Finite Group Databasebeta

← C2×#13115839 #15889709

Subgroups — C2×#13115839 #15889709

Order 3328.

2526 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#6105C211yesmaximal, characteristic, complemented
2C22×#88178C211yesmaximal, characteristic, complemented
3C22×#88050C211yesmaximal, characteristic, complemented
4C22×#88072C211yesmaximal, characteristic, complemented
5C22×C22⋊#1055C211yesmaximal, characteristic, complemented
6#13115839C218yesmaximal, direct product factor
7C2×#11718121C211yesmaximal, characteristic, complemented
8C22×C13×#121C211yesmaximal, characteristic
9C23×D4×C13—21no—
10C23×D4×C13C2211yescharacteristic
11C24×C4×C13C2211yescharacteristic
12C23×Dic26C2211yescharacteristic
13C2×C13×#121—24no—
14C2×C13×#121—22no—
15C2×C13×#121—24no—
16C2×C13×#121C2214yes—
17C22×C13×C4⋊C4C2211yescharacteristic
18C22×C13×C22⋊C4—21no—
19C22×C22⋊Dic13—21no—
20C22×#6105—24no—
21C22×#6105—24no—
22C22×#6105—24no—
23C22×#6105C2214yescomplemented
24C22×C4⋊Dic13C2211yescharacteristic
25C2×#88178—22no—
26C2×#88178—24no—
27C2×#88178—24no—
28C2×#88178C2214yescomplemented
29C22×#6101—21no—
30C23×#1055C2211yescharacteristic
31C23×#1055C2211yescharacteristic
32C2×#88072—24no—
33C2×#88072C2214yescomplemented
34C2×C22⋊#1055—24no—
35C2×C22⋊#1055C2214yescomplemented
36C2×#88050—24no—
37C2×#88050C2214yescomplemented
38#11718121C2214yescomplemented
39C25×C13—21no—
40C25×C13D411yescharacteristic
41C22×D4×C13—28no—
42C22×D4×C13—24no—
43C22×D4×C13—22no—
44C22×D4×C13—22no—
45C22×D4×C13—22no—
46C22×D4×C13—21no—
47C22×D4×C13—22no—
48C22×D4×C13—22no—
49C22×D4×C13—44no—
50C22×D4×C13—44no—
51C22×D4×C13D414yes—
52C22×D4×C13C2312yes—
53C23×C4×C13—21no—
54C23×C4×C13—22no—
55C23×C4×C13—22no—
56C23×C4×C13—22no—
57C23×C4×C13—22no—
58C23×C4×C13—22no—
59C23×C4×C13—22no—
60C23×C4×C13D411yescharacteristic
61C23×C4×C13D412yes—
62C23×C4×C13C2312yes—
63C23×C4×C13C2311yescharacteristic
64C23×Dic13—21no—
65C23×Dic13—21no—
66C22×Dic26—22no—
67C22×Dic26—22no—
68C22×Dic26—22no—
69C22×Dic26—22no—
70C22×Dic26—22no—
71C22×Dic26—21no—
72C22×Dic26D414yescomplemented
73C22×Dic26C2312yes—
74C13×#121—216no—
75C13×#121—28no—
76C13×#121—28no—
77C2×C13×C4⋊C4—21no—
78C2×C13×C4⋊C4—22no—
79C2×C13×C4⋊C4D414yes—
80C2×C13×C4⋊C4C2312yes—
81C2×C13×C22⋊C4—28no—
82C2×C13×C22⋊C4—24no—
83C2×C22⋊Dic13—28no—
84C2×C22⋊Dic13—24no—
85C2×#6105—232no—
86C2×#6105—28no—
87C2×#6105—28no—
88C2×#6105—28no—
89C2×C4⋊Dic13—22no—
90C2×C4⋊Dic13—21no—
91C2×C4⋊Dic13D414yescomplemented
92C2×C4⋊Dic13C2312yes—
93#88178—216no—
94#88178—28no—
95#88178—28no—
96C2×#6101—28no—
97C2×#6101—24no—
98C22×#1055—22no—
99C22×#1055—22no—
100C22×#1055—22no—
101C22×#1055—22no—
102C22×#1055—22no—
103C22×#1055D414yescomplemented
104C22×#1055C2312yes—
105C22×#1055C2312yes—
106#88072—28no—
107C22⋊#1055—28no—
108#88050—28no—
109C2×#2000—131nomaximal, Sylow
110C24×C13—22no—
111C24×C13—22no—
112C24×C13—22no—
113C24×C13—22no—
114C24×C13—22no—
115C24×C13—22no—
116C24×C13—24no—
117C24×C13—22no—
118C24×C13—42no—
119C24×C13—42no—
120C24×C13—42no—
121C24×C13—42no—
122C24×C13—42no—
123C24×C13—42no—
124C24×C13SD1614yescomplemented
125C24×C13C2×D412yes—
126C24×C13C2×D411yescharacteristic
127C2×D4×C13—216no—
128C2×D4×C13—28no—
129C2×D4×C13—24no—
130C2×D4×C13—28no—
131C2×D4×C13—22no—
132C2×D4×C13—22no—
133C2×D4×C13—22no—
134C2×D4×C13—22no—
135C2×D4×C13—22no—
136C2×D4×C13—48no—
137C2×D4×C13—432no—
138C2×D4×C13—48no—
139C2×D4×C13—42no—
140C2×D4×C13—44no—
141C2×D4×C13—42no—
142C2×D4×C13—44no—
143C2×D4×C13C2×D414yes—
144C22×C4×C13—22no—
145C22×C4×C13—22no—
146C22×C4×C13—24no—
147C22×C4×C13—22no—
148C22×C4×C13—22no—
149C22×C4×C13—24no—
150C22×C4×C13—24no—
151C22×C4×C13—24no—
152C22×C4×C13—22no—
153C22×C4×C13—22no—
154C22×C4×C13—22no—
155C22×C4×C13—22no—
156C22×C4×C13—22no—
157C22×C4×C13—22no—
158C22×C4×C13—42no—
159C22×C4×C13—42no—
160C22×C4×C13—44no—
161C22×C4×C13—44no—
162C22×C4×C13—44no—
163C22×C4×C13—44no—
164C22×C4×C13C2×D412yes—
165C22×C4×C13C2×D412yes—
166C22×C4×C13C2×D412yes—
167C22×C4×C13C2×D412yes—
168C22×C4×C13C2×D411yescharacteristic
169C22×C4×C13C2×D412yes—
170C22×C4×C13C2×D412yes—
171C22×C4×C13#4112yes—
172C22×C4×C13C2411yescharacteristic
173C22×Dic13—24no—
174C22×Dic13—24no—
175C22×Dic13—22no—
176C22×Dic13—22no—
177C22×Dic13—42no—
178C22×Dic13—42no—
179C22×Dic13—42no—
180C22×Dic13—42no—
181C2×Dic26—28no—
182C2×Dic26—28no—
183C2×Dic26—22no—
184C2×Dic26—22no—
185C2×Dic26—22no—
186C2×Dic26—22no—
187C2×Dic26—24no—
188C2×Dic26—22no—
189C2×Dic26—42no—
190C2×Dic26—44no—
191C2×Dic26—42no—
192C2×Dic26—44no—
193C2×Dic26C2×D414yescomplemented
194C13×C4⋊C4—24no—
195C13×C4⋊C4—22no—
196C13×C4⋊C4C2×D414yes—
197C13×C22⋊C4—216no—
198C22⋊Dic13—216no—
199#6105—232no—
200C4⋊Dic13—24no—
201C4⋊Dic13—22no—
202C4⋊Dic13C2×D414yescomplemented
203#6101—216no—
204C2×#1055—28no—
205C2×#1055—22no—
206C2×#1055—22no—
207C2×#1055—22no—
208C2×#1055—22no—
209C2×#1055—22no—
210C2×#1055—44no—
211C2×#1055C2×D414yescomplemented
212C23×SD16—131no—
213C22×C22⋊Q8—131no—
214C22×#121D1311yescharacteristic
215C22×#103—131no—
216C22×#102—131no—
217C22×C22⋊C8—131no—
218#2000—138no—
219C2×#340—131no—
220C23×C13—22no—
221C23×C13—22no—
222C23×C13—22no—
223C23×C13—22no—
224C23×C13—22no—
225C23×C13—22no—
226C23×C13—28no—
227C23×C13—28no—
228C23×C13—28no—
229C23×C13—24no—
230C23×C13—44no—
231C23×C13—44no—
232C23×C13—44no—
233C23×C13—44no—
234C23×C13—42no—
235C23×C13—42no—
236C23×C13—42no—
237C23×C13—42no—
238C23×C13—42no—
239C23×C13—42no—
240C23×C13—48no—
241C23×C13—48no—
242C23×C13—48no—
243C23×C13—84no—
244C23×C13—84no—
245C23×C13—84no—
246C23×C13—84no—
247C23×C13#12012yes—
248C23×C13#12012yes—
249C23×C13#12112yes—
250C23×C13C2×SD1612yes—
251C23×C13C2×SD1614yescomplemented
252C23×C13#13512yes—
253C23×C13AGL(1,ℤ/8)12yes—
254C23×C13#13712yes—
255C23×C13C22×D411yescharacteristic
256D4×C13—24no—
257D4×C13—24no—
258D4×C13—432no—
259D4×C13—42no—
260D4×C13—42no—
261D4×C13—48no—
262C2×C4×C13—22no—
263C2×C4×C13—22no—
264C2×C4×C13—24no—
265C2×C4×C13—22no—
266C2×C4×C13—42no—
267C2×C4×C13—42no—
268C2×C4×C13—44no—
269C2×C4×C13—48no—
270C2×C4×C13—42no—
271C2×C4×C13—42no—
272C2×C4×C13—42no—
273C2×C4×C13—42no—
274C2×C4×C13—48no—
275C2×C4×C13#12014yes—
276C2×C4×C13#12012yes—
277C2×C4×C13#12012yes—
278C2×C4×C13#12114yes—
279C2×C4×C13#12114yes—
280C2×C4×C13#12112yes—
281C2×C4×C13#12714yes—
282C2×C4×C13C22×D411yescharacteristic
283C2×C4×C13C22×D411yescharacteristic
284C2×C4×C13C22×D411yescharacteristic
285C2×C4×C13C22×D411yescharacteristic
286C2×C4×C13C22×D411yescharacteristic
287C2×C4×C13C2×#4111yescharacteristic
288C2×Dic13—24no—
289C2×Dic13—24no—
290C2×Dic13—42no—
291C2×Dic13—42no—
292C2×Dic13—42no—
293C2×Dic13—42no—
294C2×Dic13—48no—
295C2×Dic13—48no—
296Dic26—24no—
297Dic26—24no—
298Dic26—42no—
299Dic26—42no—
300Dic26—48no—
301#1055—24no—
302#1055—42no—
303C23×Q8—131no—
304C23×D4—21no—
305C23×D4C2×D1311yescharacteristic
306C24×C4C2×D1311yescharacteristic
307C22×SD16—134no—
308C22×SD16—264no—
309C22×SD16—264no—
310C22×SD16—264no—
311C23×C8—131no—
312C23×C8—131no—
313C2×C22⋊Q8—134no—
314C2×C22⋊Q8—264no—
315C2×C22⋊Q8—264no—
316C2×C22⋊Q8—262no—
317C2×#121—24no—
318C2×#121—22no—
319C2×#121—24no—
320C2×#121C2×D1314yes—
321C22×C4⋊C4—131no—
322C22×C4⋊C4—261no—
323C22×C4⋊C4C2×D1311yescharacteristic
324C22×C22⋊C4—21no—
325C22×C22⋊C4—261no—
326C2×#103—134no—
327C2×#103—264no—
328C2×#102—134no—
329C2×#102—264no—
330C2×C22⋊C8—134no—
331C2×C22⋊C8—264no—
332#340—134no—
333C22×C13—24no—
334C22×C13—24no—
335C22×C13—24no—
336C22×C13—42no—
337C22×C13—42no—
338C22×C13—42no—
339C22×C13—416no—
340C22×C13—42no—
341C22×C13—44no—
342C22×C13—44no—
343C22×C13—44no—
344C22×C13—82no—
345C22×C13—82no—
346C22×C13—816no—
347C22×C13—82no—
348C22×C13—82no—
349C22×C13#39214yes—
350C22×C13#44814yes—
351C22×C13#44914yes—
352C22×C13#45014yes—
353C22×C13#45214yes—
354C22×C13#46514yes—
355C22×C13#46814yes—
356C22×C13C2×#12011yescharacteristic
357C22×C13C2×#12011yescharacteristic
358C22×C13C2×#12111yescharacteristic
359C22×C13C22×SD1611yescharacteristic
360C22×C13C2×#13511yescharacteristic
361C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
362C22×C13C2×#13711yescharacteristic
363C4×C13—42no—
364C4×C13—44no—
365C4×C13#39214yes—
366C4×C13C2×#12011yescharacteristic
367C4×C13C2×#12111yescharacteristic
368C4×C13C2×#12111yescharacteristic
369C4×C13C2×#12711yescharacteristic
370Dic13—44no—
371Dic13—44no—
372C25—21no—
373C25C22⋊D1311yescharacteristic
374C22×Q8—134no—
375C22×Q8—132no—
376C22×Q8—262no—
377C22×Q8—261no—
378C22×Q8—262no—
379C22×Q8—262no—
380C22×Q8—262no—
381C22×Q8—262no—
382C22×D4—28no—
383C22×D4—24no—
384C22×D4—22no—
385C22×D4—22no—
386C22×D4—22no—
387C22×D4—22no—
388C22×D4—21no—
389C22×D4—22no—
390C22×D4—44no—
391C22×D4—44no—
392C22×D4C22⋊D1314yes—
393C22×D4C22×D1312yes—
394C23×C4—21no—
395C23×C4—22no—
396C23×C4—22no—
397C23×C4—22no—
398C23×C4—22no—
399C23×C4—22no—
400C23×C4—22no—
401C23×C4—261no—
402C23×C4—261no—
403C23×C4C22⋊D1312yes—
404C23×C4C22⋊D1311yescharacteristic
405C23×C4C22×D1312yes—
406C23×C4C22×D1311yescharacteristic
407C2×SD16—268no—
408C2×SD16—268no—
409C2×SD16—2632no—
410C2×SD16—268no—
411C22×C8—134no—
412C22×C8—132no—
413C22×C8—132no—
414C22×C8—262no—
415C22×C8—262no—
416C22×C8—262no—
417C22×C8—262no—
418C22×C8—262no—
419C22⋊Q8—268no—
420C22⋊Q8—268no—
421C22⋊Q8—2616no—
422#121—216no—
423#121—28no—
424#121—28no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—134no—
428C2×C4⋊C4—132no—
429C2×C4⋊C4—268no—
430C2×C4⋊C4—262no—
431C2×C4⋊C4—261no—
432C2×C4⋊C4—264no—
433C2×C4⋊C4C22⋊D1314yes—
434C2×C4⋊C4C22×D1312yes—
435C2×C22⋊C4—28no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—268no—
438C2×C22⋊C4—264no—
439#103—268no—
440#102—268no—
441C22⋊C8—268no—
442C2×C13—44no—
443C2×C13—84no—
444C2×C13#200018yescomplemented
445C2×C13C2×#39211yescharacteristic
446C2×C13C2×#45011yescharacteristic
447C2×C13C2×#44811yescharacteristic
448C2×C13C2×#44911yescharacteristic
449C2×C13C2×#45211yescharacteristic
450C2×C13C2×#46511yescharacteristic
451C2×C13C2×#46811yescharacteristic
452C24—22no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24—22no—
458C24—24no—
459C24—22no—
460C24—42no—
461C24—42no—
462C24—42no—
463C24—42no—
464C24—42no—
465C24—42no—
466C24#610514yescomplemented
467C24C2×C22⋊D1312yes—
468C24C2×C22⋊D1311yescharacteristic
469C2×Q8—134no—
470C2×Q8—264no—
471C2×Q8—262no—
472C2×Q8—268no—
473C2×Q8—268no—
474C2×Q8—262no—
475C2×Q8—262no—
476C2×Q8—262no—
477C2×Q8—262no—
478C2×Q8—522no—
479C2×Q8—524no—
480C2×Q8—524no—
481C2×Q8—522no—
482C2×D4—216no—
483C2×D4—28no—
484C2×D4—28no—
485C2×D4—24no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—48no—
492C2×D4—432no—
493C2×D4—48no—
494C2×D4—42no—
495C2×D4—44no—
496C2×D4—42no—
497C2×D4—44no—
498C2×D4C2×C22⋊D1314yes—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—24no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—44no—
516C22×C4—44no—
517C22×C4—44no—
518C22×C4—44no—
519C22×C4—264no—
520C22×C4—264no—
521C22×C4—262no—
522C22×C4—262no—
523C22×C4—522no—
524C22×C4—522no—
525C22×C4—522no—
526C22×C4—522no—
527C22×C4D4×D1312yes—
528C22×C4D4×D1312yes—
529C22×C4D4×D1312yes—
530C22×C4#612912yes—
531C22×C4C2×C22⋊D1312yes—
532C22×C4C2×C22⋊D1311yescharacteristic
533C22×C4C2×C22⋊D1312yes—
534C22×C4C2×C22⋊D1312yes—
535C22×C4C23×D1311yescharacteristic
536SD16—2632no—
537C2×C8—134no—
538C2×C8—268no—
539C2×C8—262no—
540C2×C8—262no—
541C2×C8—262no—
542C2×C8—262no—
543C2×C8—262no—
544C2×C8—524no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—134no—
548C4⋊C4—2616no—
549C4⋊C4—264no—
550C4⋊C4—262no—
551C4⋊C4C2×C22⋊D1314yes—
552C22⋊C4—216no—
553C22⋊C4—2616no—
554C13C2×#200011yescharacteristic, complemented, Sylow
555C23—28no—
556C23—28no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—28no—
564C23—24no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—48no—
576C23—48no—
577C23—48no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23#8818612yes—
583C23C2×#610512yes—
584C23C2×#610514yescomplemented
585C23#8819112yes—
586C23#8819312yes—
587C23#8820412yes—
588C23#8820512yes—
589C23#8820712yes—
590C23C22×C22⋊D1311yescharacteristic
591Q8—264no—
592Q8—264no—
593Q8—522no—
594Q8—522no—
595Q8—528no—
596D4—24no—
597D4—24no—
598D4—432no—
599D4—42no—
600D4—42no—
601D4—48no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—48no—
615C2×C4—264no—
616C2×C4—264no—
617C2×C4—528no—
618C2×C4—522no—
619C2×C4—522no—
620C2×C4—522no—
621C2×C4—522no—
622C2×C4—528no—
623C2×C4#8819114yes—
624C2×C4#8819112yes—
625C2×C4#8819214yes—
626C2×C4#8819314yes—
627C2×C4#8819312yes—
628C2×C4#8819414yes—
629C2×C4#8820712yes—
630C2×C4C2×D4×D1311yescharacteristic
631C2×C4C2×D4×D1311yescharacteristic
632C2×C4C2×D4×D1311yescharacteristic
633C2×C4C2×#612911yescharacteristic
634C2×C4C22×C22⋊D1311yescharacteristic
635C2×C4C22×C22⋊D1311yescharacteristic
636C8—264no—
637C8—522no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—416no—
644C22—42no—
645C22—42no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—816no—
653C22—82no—
654C22#1171869814yes—
655C22#1171870114yes—
656C22#1171870214yes—
657C22#1171870314yes—
658C22#1171887914yes—
659C22#1171888414yes—
660C22#1171890014yes—
661C22C2×#8818611yescharacteristic
662C22C22×#610511yescharacteristic
663C22C2×#8819111yescharacteristic
664C22C2×#8819311yescharacteristic
665C22C2×#8820411yescharacteristic
666C22C2×#8820511yescharacteristic
667C22C2×#8820711yescharacteristic
668C4—42no—
669C4—44no—
670C4—524no—
671C4—524no—
672C4#1171888414yes—
673C4C2×#8819111yescharacteristic
674C4C2×#8819211yescharacteristic
675C4C2×#8819311yescharacteristic
676C4C2×#8819411yescharacteristic
677C2—44no—
678C2—84no—
679C2#1311583918yesdirect product factor
680C2C2×#1171888411yescharacteristic
681C2C2×#1171887911yescharacteristic
682C2C2×#1171870111yescharacteristic
683C2C2×#1171869811yescharacteristic
684C2C2×#1171890011yescharacteristic
685C2C2×#1171870211yescharacteristic
686C2C2×#1171870311yescharacteristic
subgroupG/H
centerC24C2×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
Fitting, prime radicalC22×C13×#121C2
socleC24×C13C2×D4
hypercenterC22×#121D13
hypocenterC13C2×#2000
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C23×#6105C2semidirect
2. C22×#88178C2semidirect
3. C22×#88050C2semidirect
4. C22×#88072C2semidirect
5. C22×C22⋊#1055C2semidirect
6. #13115839C2direct (N × H)
7. C2×#11718121C2semidirect
23. C22×#6105C22semidirect
28. C2×#88178C22semidirect
33. C2×#88072C22semidirect
35. C2×C22⋊#1055C22semidirect
37. C2×#88050C22semidirect
38. #11718121C22semidirect
72. C22×Dic26D4semidirect
91. C2×C4⋊Dic13D4semidirect
103. C22×#1055D4semidirect
124. C24×C13SD16semidirect
193. C2×Dic26C2×D4semidirect
202. C4⋊Dic13C2×D4semidirect
211. C2×#1055C2×D4semidirect
251. C23×C13C2×SD16semidirect
444. C2×C13#2000semidirect
466. C24#6105semidirect
554. C13C2×#2000semidirect
584. C23C2×#6105semidirect
679. C2#13115839direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#61052
C22×#881782
C22×#880502
C22×#880722
C22×C22⋊#10552
#131158392
C2×#117181212
C22×C13×#1212
C2×#200013