523C22×C4—528no—
524C22×C4—528no—
525C22×C4—524no—
526C22×C4—524no—
527C22×C4—524no—
528C22×C4—524no—
529C22×C4C2×C4×D1312yes—
530C22×C4C2×C4×D1312yes—
531C22×C4#612914yes—
532C4⋊C4—528no—
533C22⋊C4—28no—
534C22⋊C4—44no—
535C22⋊C4—268no—
536C22⋊C4—268no—
537C42—524no—
538C42—524no—
539C13#1399211yescharacteristic, complemented, Sylow
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23#8812614yes—
557C23#8813514yes—
558C23#8813814yes—
559C23#8813914yes—
560C23#8814714yes—
561C23#8814814yes—
562C23#8815214yes—
563C23C22×C4×D1311yescharacteristic
564C23C2×#612711yescharacteristic
565C23C2×#612711yescharacteristic
566C23C2×D4×D1311yescharacteristic
567C23C2×D4×D1311yescharacteristic
568C23C2×#612911yescharacteristic
569C23C2×#612912yes—
570C23#8825112yes—
571C23C2×#613111yescharacteristic
572C23#41×D1312yes—
573C23#8825712yes—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—134no—
600C2×C4—262no—
601C2×C4—264no—
602C2×C4—262no—
603C2×C4—264no—
604C2×C4—264no—
605C2×C4—264no—
606C2×C4—262no—
607C2×C4—262no—
608C2×C4—264no—
609C2×C4—264no—
610C2×C4—262no—
611C2×C4—264no—
612C2×C4—264no—
613C2×C4—262no—
614C2×C4—264no—
615C2×C4—262no—
616C2×C4—264no—
617C2×C4—5216no—
618C2×C4—522no—
619C2×C4—522no—
620C2×C4—522no—
621C2×C4—522no—
622C2×C4—522no—
623C2×C4—522no—
624C2×C4—522no—
625C2×C4—521no—
626C2×C4—521no—
627C2×C4—522no—
628C2×C4—522no—
629C2×C4—522no—
630C2×C4—522no—
631C2×C4—521no—
632C2×C4—5216no—
633C2×C4—521no—
634C2×C4—522no—
635C22—24no—
636C22—28no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—28no—
645C22#1171833018yes—
646C22C2×#8812611yescharacteristic
647C22C2×#8813511yescharacteristic
648C22C2×#8813811yescharacteristic
649C22C2×#8813911yescharacteristic
650C22#1171920012yes—
651C22C2×#8814711yescharacteristic
652C22C2×#8814811yescharacteristic
653C22C2×#8815211yescharacteristic
654C22#1171923612yes—
655C22C4×#612912yes—
656C22#1171924612yes—
657C22#1171925112yes—
658C22#1171926112yes—
659C22#1171929111yescharacteristic
660C22#1171929411yescharacteristic
661C22#1171929612yes—
662C22#1171930512yes—
663C22#1171930712yes—
664C22#1171934212yes—
665C22#1171934712yes—
666C22#1171935311yescharacteristic
667C22D13×#12311yescharacteristic
668C22#1171936012yes—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—82no—
673C4—264no—
674C4—524no—
675C4—524no—
676C4—522no—
677C4—522no—
678C4—522no—
679C2—28no—
680C2C2×#1171833011yescharacteristic
681C2#1311736111yescharacteristic
682C2#1311751011yescharacteristic
683C2#1311769012yes—
684C2#1311782612yes—
685C2#1311911611yescharacteristic
686C2#1311912211yescharacteristic
687C2#1311914311yescharacteristic
688C2#1311916012yes—
689C2#1311922712yes—
690C2#1311972911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×C22⋊#95C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC22⋊#95D13
hypocenterC13#13992
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
6. #13113097C2semidirect
7. #13113094C2semidirect
8. #13113067C2semidirect
9. #13113061C2semidirect
38. #11718604C4semidirect
98. #88071C2×C4semidirect
99. #88071C2×C4semidirect
539. C13#13992semidirect

Maximal subgroups

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

subgroupindex
C2×#117188712
C2×#117186042
C2×C22⋊C4×Dic132
C22×#880712
#131130972
#131130942
#131130672
#131130612
#131130582
C13×C22⋊#952
#1399213