Finite Group Databasebeta

← #16029529

Subgroups — #16029529

Order 3328.

2002 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13121035C214yesmaximal, complemented
2#13121011C214yesmaximal, complemented
3#13120950C211yesmaximal, characteristic, complemented
4#13120809C212yesmaximal, complemented
5#13120644C212yesmaximal
6#13120643C212yesmaximal
7#13118113C214yesmaximal, complemented
8#13118107C214yesmaximal, complemented
9C22⋊Q8×Dic13C212yesmaximal, complemented
10#13117364C212yesmaximal, complemented
11#13117218C212yesmaximal
12C2×#11718327C211yesmaximal, characteristic
13C13×#2659C211yesmaximal, characteristic, complemented
14C23×C4×D13C2211yescharacteristic
15C2×C13×C22⋊Q8C2212yes—
16C2×Q8×Dic13C2212yescomplemented
17C22×#6103C2211yescharacteristic
18C22×#6103C2211yescharacteristic
19C2×#88178C2212yes—
20C22×C4×Dic13C2211yescharacteristic
21C22×C4×Dic13C2211yescharacteristic
22C2×#88146C2212yes—
23C2×#88146C2212yes—
24C2×#88141C2212yescomplemented
25C13×#398C2214yescomplemented
26C13×#395C2214yescomplemented
27C13×#380C2212yes—
28C13×#380C2212yes—
29C2×C13×#95C2211yescharacteristic
30#11718892C2214yes—
31#11718891C2212yescomplemented
32#11718873C2212yes—
33#11718873C2212yes—
34C2×#88071C2211yescharacteristic
35#11718655C2214yes—
36#11718655C2214yes—
37#11718655C2214yes—
38#11718655C2214yes—
39#11718653C2214yes—
40#11718648C2212yes—
41#11718648C2212yes—
42#11718647C2214yescomplemented
43#11718646C2212yescomplemented
44#11718645C2214yescomplemented
45#11718642C2214yescomplemented
46C4⋊C4×Dic13C2212yescomplemented
47C4⋊C4×Dic13C2214yescomplemented
48C4⋊C4×Dic13C2212yescomplemented
49#11718635C2212yescomplemented
50#11718634C2212yescomplemented
51#11718616C2214yes—
52#11718611C2214yes—
53#11718607C2214yes—
54C22⋊C4×Dic13C2214yes—
55#11718338C2214yes—
56#11718334C2214yes—
57#11718327C2214yes—
58#11718327C2214yes—
59#11718327C2214yes—
60#11718325C2214yes—
61#11718322C2214yescomplemented
62#11718322C2214yescomplemented
63#11718321C2214yescomplemented
64#11718317C2214yescomplemented
65#11718313C2214yescomplemented
66#11718310C2214yescomplemented
67C24×D13C2311yescharacteristic
68C22×Q8×C13C2312yes—
69C23×C4×C13C2311yescharacteristic
70C23×C4×C13C2311yescharacteristic
71C23×C4×C13C2311yescharacteristic
72C23×Dic13C2311yescharacteristic
73C23×Dic13C2311yescharacteristic
74C23×Dic13C2311yescharacteristic
75C22×C4×D13—42no—
76C22×C4×D13—42no—
77C22×C4×D13Q818yescomplemented
78C22×C4×D13C2312yes—
79C22×C4×D13C2312yes—
80C22×Dic26C2312yes—
81C13×C22⋊Q8—28no—
82C2×C13×C4⋊C4C2314yes—
83C2×C13×C4⋊C4C2312yes—
84C2×C13×C4⋊C4C2312yes—
85C2×C13×C4⋊C4C2312yes—
86C2×C13×C4⋊C4C2312yes—
87C2×C13×C4⋊C4C2312yes—
88C2×C13×C22⋊C4C2314yes—
89C2×C22⋊Dic13C2314yes—
90Q8×Dic13—44no—
91C2×#6103—24no—
92C2×#6103—24no—
93C2×#6103C2314yes—
94C2×#6103C2314yes—
95C2×#6103C2314yes—
96C2×#6103C2314yes—
97C2×#6103C2314yes—
98C2×#6103C2314yes—
99C2×C4⋊Dic13C2312yes—
100C2×C4⋊Dic13C2312yes—
101C2×C4⋊Dic13C2312yes—
102C2×C4⋊Dic13C2312yes—
103C2×C4⋊Dic13C2314yes—
104C2×C4⋊Dic13C2312yes—
105C2×C4⋊Dic13C2314yes—
106#88178—28no—
107C2×#6101C2314yes—
108C2×#6101C2314yes—
109C2×#6101C2314yes—
110C2×C4×Dic13—42no—
111C2×C4×Dic13—42no—
112C2×C4×Dic13C2314yes—
113C2×C4×Dic13C2312yes—
114C2×C4×Dic13C2314yes—
115C2×C4×Dic13C2312yes—
116C2×C4×Dic13C2312yes—
117C2×C4×Dic13C2312yes—
118#88146—44no—
119#88146—44no—
120#88141—44no—
121C13×#95C2314yes—
122C13×#95C2314yes—
123C13×#95C2314yes—
124#88071C2314yes—
125#88071C2314yes—
126#88071C2314yes—
127#88071C2314yes—
128#88071C2314yes—
129#88071C2314yes—
130#88071C2314yes—
131#88071C2314yes—
132#88071C2314yes—
133#34578—131nomaximal, Sylow
134C24×C13C2411yescharacteristic
135C23×D13—22no—
136C23×D13—22no—
137C23×D13—41no—
138C23×D13—41no—
139C23×D13—41no—
140C23×D13—41no—
141C23×D13C2×Q814yes—
142C23×D13C2412yes—
143C2×Q8×C13—24no—
144C2×Q8×C13—42no—
145C2×Q8×C13—42no—
146C22×C4×C13—41no—
147C22×C4×C13—41no—
148C22×C4×C13—41no—
149C22×C4×C13—41no—
150C22×C4×C13—41no—
151C22×C4×C13—41no—
152C22×C4×C13C2×Q814yes—
153C22×C4×C13#4114yes—
154C22×C4×C13#4114yes—
155C22×C4×C13C2414yes—
156C22×C4×C13C2414yes—
157C22×C4×C13C2412yes—
158C22×C4×C13C2412yes—
159C22×C4×C13C2412yes—
160C22×Dic13—22no—
161C22×Dic13—22no—
162C22×Dic13—22no—
163C22×Dic13—22no—
164C22×Dic13—41no—
165C22×Dic13—41no—
166C22×Dic13—41no—
167C22×Dic13—41no—
168C22×Dic13C2×Q814yes—
169C22×Dic13C2412yes—
170C22×Dic13C2412yes—
171C22×Dic13C2414yes—
172C22×Dic13C2414yes—
173C22×Dic13C2411yescharacteristic
174C22×Dic13C2411yescharacteristic
175C2×C4×D13—24no—
176C2×C4×D13—24no—
177C2×C4×D13—416no—
178C2×C4×D13—42no—
179C2×C4×D13—42no—
180C2×C4×D13C2×Q818yes—
181C2×C4×D13C2×Q818yes—
182C2×Dic26—24no—
183C2×Dic26—42no—
184C2×Dic26—42no—
185C13×C4⋊C4—24no—
186C13×C4⋊C4—28no—
187C13×C4⋊C4—42no—
188C13×C4⋊C4—42no—
189C13×C4⋊C4—42no—
190C13×C4⋊C4—42no—
191C13×C22⋊C4—28no—
192C22⋊Dic13—28no—
193#6103—28no—
194#6103—28no—
195#6103—44no—
196#6103—44no—
197#6103—44no—
198#6103—44no—
199C4⋊Dic13—24no—
200C4⋊Dic13—42no—
201C4⋊Dic13—42no—
202C4⋊Dic13—44no—
203C4⋊Dic13—42no—
204C4⋊Dic13—44no—
205C4⋊Dic13—42no—
206#6101—28no—
207#6101—44no—
208#6101—44no—
209C4×Dic13—28no—
210C4×Dic13—24no—
211C4×Dic13—28no—
212C4×Dic13—24no—
213C4×Dic13—42no—
214C4×Dic13—42no—
215#2667—134no—
216#2665—134no—
217#2659—131no—
218#2659D1311yescharacteristic, complemented
219#2646—132no—
220#2626—132no—
221#2626—132no—
222#2396—134no—
223#2396—134no—
224C4×C22⋊Q8—132no—
225C4×C22⋊Q8—132no—
226#2280—132no—
227C2×C4×C22⋊C4—131no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13C22×Q812yes—
233C23×C13C2×#4112yes—
234C23×C13C2×#4112yes—
235C23×C13C2511yescharacteristic
236C22×D13—28no—
237C22×D13—22no—
238C22×D13—22no—
239C22×D13—22no—
240C22×D13—22no—
241C22×D13—22no—
242C22×D13—22no—
243C22×D13—44no—
244C22×D13—44no—
245C22×D13—44no—
246C22×D13—44no—
247C22×D13—44no—
248C22×D13—44no—
249C22×D13C22×Q814yes—
250Q8×C13—48no—
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—24no—
258C2×C4×C13—24no—
259C2×C4×C13—24no—
260C2×C4×C13—24no—
261C2×C4×C13—24no—
262C2×C4×C13—24no—
263C2×C4×C13—44no—
264C2×C4×C13—44no—
265C2×C4×C13—44no—
266C2×C4×C13C22×Q814yes—
267C2×C4×C13C2×#4114yes—
268C2×C4×C13C2×#4114yes—
269C2×Dic13—21no—
270C2×Dic13—21no—
271C2×Dic13—28no—
272C2×Dic13—24no—
273C2×Dic13—24no—
274C2×Dic13—22no—
275C2×Dic13—22no—
276C2×Dic13—22no—
277C2×Dic13—22no—
278C2×Dic13—24no—
279C2×Dic13—24no—
280C2×Dic13—22no—
281C2×Dic13—24no—
282C2×Dic13—22no—
283C2×Dic13—24no—
284C2×Dic13—44no—
285C2×Dic13—44no—
286C2×Dic13C22×Q812yes—
287C2×Dic13C22×Q812yes—
288C2×Dic13H(5,𝔽2)12yes—
289C2×Dic13H(5,𝔽2)12yes—
290C4×D13—216no—
291C4×D13—48no—
292Dic26—48no—
293C24×C4—131no—
294C2×C22⋊Q8—132no—
295C2×C22⋊Q8C2×D1312yes—
296C2×C4×Q8—132no—
297C2×C4×Q8—132no—
298C2×#117—132no—
299C2×#117—132no—
300C22×C22⋊C4—131no—
301C22×C22⋊C4—131no—
302C22×C42—131no—
303C22×C42—131no—
304#400—134no—
305#400—134no—
306#399—134no—
307#399—134no—
308#398—134no—
309#398—134no—
310#398—134no—
311#398C2×D1314yescomplemented
312#396—134no—
313#396—134no—
314#396—134no—
315#396—134no—
316#395—134no—
317#395C2×D1314yescomplemented
318#393—134no—
319#393—134no—
320#391—132no—
321#391—132no—
322#389—132no—
323#389—132no—
324#387—134no—
325#387—134no—
326#385—134no—
327#385—134no—
328#381—134no—
329#380—132no—
330#380—132no—
331#380—132no—
332#380—132no—
333#380C2×D1312yes—
334#380C2×D1312yes—
335#379—134no—
336C4×C4⋊C4—134no—
337C4×C4⋊C4—132no—
338C4×C4⋊C4—132no—
339C4×C4⋊C4—134no—
340C4×C22⋊C4—134no—
341C4×C22⋊C4—134no—
342C4×C22⋊C4—134no—
343C4×C22⋊C4—134no—
344C2×#95—131no—
345C2×#95C2×D1311yescharacteristic
346C22×C13—22no—
347C22×C13—22no—
348C22×C13—22no—
349C22×C13—22no—
350C22×C13—22no—
351C22×C13—22no—
352C22×C13#53314yes—
353C22×C13C23×Q811yescharacteristic
354C22×C13C22×#4111yescharacteristic
355C22×C13C22×#4111yescharacteristic
356C22×C13C2×H(5,𝔽2)11yescharacteristic
357C22×C13C2×H(5,𝔽2)11yescharacteristic
358C22×C13#58511yescharacteristic
359C22×C13#58511yescharacteristic
360C2×D13—24no—
361C2×D13—24no—
362C2×D13—24no—
363C2×D13—416no—
364C2×D13—42no—
365C2×D13—42no—
366C2×D13—42no—
367C2×D13—42no—
368C4×C13—24no—
369C4×C13—24no—
370C4×C13—24no—
371C4×C13—44no—
372C4×C13—44no—
373Dic13—22no—
374Dic13—44no—
375Dic13—44no—
376Dic13—42no—
377Dic13—42no—
378Dic13#54314yes—
379C25—131no—
380C22×Q8—132no—
381C22×Q8C22×D1312yes—
382C23×C4—131no—
383C23×C4—132no—
384C23×C4—138no—
385C23×C4—132no—
386C23×C4—131no—
387C23×C4—131no—
388C23×C4—522no—
389C23×C4—522no—
390C23×C4C22×D1311yescharacteristic
391C23×C4C22×D1311yescharacteristic
392C23×C4C22×D1311yescharacteristic
393C22⋊Q8—28no—
394C22⋊Q8—268no—
395C4×Q8—524no—
396C4×Q8—524no—
397#117—524no—
398#117—524no—
399C2×C4⋊C4—134no—
400C2×C4⋊C4—132no—
401C2×C4⋊C4—132no—
402C2×C4⋊C4—132no—
403C2×C4⋊C4—134no—
404C2×C4⋊C4—132no—
405C2×C4⋊C4—134no—
406C2×C4⋊C4—134no—
407C2×C4⋊C4—132no—
408C2×C4⋊C4—134no—
409C2×C4⋊C4C22×D1314yes—
410C2×C4⋊C4C22×D1312yes—
411C2×C4⋊C4C22×D1312yes—
412C2×C4⋊C4C22×D1312yes—
413C2×C4⋊C4C22×D1312yes—
414C2×C4⋊C4C22×D1312yes—
415C2×C22⋊C4—134no—
416C2×C22⋊C4—134no—
417C2×C22⋊C4—134no—
418C2×C22⋊C4—134no—
419C2×C22⋊C4—134no—
420C2×C22⋊C4—134no—
421C2×C22⋊C4—134no—
422C2×C22⋊C4—264no—
423C2×C22⋊C4—264no—
424C2×C22⋊C4C22×D1314yes—
425C2×C42—134no—
426C2×C42—134no—
427C2×C42—132no—
428C2×C42—132no—
429C2×C42—132no—
430C2×C42—132no—
431C2×C42—522no—
432C2×C42—522no—
433#95—134no—
434#95—134no—
435#95—134no—
436#95—134no—
437#95—134no—
438#95—134no—
439#95—134no—
440#95—134no—
441#95—134no—
442#95C22×D1314yes—
443#95C22×D1314yes—
444#95C22×D1314yes—
445C2×C13—24no—
446C2×C13C2×#53311yescharacteristic
447C2×C13C2×#54311yescharacteristic
448C2×C13#345811yescharacteristic
449C2×C13#345911yescharacteristic
450C2×C13#345911yescharacteristic
451C2×C13#346011yescharacteristic
452C2×C13#346011yescharacteristic
453D13—44no—
454C24—132no—
455C24—134no—
456C24—262no—
457C24—262no—
458C24—521no—
459C24—521no—
460C24—521no—
461C24—521no—
462C24C23×D1311yescharacteristic
463C2×Q8—24no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—264no—
467C2×Q8—522no—
468C2×Q8—522no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—131no—
476C22×C4—134no—
477C22×C4—131no—
478C22×C4—138no—
479C22×C4—138no—
480C22×C4—132no—
481C22×C4—132no—
482C22×C4—134no—
483C22×C4—134no—
484C22×C4—264no—
485C22×C4—264no—
486C22×C4—262no—
487C22×C4—262no—
488C22×C4—262no—
489C22×C4—262no—
490C22×C4—522no—
491C22×C4—522no—
492C22×C4—5216no—
493C22×C4—521no—
494C22×C4—521no—
495C22×C4—521no—
496C22×C4—521no—
497C22×C4#612914yes—
498C22×C4Q8×D1314yes—
499C22×C4#613114yes—
500C22×C4C23×D1314yes—
501C22×C4C23×D1314yes—
502C22×C4C23×D1312yes—
503C22×C4C23×D1312yes—
504C22×C4C23×D1312yes—
505C4⋊C4—24no—
506C4⋊C4—28no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—268no—
512C4⋊C4—264no—
513C4⋊C4—522no—
514C4⋊C4—522no—
515C4⋊C4—524no—
516C4⋊C4—522no—
517C4⋊C4—524no—
518C4⋊C4—524no—
519C4⋊C4—522no—
520C4⋊C4—524no—
521C22⋊C4—28no—
522C22⋊C4—268no—
523C22⋊C4—268no—
524C22⋊C4—268no—
525C22⋊C4—524no—
526C22⋊C4—524no—
527C22⋊C4—524no—
528C22⋊C4—524no—
529C42—264no—
530C42—264no—
531C42—268no—
532C42—268no—
533C42—522no—
534C42—522no—
535C13#3457811yescharacteristic, complemented, Sylow
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—134no—
541C23—268no—
542C23—262no—
543C23—262no—
544C23—262no—
545C23—262no—
546C23—262no—
547C23—262no—
548C23—524no—
549C23—524no—
550C23—524no—
551C23—524no—
552C23—524no—
553C23—524no—
554C23C2×#612912yes—
555C23C2×Q8×D1312yes—
556C23C2×#613112yes—
557C23C24×D1311yescharacteristic
558Q8—48no—
559Q8—528no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—132no—
576C2×C4—132no—
577C2×C4—132no—
578C2×C4—132no—
579C2×C4—2616no—
580C2×C4—268no—
581C2×C4—261no—
582C2×C4—261no—
583C2×C4—262no—
584C2×C4—262no—
585C2×C4—264no—
586C2×C4—264no—
587C2×C4—262no—
588C2×C4—262no—
589C2×C4—264no—
590C2×C4—264no—
591C2×C4—264no—
592C2×C4—264no—
593C2×C4—262no—
594C2×C4—262no—
595C2×C4—528no—
596C2×C4—524no—
597C2×C4—524no—
598C2×C4C2×#612914yes—
599C2×C4C2×Q8×D1314yes—
600C2×C4C2×#613114yes—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—264no—
608C22—264no—
609C22—264no—
610C22—5216no—
611C22—522no—
612C22—522no—
613C22—522no—
614C22—522no—
615C22#1171943314yes—
616C22C22×#612911yescharacteristic
617C22C22×Q8×D1311yescharacteristic
618C22C22×#613111yescharacteristic
619C22D13×H(5,𝔽2)11yescharacteristic
620C22D13×H(5,𝔽2)11yescharacteristic
621C22#1171965011yescharacteristic
622C22#1171965211yescharacteristic
623C4—24no—
624C4—24no—
625C4—24no—
626C4—44no—
627C4—44no—
628C4—134no—
629C4—262no—
630C4—524no—
631C4—522no—
632C4—524no—
633C4—522no—
634C2—24no—
635C2—524no—
636C2C2×#1171943311yescharacteristic
637C2D13×#54311yescharacteristic
638C2#1312824011yescharacteristic
639C2#1312824211yescharacteristic
640C2#1312824811yescharacteristic
641C2#1312825311yescharacteristic
642C2#1312825911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D13
commutator, socle, cosocleC23×C13C25
FittingC13×#2659C2
prime radicalC24×D13C23
hypercenter#2659D13
hypocenterC13#34578
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #2659

Complements

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

NH ≅ G/Ntype
1. #13121035C2semidirect
2. #13121011C2semidirect
3. #13120950C2semidirect
4. #13120809C2semidirect
7. #13118113C2semidirect
8. #13118107C2semidirect
9. C22⋊Q8×Dic13C2semidirect
10. #13117364C2semidirect
13. C13×#2659C2semidirect
16. C2×Q8×Dic13C22semidirect
24. C2×#88141C22semidirect
25. C13×#398C22semidirect
26. C13×#395C22semidirect
31. #11718891C22semidirect
42. #11718647C22semidirect
43. #11718646C22semidirect
44. #11718645C22semidirect
45. #11718642C22semidirect
46. C4⋊C4×Dic13C22semidirect
47. C4⋊C4×Dic13C22semidirect
48. C4⋊C4×Dic13C22semidirect
49. #11718635C22semidirect
50. #11718634C22semidirect
61. #11718322C22semidirect
62. #11718322C22semidirect
63. #11718321C22semidirect
64. #11718317C22semidirect
65. #11718313C22semidirect
66. #11718310C22semidirect
77. C22×C4×D13Q8semidirect
218. #2659D13semidirect
311. #398C2×D13semidirect
317. #395C2×D13semidirect
535. C13#34578semidirect

Maximal subgroups

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

subgroupindex
#131210352
#131210112
#131209502
#131208092
#131206442
#131206432
#131181132
#131181072
C22⋊Q8×Dic132
#131173642
#131172182
C2×#117183272
C13×#26592
#3457813