Finite Group Databasebeta

← #15890411

Subgroups — #15890411

Order 3328.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88184C211yesmaximal, characteristic
2C2×#11718656C211yesmaximal, characteristic, complemented
3C2×#11718662C211yesmaximal, characteristic, complemented
4C2×#11718662C211yesmaximal, characteristic, complemented
5C2×#11718868C211yesmaximal, characteristic, complemented
6#13116025C214yesmaximal, complemented
7#13116018C214yesmaximal, complemented
8#13114557C211yesmaximal, characteristic, complemented
9C13×#2280C211yesmaximal, characteristic, complemented
10C24×C4×C13C2211yescharacteristic
11C23×C22⋊D13—21no—
12C23×D52C2211yescharacteristic
13C2×C13×#117C2211yescharacteristic
14C2×C13×#117C2211yescharacteristic
15C2×#88184—22no—
16C2×#88184—24no—
17C2×#88184—24no—
18C2×#88184C2214yes—
19C22×#6103—21no—
20C22×C4⋊Dic13C2211yescharacteristic
21C22×#6099C2211yescharacteristic
22C22×#6099C2211yescharacteristic
23C13×#381—21no—
24C13×#380C2212yescomplemented
25C13×#380C2212yescomplemented
26C13×#379—21no—
27C4×C13×C22⋊C4—22no—
28#11718868—24no—
29C2×C22⋊#1055C2214yescomplemented
30#11718662—24no—
31#11718662—24no—
32#11718656—24no—
33C2×#88049C2212yescomplemented
34C2×#88049C2214yescomplemented
35C2×#88049C2212yescomplemented
36C2×#88048C2212yescomplemented
37C2×#88047C2212yescomplemented
38#11718121C2214yescomplemented
39C25×C13D411yescharacteristic
40C24×D13—21no—
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×C13D412yes—
48C23×C4×C13D411yescharacteristic
49C23×C4×C13C2312yes—
50C23×C4×C13C2311yescharacteristic
51C22×C22⋊D13—28no—
52C22×C22⋊D13—24no—
53C22×C22⋊D13—48no—
54C23×Dic13—21no—
55C22×D52—22no—
56C22×D52—22no—
57C22×D52—22no—
58C22×D52—22no—
59C22×D52—22no—
60C22×D52—21no—
61C22×D52D414yes—
62C22×D52C2312yes—
63C13×#117—22no—
64C13×#117—22no—
65C13×#117—22no—
66C13×#117—22no—
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2312yes—
70C2×C13×C22⋊C4—22no—
71C2×C13×C22⋊C4—22no—
72C2×C13×C22⋊C4—21no—
73C2×C13×C22⋊C4—21no—
74C2×C42×C13—21no—
75C2×C42×C13—21no—
76#88184—216no—
77#88184—28no—
78#88184—28no—
79C2×#6103—28no—
80C2×#6103—24no—
81C2×C4⋊Dic13—22no—
82C2×C4⋊Dic13—21no—
83C2×C4⋊Dic13D414yes—
84C2×C4⋊Dic13C2312yes—
85C2×#6099—22no—
86C2×#6099—22no—
87C2×#6099—22no—
88C2×#6099—22no—
89C2×#6099—22no—
90C2×#6099D414yescomplemented
91C22×#1055C2312yes—
92C22×#1055C2312yes—
93C13×#95—22no—
94C13×#95—22no—
95C22⋊#1055—28no—
96#88049—24no—
97#88049—24no—
98#88049—28no—
99#88048—24no—
100#88047—24no—
101#23480—131nomaximal, Sylow
102C24×C13—22no—
103C24×C13—24no—
104C24×C13—22no—
105C24×C13—22no—
106C24×C13—42no—
107C24×C13C2×D412yes—
108C24×C13C2×D411yescharacteristic
109C23×D13—24no—
110C23×D13—22no—
111C23×D13—44no—
112C23×D13—44no—
113C23×D13—42no—
114C23×D13—42no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—22no—
118C22×C4×C13—24no—
119C22×C4×C13—24no—
120C22×C4×C13—24no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—22no—
130C22×C4×C13—44no—
131C22×C4×C13—44no—
132C22×C4×C13—44no—
133C22×C4×C13—44no—
134C22×C4×C13C2×D412yes—
135C22×C4×C13C2×D412yes—
136C22×C4×C13C2×D412yes—
137C22×C4×C13C2×D412yes—
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13#4112yes—
140C22×C4×C13#4112yes—
141C22×C4×C13#4112yes—
142C22×C4×C13C2411yescharacteristic
143C2×C22⋊D13—216no—
144C2×C22⋊D13—432no—
145C2×C22⋊D13—48no—
146C2×C22⋊D13—48no—
147C22×Dic13—24no—
148C22×Dic13—22no—
149C22×Dic13—44no—
150C2×D52—28no—
151C2×D52—28no—
152C2×D52—22no—
153C2×D52—22no—
154C2×D52—22no—
155C2×D52—22no—
156C2×D52—24no—
157C2×D52—22no—
158C2×D52—44no—
159C2×D52—42no—
160C2×D52—42no—
161C2×D52—44no—
162C2×D52C2×D414yes—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—22no—
168C13×C4⋊C4—22no—
169C13×C22⋊C4—44no—
170C13×C22⋊C4—42no—
171C13×C22⋊C4—42no—
172C13×C22⋊C4—44no—
173C42×C13—42no—
174C42×C13—42no—
175#6103—216no—
176C4⋊Dic13—24no—
177C4⋊Dic13—22no—
178C4⋊Dic13C2×D414yes—
179#6099—28no—
180#6099—44no—
181C2×#1055—22no—
182C2×#1055—22no—
183C2×#1055—22no—
184C2×#1055—22no—
185C2×#1055—22no—
186C2×#1055C2×D414yescomplemented
187C22×#121—131no—
188C2×#428—131no—
189C2×#418—131no—
190C2×#418—131no—
191C2×#407—131no—
192#2280D1311yescharacteristic, complemented
193#2034—134no—
194#2033—134no—
195#1736—131no—
196C23×C13—22no—
197C23×C13—22no—
198C23×C13—28no—
199C23×C13—22no—
200C23×C13—22no—
201C23×C13—22no—
202C23×C13—22no—
203C23×C13—22no—
204C23×C13—22no—
205C23×C13—48no—
206C23×C13—48no—
207C23×C13—44no—
208C23×C13—44no—
209C23×C13#12012yes—
210C23×C13#12112yes—
211C23×C13#12312yes—
212C23×C13AGL(1,ℤ/8)14yes—
213C23×C13C22×D411yescharacteristic
214C22×D13—24no—
215C22×D13—48no—
216C22×D13—48no—
217C22×D13—42no—
218C22×D13—42no—
219C22×D13—44no—
220C22×D13—44no—
221C22×D13—48no—
222C22×D13—88no—
223C22×D13—88no—
224C2×C4×C13—22no—
225C2×C4×C13—22no—
226C2×C4×C13—22no—
227C2×C4×C13—44no—
228C2×C4×C13—42no—
229C2×C4×C13—42no—
230C2×C4×C13—42no—
231C2×C4×C13—42no—
232C2×C4×C13—48no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—42no—
236C2×C4×C13—42no—
237C2×C4×C13—42no—
238C2×C4×C13—42no—
239C2×C4×C13—42no—
240C2×C4×C13#12012yes—
241C2×C4×C13#12114yes—
242C2×C4×C13#12114yes—
243C2×C4×C13#12112yes—
244C2×C4×C13#12314yes—
245C2×C4×C13#12312yes—
246C2×C4×C13#12414yes—
247C2×C4×C13C22×D411yescharacteristic
248C2×C4×C13C22×D411yescharacteristic
249C2×C4×C13C22×D411yescharacteristic
250C2×C4×C13C2×#4111yescharacteristic
251C2×C4×C13C2×#4111yescharacteristic
252C2×C4×C13C2×#4111yescharacteristic
253C22⋊D13—432no—
254C2×Dic13—24no—
255C2×Dic13—48no—
256C2×Dic13—44no—
257D52—24no—
258D52—24no—
259D52—42no—
260D52—42no—
261D52—48no—
262#1055—24no—
263#1055—42no—
264C23×D4—131no—
265C23×D4—261no—
266C24×C4C2×D1311yescharacteristic
267C22×M16—131no—
268C22×M16—131no—
269C2×#121—134no—
270C2×#121—264no—
271C2×#121—264no—
272C2×#121—262no—
273C2×#117C2×D1311yescharacteristic
274C2×#117C2×D1311yescharacteristic
275C22×C4⋊C4—131no—
276C22×C22⋊C4—261no—
277#428—264no—
278C2×#107—132no—
279C2×#106—132no—
280#418—264no—
281#418—264no—
282C2×#102—132no—
283C2×#102—134no—
284C2×#102—132no—
285#407—264no—
286C2×C22⋊C8—134no—
287#381—21no—
288#380C2×D1312yescomplemented
289#380C2×D1312yescomplemented
290#379—21no—
291C4×C22⋊C4—22no—
292#340—134no—
293C22×C13—24no—
294C22×C13—22no—
295C22×C13—22no—
296C22×C13—24no—
297C22×C13—24no—
298C22×C13—24no—
299C22×C13—24no—
300C22×C13—416no—
301C22×C13—42no—
302C22×C13—42no—
303C22×C13—42no—
304C22×C13—42no—
305C22×C13#39414yes—
306C22×C13C2×#12011yescharacteristic
307C22×C13C2×#12111yescharacteristic
308C22×C13C2×#12311yescharacteristic
309C22×C13C2×AGL(1,ℤ/8)12yes—
310C22×C13#57512yes—
311C2×D13—44no—
312C2×D13—44no—
313C2×D13—44no—
314C2×D13—816no—
315C2×D13—84no—
316C2×D13—84no—
317C4×C13—42no—
318C4×C13—81no—
319C4×C13—81no—
320C4×C13—82no—
321C4×C13#39414yes—
322C4×C13C2×#12111yescharacteristic
323C4×C13C2×#12111yescharacteristic
324C4×C13C2×#12311yescharacteristic
325C4×C13C2×#12411yescharacteristic
326Dic13—44no—
327C25—261no—
328C25C22⋊D1311yescharacteristic
329C22×D4—134no—
330C22×D4—132no—
331C22×D4—268no—
332C22×D4—264no—
333C22×D4—262no—
334C22×D4—262no—
335C22×D4—262no—
336C22×D4—262no—
337C22×D4—262no—
338C22×D4—261no—
339C22×D4—528no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—22no—
343C23×C4—22no—
344C23×C4—22no—
345C23×C4—22no—
346C23×C4—261no—
347C23×C4C22⋊D1311yescharacteristic
348C23×C4C22⋊D1312yes—
349C23×C4C22×D1312yes—
350C23×C4C22×D1311yescharacteristic
351C2×M16—134no—
352C2×M16—262no—
353C2×M16—262no—
354C2×M16—262no—
355C2×M16—262no—
356C2×M16—262no—
357C22×C8—132no—
358C22×C8—132no—
359#121—268no—
360#121—268no—
361#121—2616no—
362#117—22no—
363#117—22no—
364#117—22no—
365#117—22no—
366C2×C4⋊C4—134no—
367C2×C4⋊C4—132no—
368C2×C4⋊C4—262no—
369C2×C4⋊C4—261no—
370C2×C4⋊C4C22×D1312yes—
371C2×C4⋊C4C22×D1311yescharacteristic
372C2×C4⋊C4C22×D1311yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—268no—
378C2×C22⋊C4—264no—
379C2×C42—21no—
380C2×C42—21no—
381#107—264no—
382#106—264no—
383#102—268no—
384#102—264no—
385#102—264no—
386C22⋊C8—268no—
387#95—22no—
388#95—22no—
389C2×C13—24no—
390C2×C13—44no—
391C2×C13C2×#39411yescharacteristic
392C2×C13#300311yescharacteristic
393C2×C13#300511yescharacteristic
394C2×C13#305512yes—
395C2×C13#309412yes—
396D13—84no—
397C24—22no—
398C24—24no—
399C24—22no—
400C24—22no—
401C24—42no—
402C24—264no—
403C24—262no—
404C24—524no—
405C24—524no—
406C24—522no—
407C24—522no—
408C24C2×C22⋊D1312yes—
409C24C2×C22⋊D1311yescharacteristic
410C2×D4—134no—
411C2×D4—2616no—
412C2×D4—268no—
413C2×D4—268no—
414C2×D4—264no—
415C2×D4—262no—
416C2×D4—262no—
417C2×D4—262no—
418C2×D4—262no—
419C2×D4—262no—
420C2×D4—5232no—
421C2×D4—528no—
422C2×D4—528no—
423C2×D4—524no—
424C2×D4—524no—
425C2×D4—522no—
426C2×D4—522no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—264no—
447C22×C4—262no—
448C22×C4—524no—
449C22×C4#612712yes—
450C22×C4#612712yes—
451C22×C4D4×D1312yes—
452C22×C4D4×D1312yes—
453C22×C4D4×D1312yes—
454C22×C4#612912yes—
455C22×C4C2×C22⋊D1312yes—
456C22×C4C2×C22⋊D1311yescharacteristic
457C22×C4C23×D1311yescharacteristic
458M16—268no—
459M16—524no—
460C2×C8—134no—
461C2×C8—262no—
462C2×C8—262no—
463C2×C8—262no—
464C2×C8—262no—
465C2×C8—262no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—134no—
473C4⋊C4—264no—
474C4⋊C4—262no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—2616no—
480C42—42no—
481C42—42no—
482C13#2348011yescharacteristic, complemented, Sylow
483C23—22no—
484C23—22no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—48no—
493C23—48no—
494C23—44no—
495C23—44no—
496C23—264no—
497C23—528no—
498C23—528no—
499C23—522no—
500C23—522no—
501C23—524no—
502C23—524no—
503C23—528no—
504C23—1048no—
505C23—1048no—
506C23#8818312yes—
507C23#8819112yes—
508C23#8819312yes—
509C23#8820314yes—
510C23C22×C22⋊D1311yescharacteristic
511D4—264no—
512D4—264no—
513D4—5232no—
514D4—522no—
515D4—522no—
516D4—528no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—264no—
534C2×C4—528no—
535C2×C4—524no—
536C2×C4#8813714yes—
537C2×C4#8813814yes—
538C2×C4#8813814yes—
539C2×C4#8813914yes—
540C2×C4#8818312yes—
541C2×C4#8819112yes—
542C2×C4#8819312yes—
543C2×C4C2×#612711yescharacteristic
544C2×C4C2×#612711yescharacteristic
545C2×C4C2×D4×D1311yescharacteristic
546C2×C4C2×D4×D1311yescharacteristic
547C2×C4C2×D4×D1311yescharacteristic
548C2×C4C2×#612911yescharacteristic
549C8—264no—
550C8—522no—
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—524no—
564C22—524no—
565C22—524no—
566C22—10416no—
567C22—1044no—
568C22—1044no—
569C22#1171862014yes—
570C22C2×#8818311yescharacteristic
571C22C2×#8819111yescharacteristic
572C22C2×#8819311yescharacteristic
573C22C2×#8820312yes—
574C22#1171952712yes—
575C4—42no—
576C4—82no—
577C4—81no—
578C4—81no—
579C4—524no—
580C4#1171862014yes—
581C4C2×#8813711yescharacteristic
582C4C2×#8813811yescharacteristic
583C4C2×#8813811yescharacteristic
584C4C2×#8813911yescharacteristic
585C2—24no—
586C2—44no—
587C2—1044no—
588C2C2×#1171862011yescharacteristic
589C2#1312406211yescharacteristic
590C2#1312407611yescharacteristic
591C2#1312445712yes—
592C2#1312477612yes—
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2280C2
socleC23×C13C22×D4
prime radicalC22×#88184C2
hypercenter#2280D13
hypocenterC13#23480
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C2×#11718656C2semidirect
3. C2×#11718662C2semidirect
4. C2×#11718662C2semidirect
5. C2×#11718868C2semidirect
6. #13116025C2semidirect
7. #13116018C2semidirect
8. #13114557C2semidirect
9. C13×#2280C2semidirect
24. C13×#380C22semidirect
25. C13×#380C22semidirect
29. C2×C22⋊#1055C22semidirect
33. C2×#88049C22semidirect
34. C2×#88049C22semidirect
35. C2×#88049C22semidirect
36. C2×#88048C22semidirect
37. C2×#88047C22semidirect
38. #11718121C22semidirect
90. C2×#6099D4semidirect
186. C2×#1055C2×D4semidirect
192. #2280D13semidirect
288. #380C2×D13semidirect
289. #380C2×D13semidirect
482. C13#23480semidirect

Maximal subgroups

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

subgroupindex
C22×#881842
C2×#117186562
C2×#117186622
C2×#117188682
#131160252
#131160182
#131145572
C13×#22802
#2348013