Finite Group Databasebeta

← #15899940

Subgroups — #15899940

Order 3328.

1580 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718712C212yesmaximal, complemented
2C2×#11718889C211yesmaximal, characteristic, complemented
3C22×C22⋊#1055C211yesmaximal, characteristic
4#13123123C211yesmaximal, characteristic
5#13116036C212yesmaximal, complemented
6#13115844C212yesmaximal, complemented
7#13115260C212yesmaximal, complemented
8#13115143C211yesmaximal, characteristic, complemented
9#13115024C211yesmaximal, characteristic, complemented
10#13114771C211yesmaximal, characteristic, complemented
11C22×C13×C22⋊Q8C211yesmaximal, characteristic
12C23×Q8×C13C2211yescharacteristic
13C24×C4×C13C2211yescharacteristic
14C2×C13×C22⋊Q8—21no—
15C2×C13×C22⋊Q8—21no—
16C2×C13×C22⋊Q8—28no—
17C2×C13×C22⋊Q8C2212yes—
18C2×C13×C22⋊Q8C2212yes—
19C22×C13×C4⋊C4—21no—
20C22×C13×C4⋊C4C2211yescharacteristic
21C22×C13×C22⋊C4—21no—
22C22×#6107C2212yescomplemented
23C2×#88178C2211yescharacteristic
24C2×#88178C2211yescharacteristic
25C2×#88178C2211yescharacteristic
26C23×#1055C2212yes—
27#11718908—21no—
28#11718889—24no—
29C2×#88075C2212yescomplemented
30C2×#88075C2212yescomplemented
31C2×#88075C2212yescomplemented
32#11718873C2211yescharacteristic
33#11718873C2211yescharacteristic
34#11718873C2211yescharacteristic
35#11718871—21no—
36#11718871—21no—
37#11718871—21no—
38C2×C22⋊#1055—24no—
39C2×C22⋊#1055C2211yescharacteristic
40C2×C22⋊#1055C2212yes—
41C2×C22⋊#1055C2211yescharacteristic
42#11718712—28no—
43C2×#88050C2212yescomplemented
44C2×#88047C2212yescomplemented
45#11718121C2212yescomplemented
46#11718121C2212yescomplemented
47C25×C13D411yescharacteristic
48C22×Q8×C13—24no—
49C22×Q8×C13—21no—
50C22×Q8×C13—22no—
51C22×Q8×C13—22no—
52C22×Q8×C13—42no—
53C22×Q8×C13C2312yes—
54C23×C4×C13—21no—
55C23×C4×C13—21no—
56C23×C4×C13—24no—
57C23×C4×C13—22no—
58C23×C4×C13—22no—
59C23×C4×C13—42no—
60C23×C4×C13D411yescharacteristic
61C23×C4×C13D411yescharacteristic
62C23×C4×C13D411yescharacteristic
63C23×C4×C13C2311yescharacteristic
64C23×C4×C13C2311yescharacteristic
65C23×C4×C13C2311yescharacteristic
66C22×Dic26C2311yescharacteristic
67C13×C22⋊Q8—28no—
68C13×C22⋊Q8—28no—
69C13×C22⋊Q8—24no—
70C13×C22⋊Q8—24no—
71C13×C22⋊Q8—24no—
72C13×C22⋊Q8D418yes—
73C2×C13×C4⋊C4—28no—
74C2×C13×C4⋊C4—22no—
75C2×C13×C4⋊C4—22no—
76C2×C13×C4⋊C4—21no—
77C2×C13×C4⋊C4—24no—
78C2×C13×C4⋊C4C2312yes—
79C2×C13×C22⋊C4—28no—
80C2×C13×C22⋊C4—22no—
81C2×C13×C22⋊C4—22no—
82C2×C22⋊Dic13—21no—
83C2×C22⋊Dic13—21no—
84C2×C22⋊Dic13—21no—
85C2×C22⋊Dic13—21no—
86C2×C22⋊Dic13—21no—
87C2×C22⋊Dic13—21no—
88C2×#6107—24no—
89C2×#6107—24no—
90C2×#6107—42no—
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94#88178—24no—
95#88178—22no—
96#88178—22no—
97#88178—42no—
98C2×#6101—21no—
99C2×#6101—21no—
100C2×#6101—21no—
101C22×#1055—24no—
102C22×#1055—24no—
103C22×#1055—42no—
104C22×#1055C2312yes—
105C22×#1055C2312yes—
106#88075—24no—
107#88075—24no—
108#88075—42no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112C22⋊#1055—22no—
113C22⋊#1055—24no—
114C22⋊#1055D414yes—
115#88050—24no—
116#88047—24no—
117#24380—131nomaximal, Sylow
118C24×C13—22no—
119C24×C13—22no—
120C24×C13—22no—
121C24×C13—22no—
122C24×C13—41no—
123C24×C13—41no—
124C24×C13Q1614yescomplemented
125C24×C13C2×D411yescharacteristic
126C24×C13C2×D411yescharacteristic
127C24×C13C2×D411yescharacteristic
128C2×Q8×C13—24no—
129C2×Q8×C13—22no—
130C2×Q8×C13—22no—
131C2×Q8×C13—22no—
132C2×Q8×C13—22no—
133C2×Q8×C13—22no—
134C2×Q8×C13—48no—
135C2×Q8×C13—48no—
136C2×Q8×C13—42no—
137C2×Q8×C13—42no—
138C2×Q8×C13C2×D414yes—
139C22×C4×C13—21no—
140C22×C4×C13—22no—
141C22×C4×C13—21no—
142C22×C4×C13—22no—
143C22×C4×C13—22no—
144C22×C4×C13—24no—
145C22×C4×C13—24no—
146C22×C4×C13—22no—
147C22×C4×C13—22no—
148C22×C4×C13—22no—
149C22×C4×C13—22no—
150C22×C4×C13—22no—
151C22×C4×C13—22no—
152C22×C4×C13—22no—
153C22×C4×C13—22no—
154C22×C4×C13—21no—
155C22×C4×C13—22no—
156C22×C4×C13—21no—
157C22×C4×C13—22no—
158C22×C4×C13—41no—
159C22×C4×C13—41no—
160C22×C4×C13—41no—
161C22×C4×C13—48no—
162C22×C4×C13—48no—
163C22×C4×C13—42no—
164C22×C4×C13—42no—
165C22×C4×C13—42no—
166C22×C4×C13—42no—
167C22×C4×C13C2×D412yes—
168C22×C4×C13C2×D411yescharacteristic
169C22×C4×C13C2×D411yescharacteristic
170C22×C4×C13C2×D414yes—
171C22×C4×C13C2×D411yescharacteristic
172C22×C4×C13#4112yes—
173C22×C4×C13C2411yescharacteristic
174C22×Dic13—21no—
175C22×Dic13—21no—
176C22×Dic13—21no—
177C22×Dic13—21no—
178C2×Dic26—21no—
179C2×Dic26—22no—
180C2×Dic26—21no—
181C2×Dic26—22no—
182C13×C4⋊C4—24no—
183C13×C4⋊C4—28no—
184C13×C4⋊C4—28no—
185C13×C4⋊C4—22no—
186C13×C4⋊C4C2×D414yes—
187C13×C22⋊C4—28no—
188C13×C22⋊C4—28no—
189C22⋊Dic13—42no—
190C22⋊Dic13—42no—
191C22⋊Dic13—42no—
192C22⋊Dic13—42no—
193C22⋊Dic13—42no—
194C22⋊Dic13—42no—
195#6107—48no—
196C4⋊Dic13—21no—
197C4⋊Dic13—22no—
198C4⋊Dic13—21no—
199C4⋊Dic13—22no—
200#6101—42no—
201#6101—42no—
202#6101—42no—
203C2×#1055—22no—
204C2×#1055—22no—
205C2×#1055—22no—
206C2×#1055—22no—
207C2×#1055—22no—
208C2×#1055—48no—
209C2×#1055C2×D414yes—
210C22×C22⋊Q8D1311yescharacteristic
211C2×#467—132no—
212C2×#451—131no—
213C22×C22⋊C8—131no—
214#2848—131no—
215#2036—132no—
216#2001—132no—
217#1892—132no—
218#1872—131no—
219#1852—131no—
220#1787—131no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13—21no—
235C23×C13—28no—
236C23×C13—28no—
237C23×C13—44no—
238C23×C13—44no—
239C23×C13—44no—
240C23×C13—44no—
241C23×C13—44no—
242C23×C13#12012yes—
243C23×C13#12112yes—
244C23×C13C2×Q1612yes—
245C23×C13C2×Q1612yes—
246C23×C13C2×Q1612yes—
247C23×C13#13512yes—
248C23×C13#13712yes—
249C23×C13C22×D411yescharacteristic
250Q8×C13—24no—
251Q8×C13—24no—
252Q8×C13—48no—
253Q8×C13—42no—
254Q8×C13—42no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—24no—
260C2×C4×C13—24no—
261C2×C4×C13—24no—
262C2×C4×C13—24no—
263C2×C4×C13—24no—
264C2×C4×C13—24no—
265C2×C4×C13—21no—
266C2×C4×C13—21no—
267C2×C4×C13—21no—
268C2×C4×C13—44no—
269C2×C4×C13—44no—
270C2×C4×C13—44no—
271C2×C4×C13—48no—
272C2×C4×C13—48no—
273C2×C4×C13—42no—
274C2×C4×C13—42no—
275C2×C4×C13—42no—
276C2×C4×C13—42no—
277C2×C4×C13—42no—
278C2×C4×C13—42no—
279C2×C4×C13#12012yes—
280C2×C4×C13#12112yes—
281C2×C4×C13C22×D411yescharacteristic
282C2×C4×C13C22×D412yes—
283C2×C4×C13C2×#4111yescharacteristic
284C2×C4×C13H(5,𝔽2)12yes—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—41no—
293C2×Dic13—41no—
294C2×Dic13—41no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297Dic26—42no—
298Dic26—42no—
299#1055—24no—
300#1055—42no—
301C23×Q8C2×D1311yescharacteristic
302C24×C4C2×D1311yescharacteristic
303C22×Q16—132no—
304C23×C8—132no—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8—21no—
307C2×C22⋊Q8—28no—
308C2×C22⋊Q8—131no—
309C2×C22⋊Q8—131no—
310C2×C22⋊Q8—131no—
311C2×C22⋊Q8C2×D1312yes—
312C2×C22⋊Q8C2×D1312yes—
313C22×C4⋊C4—21no—
314C22×C4⋊C4C2×D1311yescharacteristic
315C22×C22⋊C4—21no—
316#467—268no—
317#451—264no—
318C2×#107—132no—
319C2×#103—132no—
320C2×#103—132no—
321C2×#103—132no—
322C2×#103—132no—
323C2×C22⋊C8—131no—
324C2×C22⋊C8—131no—
325C2×C22⋊C8—132no—
326C2×C22⋊C8—264no—
327#385—261no—
328#385—261no—
329#385—261no—
330#380—131no—
331#380—131no—
332#380—131no—
333#379—261no—
334#340—132no—
335#340—132no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—22no—
339C22×C13—22no—
340C22×C13—22no—
341C22×C13—22no—
342C22×C13—22no—
343C22×C13—22no—
344C22×C13—22no—
345C22×C13—22no—
346C22×C13—22no—
347C22×C13—22no—
348C22×C13—22no—
349C22×C13—22no—
350C22×C13—22no—
351C22×C13—416no—
352C22×C13—41no—
353C22×C13—41no—
354C22×C13—41no—
355C22×C13—41no—
356C22×C13#45114yescomplemented
357C22×C13#46714yescomplemented
358C22×C13C2×#12011yescharacteristic
359C22×C13C2×#12111yescharacteristic
360C22×C13#53411yescharacteristic
361C22×C13C22×Q1611yescharacteristic
362C22×C13C2×#13511yescharacteristic
363C22×C13C2×#13711yescharacteristic
364C22×C13#57511yescharacteristic
365C4×C13—22no—
366C4×C13—22no—
367C4×C13—41no—
368C4×C13—44no—
369C4×C13—44no—
370C4×C13D4212yes—
371C4×C13#54612yes—
372Dic13—81no—
373Dic13—81no—
374Dic13—81no—
375Dic13—81no—
376C25C22⋊D1311yescharacteristic
377C22×Q8—24no—
378C22×Q8—21no—
379C22×Q8—22no—
380C22×Q8—22no—
381C22×Q8—42no—
382C22×Q8—131no—
383C22×Q8C22×D1312yes—
384C23×C4—21no—
385C23×C4—21no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—24no—
389C23×C4—42no—
390C23×C4C22⋊D1311yescharacteristic
391C23×C4C22⋊D1311yescharacteristic
392C23×C4C22⋊D1311yescharacteristic
393C23×C4C22×D1311yescharacteristic
394C23×C4C22×D1311yescharacteristic
395C23×C4C22×D1311yescharacteristic
396C2×Q16—264no—
397C2×Q16—264no—
398C2×Q16—522no—
399C22×C8—132no—
400C22×C8—132no—
401C22×C8—264no—
402C22×C8—264no—
403C22×C8—522no—
404C22⋊Q8—28no—
405C22⋊Q8—28no—
406C22⋊Q8—24no—
407C22⋊Q8—24no—
408C22⋊Q8—24no—
409C22⋊Q8—262no—
410C22⋊Q8—262no—
411C22⋊Q8—264no—
412C22⋊Q8—522no—
413C22⋊Q8C22⋊D1318yes—
414C2×C4⋊C4—28no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—24no—
419C2×C4⋊C4—131no—
420C2×C4⋊C4—131no—
421C2×C4⋊C4—131no—
422C2×C4⋊C4—261no—
423C2×C4⋊C4—261no—
424C2×C4⋊C4—261no—
425C2×C4⋊C4C22×D1312yes—
426C2×C22⋊C4—28no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—261no—
430C2×C22⋊C4—261no—
431C2×C22⋊C4—261no—
432C2×C22⋊C4—261no—
433C2×C22⋊C4—261no—
434C2×C22⋊C4—261no—
435#107—264no—
436#103—264no—
437#103—264no—
438#103—264no—
439#103—522no—
440C22⋊C8—134no—
441C22⋊C8—264no—
442C22⋊C8—262no—
443#95—261no—
444#95—261no—
445#95—261no—
446C2×C13—24no—
447C2×C13—24no—
448C2×C13—42no—
449C2×C13#241011yescharacteristic
450C2×C13C2×#45111yescharacteristic
451C2×C13#300211yescharacteristic
452C2×C13C2×#46711yescharacteristic
453C2×C13#305611yescharacteristic
454C2×C13#318911yescharacteristic
455C2×C13#319011yescharacteristic
456C24—22no—
457C24—22no—
458C24—22no—
459C24—22no—
460C24—41no—
461C24—41no—
462C24#610714yescomplemented
463C24C2×C22⋊D1311yescharacteristic
464C24C2×C22⋊D1311yescharacteristic
465C24C2×C22⋊D1311yescharacteristic
466C2×Q8—24no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—48no—
473C2×Q8—48no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—261no—
477C2×Q8—261no—
478C2×Q8—262no—
479C2×Q8—262no—
480C2×Q8C2×C22⋊D1314yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—261no—
510C22×C4—261no—
511C22×C4—261no—
512C22×C4—261no—
513C22×C4D4×D1312yes—
514C22×C4#612912yes—
515C22×C4C2×C22⋊D1311yescharacteristic
516C22×C4C2×C22⋊D1311yescharacteristic
517C22×C4C2×C22⋊D1311yescharacteristic
518C22×C4C2×C22⋊D1314yes—
519C22×C4C23×D1311yescharacteristic
520Q16—528no—
521C2×C8—134no—
522C2×C8—262no—
523C2×C8—262no—
524C2×C8—262no—
525C2×C8—262no—
526C2×C8—262no—
527C2×C8—528no—
528C4⋊C4—24no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—22no—
532C4⋊C4—261no—
533C4⋊C4—261no—
534C4⋊C4—262no—
535C4⋊C4—262no—
536C4⋊C4—522no—
537C4⋊C4—522no—
538C4⋊C4—522no—
539C4⋊C4C2×C22⋊D1314yes—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—522no—
543C22⋊C4—522no—
544C22⋊C4—522no—
545C22⋊C4—522no—
546C22⋊C4—522no—
547C22⋊C4—522no—
548C13#2438011yescharacteristic, complemented, Sylow
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—28no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—28no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23#8819312yes—
571C23#8819612yes—
572C23C2×#610712yes—
573C23C2×#610712yes—
574C23C2×#610712yes—
575C23#8820412yes—
576C23#8820712yes—
577C23C22×C22⋊D1311yescharacteristic
578Q8—24no—
579Q8—24no—
580Q8—48no—
581Q8—42no—
582Q8—42no—
583Q8—522no—
584Q8—522no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—521no—
610C2×C4—521no—
611C2×C4—521no—
612C2×C4—521no—
613C2×C4—521no—
614C2×C4—521no—
615C2×C4—521no—
616C2×C4—521no—
617C2×C4—521no—
618C2×C4—521no—
619C2×C4—521no—
620C2×C4—521no—
621C2×C4#8819312yes—
622C2×C4#8820712yes—
623C2×C4C2×D4×D1311yescharacteristic
624C2×C4C2×#612911yescharacteristic
625C2×C4#8825112yes—
626C2×C4C22×C22⋊D1312yes—
627C8—264no—
628C8—522no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—416no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22#1171871214yescomplemented
650C22#1171888914yescomplemented
651C22C2×#8819311yescharacteristic
652C22#1171950911yescharacteristic
653C22C2×#8819611yescharacteristic
654C22C22×#610711yescharacteristic
655C22C2×#8820411yescharacteristic
656C22#1171952711yescharacteristic
657C22C2×#8820711yescharacteristic
658C4—22no—
659C4—22no—
660C4—41no—
661C4—44no—
662C4—44no—
663C4—1041no—
664C4—1041no—
665C4—1041no—
666C4—1041no—
667C4D4×C22⋊D1312yes—
668C4#1171951112yes—
669C2—24no—
670C2—24no—
671C2—42no—
672C2#1311821011yescharacteristic
673C2C2×#1171888911yescharacteristic
674C2#1312405411yescharacteristic
675C2C2×#1171871211yescharacteristic
676C2#1312446211yescharacteristic
677C2#1312560811yescharacteristic
678C2#1312561711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC22×C13×C22⋊Q8C2
socleC23×C13C22×D4
prime radicalC25×C13D4
hypercenterC22×C22⋊Q8D13
hypocenterC13#24380
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11718712C2semidirect
2. C2×#11718889C2semidirect
5. #13116036C2semidirect
6. #13115844C2semidirect
7. #13115260C2semidirect
8. #13115143C2semidirect
9. #13115024C2semidirect
10. #13114771C2semidirect
22. C22×#6107C22semidirect
29. C2×#88075C22semidirect
30. C2×#88075C22semidirect
31. C2×#88075C22semidirect
43. C2×#88050C22semidirect
44. C2×#88047C22semidirect
45. #11718121C22semidirect
46. #11718121C22semidirect
124. C24×C13Q16semidirect
356. C22×C13#451semidirect
357. C22×C13#467semidirect
462. C24#6107semidirect
548. C13#24380semidirect
649. C22#11718712semidirect
650. C22#11718889semidirect

Maximal subgroups

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

subgroupindex
C2×#117187122
C2×#117188892
C22×C22⋊#10552
#131231232
#131160362
#131158442
#131152602
#131151432
#131150242
#131147712
C22×C13×C22⋊Q82
#2438013