Finite Group Databasebeta

← #15784848

Subgroups — #15784848

Order 3328.

1918 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718872C211yesmaximal, characteristic
2C2×#11718872C211yesmaximal, characteristic
3C2×#11718873C211yesmaximal, characteristic
4C2×C4×C22⋊Dic13C211yesmaximal, characteristic
5#13113098C212yesmaximal, complemented
6#13113093C212yesmaximal, complemented
7#13113070C212yesmaximal, complemented
8#13113061C212yesmaximal, complemented
9#13113059C211yesmaximal, characteristic
10#13113059C211yesmaximal, characteristic
11C22×C13×#95C211yesmaximal, characteristic
12C24×C4×C13C2211yescharacteristic
13C24×C4×C13C2211yescharacteristic
14C24×C4×C13C2211yescharacteristic
15C22×C22⋊Dic13C2211yescharacteristic
16C22×C22⋊Dic13C2211yescharacteristic
17C22×C22⋊Dic13C2211yescharacteristic
18C22×C22⋊Dic13C2211yescharacteristic
19C22×C4⋊Dic13C2212yes—
20C22×C4×Dic13C2212yes—
21C2×C13×#95C418yescomplemented
22C2×C13×#95C2211yescharacteristic
23C2×C13×#95C2211yescharacteristic
24C2×C13×#95C2211yescharacteristic
25C2×C13×#95C2211yescharacteristic
26#11718873—24no—
27#11718872—24no—
28#11718872—24no—
29C4×C22⋊Dic13—24no—
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36C2×#88071C2212yes—
37C2×#88071C2212yes—
38C2×#88071C2212yes—
39C2×#88071C2212yes—
40C25×C13C2311yescharacteristic
41C23×C4×C13—22no—
42C23×C4×C13—22no—
43C23×C4×C13—22no—
44C23×C4×C13—22no—
45C23×C4×C13—22no—
46C23×C4×C13—22no—
47C23×C4×C13—22no—
48C23×C4×C13—22no—
49C23×C4×C13—22no—
50C23×C4×C13—22no—
51C23×C4×C13—24no—
52C23×C4×C13—24no—
53C23×C4×C13—42no—
54C23×C4×C13—42no—
55C23×C4×C13C2×C414yes—
56C23×C4×C13C2×C414yes—
57C23×C4×C13C2×C414yes—
58C23×C4×C13C2311yescharacteristic
59C23×C4×C13C2311yescharacteristic
60C23×C4×C13C2311yescharacteristic
61C23×C4×C13C2311yescharacteristic
62C23×C4×C13C2311yescharacteristic
63C23×C4×C13C2311yescharacteristic
64C23×Dic13C2312yes—
65C23×Dic13C2312yes—
66C23×Dic13C2312yes—
67C23×Dic13C2312yes—
68C2×C22⋊Dic13—22no—
69C2×C22⋊Dic13—22no—
70C2×C22⋊Dic13—22no—
71C2×C22⋊Dic13—22no—
72C2×C22⋊Dic13—22no—
73C2×C22⋊Dic13—22no—
74C2×C22⋊Dic13—22no—
75C2×C22⋊Dic13—22no—
76C2×C22⋊Dic13—22no—
77C2×C22⋊Dic13—22no—
78C2×C22⋊Dic13D414yes—
79C2×C22⋊Dic13D414yes—
80C2×C4⋊Dic13—24no—
81C2×C4⋊Dic13—24no—
82C2×C4⋊Dic13—42no—
83C2×C4×Dic13—24no—
84C2×C4×Dic13—24no—
85C2×C4×Dic13—42no—
86C13×#95C2×C414yes—
87C13×#95C2×C414yes—
88C13×#95C2×C414yes—
89C13×#95C2×C414yes—
90#88071—24no—
91#88071—24no—
92#88071—24no—
93#88071—24no—
94#88071—24no—
95#88071—24no—
96#88071—24no—
97#88071—24no—
98#88071—42no—
99#88071—42no—
100#13874—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—21no—
114C22×C4×C13—21no—
115C22×C4×C13—21no—
116C22×C4×C13—21no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—28no—
120C22×C4×C13—28no—
121C22×C4×C13—28no—
122C22×C4×C13—28no—
123C22×C4×C13—28no—
124C22×C4×C13—21no—
125C22×C4×C13—21no—
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—22no—
135C22×C4×C13—22no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—22no—
143C22×C4×C13—22no—
144C22×C4×C13—28no—
145C22×C4×C13—28no—
146C22×C4×C13—44no—
147C22×C4×C13—44no—
148C22×C4×C13—48no—
149C22×C4×C13—48no—
150C22×C4×C13—48no—
151C22×C4×C13—48no—
152C22×C4×C13—44no—
153C22×C4×C13—44no—
154C22×C4×C13—44no—
155C22×C4×C13—44no—
156C22×C4×C13C22×C412yes—
157C22×C4×C13C22×C412yes—
158C22×C4×C13C22×C412yes—
159C22×C4×C13C22×C412yes—
160C22×C4×C13C22×C412yes—
161C22×C4×C13C22×C412yes—
162C22×C4×C13#4114yes—
163C22×C4×C13#4114yes—
164C22×Dic13—22no—
165C22×Dic13—22no—
166C22×Dic13—22no—
167C22×Dic13—22no—
168C22×Dic13—22no—
169C22×Dic13—22no—
170C22×Dic13—22no—
171C22×Dic13—22no—
172C22×Dic13—22no—
173C22×Dic13—22no—
174C22×Dic13—22no—
175C22×Dic13—22no—
176C22×Dic13—22no—
177C22×Dic13—22no—
178C22×Dic13—22no—
179C22×Dic13—22no—
180C22×Dic13—22no—
181C22×Dic13—22no—
182C22×Dic13—22no—
183C22×Dic13—22no—
184C22×Dic13—22no—
185C22×Dic13—22no—
186C22×Dic13—22no—
187C22×Dic13—22no—
188C22×Dic13—22no—
189C22×Dic13—22no—
190C22×Dic13C2×D414yes—
191C22×Dic13C2×D414yes—
192C22⋊Dic13—28no—
193C22⋊Dic13—28no—
194C22⋊Dic13—44no—
195C22⋊Dic13—44no—
196C4⋊Dic13—48no—
197C4×Dic13—48no—
198C2×#381—131no—
199C2×#381—131no—
200C2×#380—131no—
201C2×C4×C22⋊C4—131no—
202C22×#95D1311yescharacteristic
203#1443—132no—
204#1442—132no—
205#1436—132no—
206#1434—132no—
207C22⋊#95—131no—
208C22⋊#95—131no—
209C23×C13—22no—
210C23×C13—22no—
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—24no—
223C23×C13—24no—
224C23×C13—24no—
225C23×C13#11714yes—
226C23×C13C4×D414yes—
227C23×C13C4×D414yes—
228C23×C13#12114yes—
229C23×C13#12314yes—
230C23×C13#12414yes—
231C23×C13#12614yes—
232C23×C13C23×C411yescharacteristic
233C23×C13C22×D411yescharacteristic
234C23×C13C22×D411yescharacteristic
235C23×C13C2×#4111yescharacteristic
236C23×C13C2×#4111yescharacteristic
237C23×C13C2×#4112yes—
238C23×C13C2×#4112yes—
239C23×C13C2×#4111yescharacteristic
240C23×C13C2×#4111yescharacteristic
241C23×C13H(5,𝔽2)12yes—
242C23×C132−1+412yes—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—22no—
247C2×C4×C13—24no—
248C2×C4×C13—24no—
249C2×C4×C13—24no—
250C2×C4×C13—24no—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—22no—
255C2×C4×C13—22no—
256C2×C4×C13—22no—
257C2×C4×C13—22no—
258C2×C4×C13—22no—
259C2×C4×C13—22no—
260C2×C4×C13—22no—
261C2×C4×C13—22no—
262C2×C4×C13—22no—
263C2×C4×C13—416no—
264C2×C4×C13—416no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13—41no—
269C2×C4×C13—41no—
270C2×C4×C13—41no—
271C2×C4×C13—41no—
272C2×C4×C13—41no—
273C2×C4×C13—42no—
274C2×C4×C13—42no—
275C2×C4×C13—42no—
276C2×C4×C13—42no—
277C2×C4×C13—416no—
278C2×C4×C13#11714yes—
279C2×C4×C13#11714yes—
280C2×Dic13—24no—
281C2×Dic13—24no—
282C2×Dic13—24no—
283C2×Dic13—24no—
284C2×Dic13—24no—
285C2×Dic13—24no—
286C2×Dic13—42no—
287C2×Dic13—42no—
288C2×Dic13—42no—
289C2×Dic13—42no—
290C2×Dic13—42no—
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—42no—
300C2×Dic13—42no—
301C2×Dic13—42no—
302C2×Dic13—42no—
303C2×Dic13—42no—
304C2×Dic13—42no—
305C2×Dic13—42no—
306C2×Dic13—42no—
307C2×Dic13—42no—
308C24×C4C2×D1311yescharacteristic
309C24×C4C2×D1311yescharacteristic
310C24×C4C2×D1311yescharacteristic
311C22×C4⋊C4—132no—
312C22×C22⋊C4—131no—
313C22×C22⋊C4—131no—
314C22×C22⋊C4—131no—
315C22×C22⋊C4—131no—
316C22×C42—132no—
317#381—264no—
318#381—264no—
319#380—264no—
320C4×C22⋊C4—264no—
321C2×#95—132no—
322C2×#95—132no—
323C2×#95—132no—
324C2×#95—132no—
325C2×#95—132no—
326C2×#95—132no—
327C2×#95—132no—
328C2×#95—132no—
329C2×#95—132no—
330C2×#95—132no—
331C2×#95Dic1318yescomplemented
332C2×#95C2×D1311yescharacteristic
333C2×#95C2×D1311yescharacteristic
334C2×#95C2×D1311yescharacteristic
335C2×#95C2×D1311yescharacteristic
336C22×C13—24no—
337C22×C13—24no—
338C22×C13—24no—
339C22×C13—24no—
340C22×C13—24no—
341C22×C13—28no—
342C22×C13—28no—
343C22×C13—28no—
344C22×C13—28no—
345C22×C13—24no—
346C22×C13#38818yes—
347C22×C13C2×#11711yescharacteristic
348C22×C13C2×#11711yescharacteristic
349C22×C13C2×#11711yescharacteristic
350C22×C13C2×C4×D411yescharacteristic
351C22×C13C2×C4×D411yescharacteristic
352C22×C13#51811yescharacteristic
353C22×C13#51911yescharacteristic
354C22×C13C2×#12111yescharacteristic
355C22×C13C2×#12311yescharacteristic
356C22×C13C2×#12411yescharacteristic
357C22×C13C2×#12611yescharacteristic
358C22×C13#52912yes—
359C22×C13#52912yes—
360C22×C13#53912yes—
361C22×C13#53912yes—
362C22×C13#54612yes—
363C22×C13#54612yes—
364C22×C13#54712yes—
365C22×C13#54712yes—
366C22×C13#55112yes—
367C22×C13#55112yes—
368C22×C13#55212yes—
369C22×C13#55212yes—
370C4×C13—24no—
371C4×C13—24no—
372C4×C13—42no—
373C4×C13—42no—
374C4×C13—42no—
375C4×C13—42no—
376Dic13—44no—
377Dic13—44no—
378Dic13—82no—
379Dic13—82no—
380C25C22×D1311yescharacteristic
381C23×C4—22no—
382C23×C4—22no—
383C23×C4—22no—
384C23×C4—22no—
385C23×C4—22no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—24no—
392C23×C4—24no—
393C23×C4—42no—
394C23×C4—42no—
395C23×C4—132no—
396C23×C4—132no—
397C23×C4—132no—
398C23×C4—132no—
399C23×C4C2×Dic1314yes—
400C23×C4C2×Dic1314yes—
401C23×C4C2×Dic1314yes—
402C23×C4C22×D1311yescharacteristic
403C23×C4C22×D1311yescharacteristic
404C23×C4C22×D1311yescharacteristic
405C23×C4C22×D1311yescharacteristic
406C23×C4C22×D1311yescharacteristic
407C23×C4C22×D1311yescharacteristic
408C2×C4⋊C4—264no—
409C2×C4⋊C4—264no—
410C2×C4⋊C4—522no—
411C2×C22⋊C4—134no—
412C2×C22⋊C4—134no—
413C2×C22⋊C4—262no—
414C2×C22⋊C4—262no—
415C2×C22⋊C4—262no—
416C2×C22⋊C4—262no—
417C2×C22⋊C4—262no—
418C2×C22⋊C4—262no—
419C2×C22⋊C4—262no—
420C2×C22⋊C4—262no—
421C2×C22⋊C4—262no—
422C2×C22⋊C4—262no—
423C2×C42—264no—
424C2×C42—264no—
425C2×C42—522no—
426#95—264no—
427#95—264no—
428#95—264no—
429#95—264no—
430#95—264no—
431#95—264no—
432#95—264no—
433#95—264no—
434#95—522no—
435#95—522no—
436#95C2×Dic1314yes—
437#95C2×Dic1314yes—
438#95C2×Dic1314yes—
439#95C2×Dic1314yes—
440C2×C13—28no—
441C2×C13#228311yescharacteristic
442C2×C13C2×#38811yescharacteristic
443C2×C13#231311yescharacteristic
444C2×C13#234211yescharacteristic
445C2×C13#234211yescharacteristic
446C2×C13#234411yescharacteristic
447C2×C13#234411yescharacteristic
448C2×C13#241912yes—
449C2×C13#242512yes—
450C2×C13#248812yes—
451C2×C13#249312yes—
452C24—22no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24#612712yes—
457C24#612712yes—
458C24D4×D1312yes—
459C24D4×D1312yes—
460C24#612912yes—
461C24#612912yes—
462C24C22×Dic1312yes—
463C24C23×D1311yescharacteristic
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—28no—
496C22×C4—28no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—134no—
508C22×C4—134no—
509C22×C4—262no—
510C22×C4—262no—
511C22×C4—262no—
512C22×C4—262no—
513C22×C4—262no—
514C22×C4—262no—
515C22×C4—262no—
516C22×C4—262no—
517C22×C4—262no—
518C22×C4—262no—
519C22×C4—262no—
520C22×C4—262no—
521C22×C4—262no—
522C22×C4—262no—
523C22×C4—262no—
524C22×C4—262no—
525C22×C4—262no—
526C22×C4—262no—
527C22×C4—262no—
528C22×C4—262no—
529C22×C4—262no—
530C22×C4—262no—
531C22×C4—262no—
532C22×C4—262no—
533C22×C4—262no—
534C22×C4—262no—
535C22×C4#612714yes—
536C22×C4#612714yes—
537C22×C4C22×Dic1312yes—
538C22×C4C22×Dic1312yes—
539C22×C4C22×Dic1312yes—
540C22×C4C22×Dic1312yes—
541C22×C4C22×Dic1312yes—
542C22×C4C22×Dic1312yes—
543C4⋊C4—528no—
544C22⋊C4—268no—
545C22⋊C4—268no—
546C22⋊C4—524no—
547C22⋊C4—524no—
548C42—528no—
549C13#1387411yescharacteristic, complemented, Sylow
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23#8813314yes—
567C23#8813714yes—
568C23#8813814yes—
569C23#8813914yes—
570C23#8818014yes—
571C23D4×Dic1314yes—
572C23D4×Dic1314yes—
573C23C2×#612711yescharacteristic
574C23C2×#612711yescharacteristic
575C23C2×#612712yes—
576C23C2×#612712yes—
577C23C2×D4×D1311yescharacteristic
578C23C2×D4×D1311yescharacteristic
579C23C2×#612911yescharacteristic
580C23C2×#612911yescharacteristic
581C23#8825112yes—
582C23#8825412yes—
583C23C23×Dic1311yescharacteristic
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—416no—
619C2×C4—264no—
620C2×C4—264no—
621C2×C4—264no—
622C2×C4—264no—
623C2×C4—264no—
624C2×C4—264no—
625C2×C4—522no—
626C2×C4—522no—
627C2×C4—522no—
628C2×C4—522no—
629C2×C4—522no—
630C2×C4—522no—
631C2×C4—522no—
632C2×C4—522no—
633C2×C4—522no—
634C2×C4—522no—
635C2×C4—522no—
636C2×C4—522no—
637C2×C4—522no—
638C2×C4—522no—
639C2×C4—522no—
640C2×C4—522no—
641C2×C4—522no—
642C2×C4—522no—
643C2×C4—522no—
644C2×C4—522no—
645C2×C4—522no—
646C2×C4—522no—
647C2×C4#8818014yes—
648C2×C4#8818014yes—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—28no—
653C22—28no—
654C22—28no—
655C22—24no—
656C22—24no—
657C22—28no—
658C22—24no—
659C22#1171861218yes—
660C22#1171918412yes—
661C22#1171918412yes—
662C22C2×#8813311yescharacteristic
663C22C2×#8813711yescharacteristic
664C22C2×#8813811yescharacteristic
665C22#1171919512yes—
666C22#1171919512yes—
667C22C2×#8813911yescharacteristic
668C22#1171920012yes—
669C22#1171920012yes—
670C22#1171921212yes—
671C22#1171921212yes—
672C22#1171921812yes—
673C22#1171921812yes—
674C22#1171922012yes—
675C22#1171922012yes—
676C22C2×#8818011yescharacteristic
677C22C2×#8818011yescharacteristic
678C22C2×#8818011yescharacteristic
679C22C2×D4×Dic1311yescharacteristic
680C22C2×D4×Dic1311yescharacteristic
681C22#1171950311yescharacteristic
682C22#1171951711yescharacteristic
683C4—24no—
684C4—24no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—524no—
690C4—524no—
691C4—1042no—
692C4—1042no—
693C2—28no—
694C2#1311723411yescharacteristic
695C2C2×#1171861211yescharacteristic
696C2#1311743911yescharacteristic
697C2#1311768511yescharacteristic
698C2#1311768511yescharacteristic
699C2#1311770511yescharacteristic
700C2#1311770511yescharacteristic
701C2#1311831312yes—
702C2#1311838712yes—
703C2#1311916212yes—
704C2#1311922112yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC22×C13×#95C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC22×#95D13
hypocenterC13#13874
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
5. #13113098C2semidirect
6. #13113093C2semidirect
7. #13113070C2semidirect
8. #13113061C2semidirect
21. C2×C13×#95C4semidirect
331. C2×#95Dic13semidirect
549. C13#13874semidirect

Maximal subgroups

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

subgroupindex
C2×#117188722
C2×#117188732
C2×C4×C22⋊Dic132
#131130982
#131130932
#131130702
#131130612
#131130592
C22×C13×#952
#1387413