Finite Group Databasebeta

← #15866835

Subgroups — #15866835

Order 3328.

1782 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88203C211yesmaximal, characteristic
2C2×#11718625C211yesmaximal, characteristic, complemented
3C2×#11718792C211yesmaximal, characteristic, complemented
4C2×M16×Dic13C211yesmaximal, characteristic, complemented
5#13117327C211yesmaximal, characteristic, complemented
6C2×#11718649C211yesmaximal, characteristic, complemented
7#13115714C212yesmaximal, complemented
8#13115709C212yesmaximal, complemented
9#13115699C212yesmaximal, complemented
10#13115687C212yesmaximal, complemented
11C2×C13×#417C211yesmaximal, characteristic, complemented
12C22×C13×#41C2211yescharacteristic
13C23×D52C2211yescharacteristic
14C22×C13×M16C2211yescharacteristic
15C22×C13×C4⋊C4C2211yescharacteristic
16C2×#88203—24no—
17C2×#88203—24no—
18C2×#88203C418yescomplemented
19C22×#6106C2212yes—
20C22×#6104C2212yes—
21C22×#6103—22no—
22C2×#88180C2211yescharacteristic
23C22×C4×Dic13C2211yescharacteristic
24C22×#6099C2211yescharacteristic
25C13×#417—24no—
26C2×C13×#103C2212yescomplemented
27C2×C13×#102C2212yescomplemented
28#11718891C2211yescharacteristic, complemented
29#11718891C2211yescharacteristic, complemented
30#11718882C2211yescharacteristic, complemented
31#11718882C2211yescharacteristic, complemented
32#11718881—22no—
33C4×C22⋊Dic13—22no—
34#11718792—24no—
35M16×Dic13—24no—
36C2×#88061C2212yes—
37C2×#88061C2212yes—
38C2×#88055C2212yescomplemented
39C2×C8×Dic13C2212yescomplemented
40#11718649—22no—
41#11718649C2212yes—
42#11718649C2212yes—
43#11718625—24no—
44C2×#88048C2212yescomplemented
45C2×#88047C2212yescomplemented
46C24×D13—21no—
47C2×C13×#41—22no—
48C2×C13×#41—22no—
49C2×C13×#41—22no—
50C2×C13×#41—22no—
51C2×C13×#41—22no—
52C2×C13×#41C2×C414yes—
53C22×Q8×C13C2311yescharacteristic
54C22×D4×C13—21no—
55C22×D4×C13C2311yescharacteristic
56C23×C4×C13—21no—
57C23×C4×C13—21no—
58C23×C4×C13C2311yescharacteristic
59C23×Dic13D412yes—
60C22×D52—22no—
61C22×D52—22no—
62C22×D52—22no—
63C22×D52—22no—
64C22×D52—21no—
65C22×D52—22no—
66C22×D52C2×C414yes—
67C22×D52C2312yes—
68C2×C13×M16—22no—
69C2×C13×M16—22no—
70C2×C13×M16D414yes—
71C22×C8×C13C2312yes—
72C2×C13×C4⋊C4—22no—
73C2×C13×C4⋊C4—22no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4C2312yes—
76C2×C22⋊Dic13—21no—
77C2×C22⋊Dic13—22no—
78C2×C22⋊Dic13—22no—
79#88203—232no—
80C2×#6106—24no—
81C2×#6106—24no—
82C2×#6106C2×C418yescomplemented
83C2×#6104—24no—
84C2×#6104—24no—
85C2×#6104C2×C418yescomplemented
86C2×#6103—24no—
87C2×#6103—24no—
88C2×#6103—48no—
89#88180—22no—
90#88180—22no—
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4×Dic13—22no—
94C2×C4×Dic13—22no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13D412yes—
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C2×#6099C2×C414yes—
100C2×#6099D414yes—
101C2×#6099D414yes—
102C22×#1055C2312yes—
103C13×#103—24no—
104C13×#102—24no—
105#88071—22no—
106#88071—22no—
107#88061—24no—
108#88061—24no—
109#88055—24no—
110C8×Dic13—24no—
111#88048—24no—
112#88047—24no—
113#21236—131nomaximal, Sylow
114C24×C13—21no—
115C24×C13C2×D411yescharacteristic
116C23×D13—24no—
117C23×D13—22no—
118C23×D13—44no—
119C23×D13—44no—
120C23×D13—42no—
121C23×D13—42no—
122C13×#41—28no—
123C13×#41—28no—
124C13×#41—44no—
125C13×#41—44no—
126C2×Q8×C13—21no—
127C2×Q8×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13C22×C412yes—
132C2×D4×C13—24no—
133C2×D4×C13—21no—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13—21no—
137C2×D4×C13—21no—
138C2×D4×C13—42no—
139C2×D4×C13—42no—
140C2×D4×C13C22×C412yes—
141C22×C4×C13—21no—
142C22×C4×C13—24no—
143C22×C4×C13—22no—
144C22×C4×C13—42no—
145C22×C4×C13—42no—
146C22×C4×C13—44no—
147C22×C4×C13—42no—
148C22×C4×C13C22×C412yes—
149C22×C4×C13C2×D412yes—
150C22×C4×C13C2×D412yes—
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2×D412yes—
153C22×C4×C13#4112yes—
154C22×C4×C13#4112yes—
155C22×C4×C13C2411yescharacteristic
156C22×Dic13—24no—
157C22×Dic13—24no—
158C22×Dic13—21no—
159C22×Dic13—21no—
160C22×Dic13—42no—
161C22×Dic13C2×D412yes—
162C22×Dic13C2×D412yes—
163C2×D52—28no—
164C2×D52—28no—
165C2×D52—22no—
166C2×D52—22no—
167C2×D52—22no—
168C2×D52—22no—
169C2×D52—24no—
170C2×D52—22no—
171C2×D52—44no—
172C2×D52—42no—
173C2×D52—44no—
174C2×D52—42no—
175C2×D52C22×C414yes—
176C13×M16—28no—
177C2×C8×C13—22no—
178C2×C8×C13—22no—
179C2×C8×C13C2×D414yes—
180C13×C4⋊C4—22no—
181C13×C4⋊C4—22no—
182C13×C4⋊C4—42no—
183C22⋊Dic13—44no—
184C22⋊Dic13—42no—
185C22⋊Dic13—44no—
186#6106—216no—
187#6104—216no—
188#6103—48no—
189#6103—48no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13—21no—
194C4×Dic13—24no—
195C4×Dic13—21no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—42no—
200#6099C22⋊C4116yescomplemented
201C2×#1055C22×C414yes—
202C2×#1055C2×D414yes—
203C2×#1055C2×D414yes—
204C22×AGL(1,ℤ/8)—131no—
205C2×#428—131no—
206C2×#417D1311yescharacteristic, complemented
207C2×#418—131no—
208C2×C4×M16—131no—
209#2297—131no—
210C2×#390—131no—
211#1973—132no—
212#1972—132no—
213#1970—132no—
214#1968—132no—
215C23×C13—22no—
216C23×C13—22no—
217C23×C13—22no—
218C23×C13—41no—
219C23×C13—41no—
220C23×C13—42no—
221C23×C13—42no—
222C23×C13C4×D412yes—
223C23×C13#12112yes—
224C23×C13#12712yes—
225C23×C13C22×D411yescharacteristic
226C22×D13—24no—
227C22×D13—416no—
228C22×D13—42no—
229C22×D13—42no—
230C22×D13—44no—
231C22×D13—44no—
232C22×D13—48no—
233C22×D13—84no—
234C22×D13—84no—
235C22×D13—88no—
236Q8×C13—22no—
237Q8×C13—22no—
238Q8×C13—41no—
239Q8×C13—41no—
240D4×C13—22no—
241D4×C13—22no—
242D4×C13—48no—
243D4×C13—41no—
244D4×C13—41no—
245C2×C4×C13—22no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—48no—
249C2×C4×C13—42no—
250C2×C4×C13—44no—
251C2×C4×C13—84no—
252C2×C4×C13C2×C22⋊C414yes—
253C2×C4×C13C4×D414yes—
254C2×C4×C13C4×D412yes—
255C2×C4×C13#12114yes—
256C2×C4×C13#12112yes—
257C2×C4×C13#12414yes—
258C2×C4×C13#12712yes—
259C2×C4×C13C23×C411yescharacteristic
260C2×C4×C13C22×D411yescharacteristic
261C2×C4×C13C22×D411yescharacteristic
262C2×C4×C13C22×D411yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×Dic13—22no—
266C2×Dic13—22no—
267C2×Dic13—22no—
268C2×Dic13—22no—
269C2×Dic13—22no—
270C2×Dic13—48no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—42no—
274C2×Dic13—42no—
275C2×Dic13AGL(1,ℤ/8)14yescomplemented
276D52—24no—
277D52—24no—
278D52—42no—
279D52—42no—
280D52—48no—
281C8×C13—24no—
282#1055C2×C22⋊C418yescomplemented
283C22×#41C2×D1311yescharacteristic
284C23×D4—131no—
285C2×AGL(1,ℤ/8)—138no—
286C2×AGL(1,ℤ/8)—264no—
287C2×AGL(1,ℤ/8)—264no—
288C22×SD16—132no—
289C22×D8—132no—
290C22×M16—131no—
291C22×M16C2×D1311yescharacteristic
292C2×#117—131no—
293C22×C4⋊C4C2×D1311yescharacteristic
294C22×C22⋊C4—262no—
295C22×C42—131no—
296#428—264no—
297C2×#107—132no—
298C2×#106—132no—
299#418—264no—
300#417—24no—
301C2×#103C2×D1312yescomplemented
302C2×#102—132no—
303C2×#102—132no—
304C2×#102C2×D1312yescomplemented
305C4×M16—264no—
306C2×#97—132no—
307C2×C4×C8—132no—
308#391—131no—
309#391—131no—
310#390—131no—
311#390—131no—
312#390—132no—
313#390—132no—
314#390—262no—
315#386—262no—
316C4×C22⋊C4—262no—
317C22×C13—22no—
318C22×C13—22no—
319C22×C13—24no—
320C22×C13—24no—
321C22×C13—44no—
322C22×C13—42no—
323C22×C13—81no—
324C22×C13—81no—
325C22×C13—82no—
326C22×C13#39014yes—
327C22×C13C2×C4×D411yescharacteristic
328C22×C13C2×#12111yescharacteristic
329C22×C13C2×#12711yescharacteristic
330C22×C13C2×AGL(1,ℤ/8)12yes—
331C22×C13#57512yes—
332C2×D13—48no—
333C2×D13—44no—
334C2×D13—816no—
335C2×D13—82no—
336C2×D13—82no—
337C2×D13—84no—
338C4×C13—42no—
339C4×C13—82no—
340C4×C13#39014yes—
341C4×C13C22×C22⋊C411yescharacteristic
342C4×C13C2×C4×D411yescharacteristic
343C4×C13C2×#12111yescharacteristic
344C4×C13C2×#12411yescharacteristic
345Dic13—24no—
346Dic13—42no—
347Dic13—81no—
348Dic13—81no—
349C25—261no—
350C2×#41—22no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41C4×D1314yes—
356C22×Q8C22×D1311yescharacteristic
357C22×D4—21no—
358C22×D4—134no—
359C22×D4—132no—
360C22×D4—262no—
361C22×D4—262no—
362C22×D4—262no—
363C22×D4—261no—
364C22×D4—262no—
365C22×D4—262no—
366C22×D4C22×D1311yescharacteristic
367C23×C4—21no—
368C23×C4—21no—
369C23×C4—132no—
370C23×C4C22×D1311yescharacteristic
371AGL(1,ℤ/8)—2632no—
372C2×SD16—138no—
373C2×SD16—264no—
374C2×SD16—264no—
375C2×D8—138no—
376C2×D8—264no—
377C2×D8—264no—
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—134no—
381C2×M16—134no—
382C2×M16—134no—
383C2×M16C22⋊D1314yes—
384C22×C8—132no—
385C22×C8C22×D1312yes—
386#117—262no—
387#117—262no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—131no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4C22×D1312yes—
394C2×C22⋊C4—264no—
395C2×C22⋊C4—264no—
396C2×C22⋊C4—262no—
397C2×C22⋊C4—262no—
398C2×C22⋊C4—261no—
399C2×C22⋊C4—528no—
400C2×C42—132no—
401C2×C42—131no—
402C2×C42—131no—
403C2×C42—262no—
404C2×C42—262no—
405C2×C42—261no—
406#107—264no—
407#106—264no—
408#103—24no—
409#102—24no—
410#102—264no—
411#102—264no—
412#97—264no—
413C4×C8—264no—
414#95—262no—
415#95—262no—
416C2×C13—24no—
417C2×C13—82no—
418C2×C13C2×#39011yescharacteristic
419C2×C13C4×AGL(1,ℤ/8)12yes—
420C2×C13#305512yes—
421C2×C13#315111yescharacteristic
422C2×C13#315311yescharacteristic
423D13—84no—
424C24—21no—
425C24—264no—
426C24—262no—
427C24—524no—
428C24—524no—
429C24—522no—
430C24—522no—
431C24D4×D1311yescharacteristic
432#41—28no—
433#41—28no—
434#41—44no—
435#41—44no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C2×C4×D1312yes—
442C2×D4—24no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—134no—
451C2×D4—268no—
452C2×D4—264no—
453C2×D4—268no—
454C2×D4—262no—
455C2×D4—262no—
456C2×D4—262no—
457C2×D4—262no—
458C2×D4—262no—
459C2×D4—522no—
460C2×D4—524no—
461C2×D4—524no—
462C2×D4—522no—
463C2×D4C2×C4×D1312yes—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—132no—
472C22×C4—132no—
473C22×C4—264no—
474C22×C4—264no—
475C22×C4—261no—
476C22×C4—261no—
477C22×C4—522no—
478C22×C4C2×C4×D1312yes—
479C22×C4#612712yes—
480C22×C4D4×D1312yes—
481C22×C4D4×D1312yes—
482C22×C4D4×D1311yescharacteristic
483C22×C4#612912yes—
484C22×C4C2×C22⋊D1312yes—
485C22×C4C23×D1311yescharacteristic
486SD16—2616no—
487D8—2616no—
488M16—28no—
489M16—1316no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—134no—
493C2×C8—134no—
494C2×C8—134no—
495C2×C8C2×C22⋊D1314yes—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—261no—
500C4⋊C4—261no—
501C4⋊C4—261no—
502C4⋊C4—261no—
503C22⋊C4—528no—
504C22⋊C4—528no—
505C22⋊C4—524no—
506C22⋊C4—524no—
507C22⋊C4—522no—
508C42—264no—
509C42—261no—
510C42—261no—
511C42—261no—
512C42—261no—
513C42—522no—
514C13#2123611yescharacteristic, complemented, Sylow
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—264no—
523C23—5216no—
524C23—522no—
525C23—524no—
526C23—524no—
527C23—522no—
528C23—528no—
529C23—1048no—
530C23—1044no—
531C23—1044no—
532C23#8813512yes—
533C23#8813812yes—
534C23#8819412yes—
535C23C2×D4×D1311yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545D4—264no—
546D4—264no—
547D4—522no—
548D4—522no—
549D4—528no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—134no—
558C2×C4—262no—
559C2×C4—262no—
560C2×C4—262no—
561C2×C4—262no—
562C2×C4—262no—
563C2×C4—528no—
564C2×C4—522no—
565C2×C4—522no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4C22⋊C4×D1314yes—
569C2×C4#8813512yes—
570C2×C4#8813812yes—
571C2×C4#8813914yes—
572C2×C4C4×C22⋊D1314yes—
573C2×C4#8819314yes—
574C2×C4#8819412yes—
575C2×C4C22×C4×D1311yescharacteristic
576C2×C4C2×#612711yescharacteristic
577C2×C4C2×D4×D1311yescharacteristic
578C2×C4C2×D4×D1311yescharacteristic
579C2×C4C2×#612911yescharacteristic
580C2×C4C22×C22⋊D1311yescharacteristic
581C8—24no—
582C8—138no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—44no—
588C22—42no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22—528no—
593C22—524no—
594C22—10416no—
595C22—1044no—
596C22—1042no—
597C22—1042no—
598C22#1171861814yes—
599C22C2×#8813511yescharacteristic
600C22C2×#8813811yescharacteristic
601C22D13×AGL(1,ℤ/8)12yes—
602C22#1171948212yes—
603C22C2×#8819411yescharacteristic
604C4—42no—
605C4—82no—
606C4—264no—
607C4—522no—
608C4—1041no—
609C4—1041no—
610C4#1171861814yes—
611C4C2×C22⋊C4×D1311yescharacteristic
612C4C2×#8813911yescharacteristic
613C4C2×C4×C22⋊D1311yescharacteristic
614C4C2×#8819311yescharacteristic
615C2—24no—
616C2—82no—
617C2—1044no—
618C2C2×#1171861811yescharacteristic
619C2#1312364912yes—
620C2#1312445812yes—
621C2#1312528411yescharacteristic
622C2#1312530311yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#417C2
socleC23×C13C22×D4
prime radicalC22×#88203C2
cosocleC22×C4×C13C24
hypercenterC2×#417D13
hypocenterC13#21236
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#11718625C2semidirect
3. C2×#11718792C2semidirect
4. C2×M16×Dic13C2semidirect
5. #13117327C2semidirect
6. C2×#11718649C2semidirect
7. #13115714C2semidirect
8. #13115709C2semidirect
9. #13115699C2semidirect
10. #13115687C2semidirect
11. C2×C13×#417C2semidirect
18. C2×#88203C4semidirect
26. C2×C13×#103C22semidirect
27. C2×C13×#102C22semidirect
28. #11718891C22semidirect
29. #11718891C22semidirect
30. #11718882C22semidirect
31. #11718882C22semidirect
38. C2×#88055C22semidirect
39. C2×C8×Dic13C22semidirect
44. C2×#88048C22semidirect
45. C2×#88047C22semidirect
82. C2×#6106C2×C4semidirect
85. C2×#6104C2×C4semidirect
200. #6099C22⋊C4semidirect
206. C2×#417D13semidirect
275. C2×Dic13AGL(1,ℤ/8)semidirect
282. #1055C2×C22⋊C4semidirect
301. C2×#103C2×D13semidirect
304. C2×#102C2×D13semidirect
514. C13#21236semidirect

Maximal subgroups

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

subgroupindex
C22×#882032
C2×#117186252
C2×#117187922
C2×M16×Dic132
#131173272
C2×#117186492
#131157142
#131157092
#131156992
#131156872
C2×C13×#4172
#2123613