Finite Group Databasebeta

← #16796395

Subgroups — #16796395

Order 3328.

720 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131673C211yesmaximal, characteristic, complemented
2D4×#6122C211yesmaximal, characteristic, complemented
3#13130479C211yesmaximal, characteristic, complemented
4#13130472C211yesmaximal, characteristic, complemented
5#13130457C211yesmaximal, characteristic, complemented
6#13130453C211yesmaximal, characteristic, complemented
7#13130441C211yesmaximal, characteristic, complemented
8#13130437C211yesmaximal, characteristic, complemented
9#13130432C211yesmaximal, characteristic, complemented
10D8×#1057C211yesmaximal, characteristic, complemented
11#13130400C211yesmaximal, characteristic, complemented
12#13130175C211yesmaximal, characteristic, complemented
13#13130171C211yesmaximal, characteristic, complemented
14#13130059C211yesmaximal, characteristic, complemented
15AGL(1,ℤ/8)×Dic13C211yesmaximal, characteristic
16C22×#6122—21no—
17C2×C13×AGL(1,ℤ/8)C411yescharacteristic
18#41×Dic13C2211yescharacteristic
19C2×D4×Dic13C2211yescharacteristic
20#11719169—21no—
21#11719158C2211yescharacteristic, complemented
22#11719158C2211yescharacteristic, complemented
23#11719145C2211yescharacteristic
24#11719145C2211yescharacteristic
25#11719144—21no—
26#11719144—21no—
27#11719144—21no—
28#11719144—21no—
29D4×#1057C2211yescharacteristic
30D4×#1057C2211yescharacteristic
31D4×#1057C2211yescharacteristic, complemented
32D4×#1057C2211yescharacteristic, complemented
33#11719130C2211yescharacteristic
34#11719129C2211yescharacteristic
35#11719128—21no—
36C4×#6122C2211yescharacteristic
37C4×#6122C2211yescharacteristic
38C2×#88110—21no—
39#11718893C411yescharacteristic
40#11718876C411yescharacteristic
41#11718825C411yescharacteristic
42#11718825C411yescharacteristic
43SD16×Dic13C2211yescharacteristic
44SD16×Dic13C2211yescharacteristic
45#11718813C411yescharacteristic
46#11718813C411yescharacteristic
47D8×Dic13C2211yescharacteristic
48D8×Dic13C2211yescharacteristic
49#11718782C411yescharacteristic
50M16×Dic13C2211yescharacteristic
51Q8⋊#1057C2211yescharacteristic, complemented
52Q8⋊#1057C2211yescharacteristic, complemented
53D4⋊#1057C2211yescharacteristic
54D4⋊#1057C2211yescharacteristic
55D4⋊#1057C2211yescharacteristic
56D4⋊#1057C2211yescharacteristic
57D4⋊#1057C2211yescharacteristic, complemented
58D4⋊#1057C2211yescharacteristic, complemented
59#11718256C2211yescharacteristic, complemented
60#11718255C2211yescharacteristic, complemented
61#11718254C2211yescharacteristic, complemented
62#11718253C2211yescharacteristic, complemented
63#1057⋊C8C2211yescharacteristic, complemented
64#1057⋊C8C2211yescharacteristic, complemented
65#11718251C2211yescharacteristic, complemented
66C8×#1057C2211yescharacteristic, complemented
67C2×C13×#41C2×C411yescharacteristic
68C22×D4×C13C2×C411yescharacteristic
69C23×Dic13—21no—
70C2×#6122—21no—
71C2×#6122—21no—
72C2×#6122—21no—
73C2×#6122—21no—
74C2×#6122—44no—
75C2×#6122D411yescharacteristic
76C2×#6122D411yescharacteristic
77C22×#1057—21no—
78C22×#1057—21no—
79C22×#1057—21no—
80C22×#1057—21no—
81C13×AGL(1,ℤ/8)—24no—
82C2×C13×SD16C2×C411yescharacteristic
83C2×C13×SD16C2×C411yescharacteristic
84C2×C13×D8C2×C411yescharacteristic
85C2×C13×D8C2×C411yescharacteristic
86C2×C13×M16C2×C411yescharacteristic
87C2×C22⋊Dic13—21no—
88Q8×Dic13C2×C411yescharacteristic
89Q8×Dic13C2311yescharacteristic
90D4×Dic13—21no—
91D4×Dic13—21no—
92D4×Dic13—21no—
93D4×Dic13—21no—
94D4×Dic13C2×C411yescharacteristic
95D4×Dic13C2×C411yescharacteristic
96D4×Dic13C2×C411yescharacteristic
97D4×Dic13C2311yescharacteristic
98D4×Dic13C2311yescharacteristic
99D4×Dic13C2311yescharacteristic
100#88180—21no—
101#88180C2×C411yescharacteristic
102C2×C4⋊Dic13C2×C411yescharacteristic
103C2×C4×Dic13—21no—
104C2×C4×Dic13C2311yescharacteristic
105C2×#6099C2×C411yescharacteristic
106C22⋊#1057—21no—
107C22⋊#1057—21no—
108C22⋊#1057—21no—
109C22⋊#1057—21no—
110C22⋊#1057—21no—
111C22⋊#1057—21no—
112C22⋊#1057—21no—
113#88112—21no—
114#88112—21no—
115#88112—21no—
116#88110—21no—
117#88110—21no—
118#88110—21no—
119#88110—21no—
120#88110D411yescharacteristic
121#88110D411yescharacteristic
122C4⋊#1057C2311yescharacteristic
123C4⋊#1057C2311yescharacteristic
124C4⋊#1057C2311yescharacteristic
125C4⋊#1057C2311yescharacteristic
126C4×#1057C2311yescharacteristic
127C4×#1057C2311yescharacteristic
128C4×#1057C2311yescharacteristic
129C4×#1057C2311yescharacteristic
130#88075C2×C411yescharacteristic
131#88075C2×C411yescharacteristic
132#88072C2×C411yescharacteristic
133#88072C2×C411yescharacteristic
134#88072C2×C411yescharacteristic
135#88072C2×C411yescharacteristic
136#88072C2×C411yescharacteristic
137#88072C2×C411yescharacteristic
138#88058C2×C411yescharacteristic
139#88058C2×C411yescharacteristic
140#88057C2×C411yescharacteristic
141#88057C2×C411yescharacteristic
142#88055C2×C411yescharacteristic
143#88055C2×C411yescharacteristic
144C8×Dic13C2311yescharacteristic
145C8×Dic13C2311yescharacteristic
146#18046—131nomaximal, Sylow
147C24×C13—21no—
148C13×#41—22no—
149C13×#41—41no—
150C2×Q8×C13C22×C411yescharacteristic
151C2×D4×C13—22no—
152C2×D4×C13—21no—
153C2×D4×C13—21no—
154C2×D4×C13—41no—
155C2×D4×C13C22×C411yescharacteristic
156C2×D4×C13C22×C411yescharacteristic
157C2×D4×C13C22×C411yescharacteristic
158C22×C4×C13—21no—
159C22×C4×C13C22×C411yescharacteristic
160C22×Dic13—21no—
161C22×Dic13—21no—
162C22×Dic13—21no—
163C22×Dic13—21no—
164C22×Dic13—21no—
165C22×Dic13—21no—
166C22×Dic13—21no—
167C22×Dic13—42no—
168C22×Dic13—42no—
169C22×Dic13C2×D411yescharacteristic
170C22×Dic13#4111yescharacteristic
171#6122—22no—
172#6122—41no—
173#6122—42no—
174#6122—42no—
175#6122—42no—
176#6122—42no—
177C2×#1057—21no—
178C2×#1057—21no—
179C2×#1057—21no—
180C2×#1057—21no—
181C2×#1057—21no—
182C2×#1057—21no—
183C2×#1057—42no—
184C2×#1057—42no—
185C2×#1057—42no—
186C2×#1057—42no—
187C2×#1057C2×D411yescharacteristic
188C2×#1057C2×D411yescharacteristic
189C2×#1057C2×D411yescharacteristic
190C2×#1057C2×D411yescharacteristic
191C13×SD16—22no—
192C13×SD16—22no—
193C13×D8—22no—
194C13×D8M1614yescomplemented
195C13×M16—22no—
196C2×C8×C13C22×C411yescharacteristic
197C2×C8×C13C22×C411yescharacteristic
198C22⋊Dic13—21no—
199C22⋊Dic13—21no—
200C22⋊Dic13—21no—
201C22⋊Dic13—21no—
202C22⋊Dic13—21no—
203C22⋊Dic13—21no—
204C22⋊Dic13—21no—
205C4⋊Dic13—21no—
206C4⋊Dic13—21no—
207C4⋊Dic13—21no—
208C4⋊Dic13C22×C411yescharacteristic
209C4⋊Dic13C22×C411yescharacteristic
210C4⋊Dic13C22×C411yescharacteristic
211C4⋊Dic13C22×C411yescharacteristic
212C4×Dic13—21no—
213C4×Dic13—21no—
214C4×Dic13—21no—
215C4×Dic13C22×C411yescharacteristic
216C4×Dic13C2×D411yescharacteristic
217C4×Dic13#4111yescharacteristic
218C4×Dic13C2411yescharacteristic
219#6099—41no—
220C2×#1055C22×C411yescharacteristic
221C2×#1055C22×C411yescharacteristic
222#2970—131no—
223C4×AGL(1,ℤ/8)—131no—
224D4×M16—131no—
225#1592—131no—
226#1590—131no—
227#1586—131no—
228#1585—131no—
229#1582—131no—
230#1581—131no—
231#1580—131no—
232C8×D8—131no—
233#1569—131no—
234#1490—131no—
235#1489—131no—
236#1455—131no—
237C23×C13—21no—
238C23×C13—21no—
239C23×C13—21no—
240C23×C13—42no—
241C23×C13—42no—
242C23×C13—81no—
243C23×C13C4×D411yescharacteristic
244Q8×C13—21no—
245Q8×C13—21no—
246D4×C13—21no—
247D4×C13—21no—
248D4×C13—21no—
249D4×C13—21no—
250D4×C13—42no—
251D4×C13—42no—
252D4×C13C2×M1612yes—
253D4×C13C2×M1612yescomplemented
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—42no—
258C2×C4×C13C4×D411yescharacteristic
259C2×C4×C13C23×C411yescharacteristic
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—41no—
267C2×Dic13—42no—
268C2×Dic13—42no—
269C2×Dic13—42no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—81no—
277C2×Dic13C4×D411yescharacteristic
278C2×Dic13C4×D411yescharacteristic
279C2×Dic13C22×D411yescharacteristic
280C2×Dic13C2×#4111yescharacteristic
281#1057—22no—
282#1057—21no—
283#1057—21no—
284#1057—21no—
285#1057—41no—
286#1057—41no—
287#1057—41no—
288#1057—41no—
289#1057—41no—
290#1057AGL(1,ℤ/8)12yescomplemented
291C8×C13—21no—
292C8×C13C2×M1612yescomplemented
293#1055—21no—
294#1055—21no—
295C2×AGL(1,ℤ/8)#26111yescharacteristic
296C22×M16—261no—
297C4×#41—131no—
298C2×C4×D4—131no—
299#446—131no—
300#446—131no—
301#442—131no—
302#442—131no—
303#440—131no—
304#440—131no—
305C4×SD16—131no—
306C4×SD16—131no—
307C4×D8—131no—
308C4×D8—131no—
309#436—131no—
310#436—131no—
311#435—261no—
312#435—261no—
313#435—261no—
314#435—261no—
315C8×D4—131no—
316C8×D4—131no—
317C8×D4—131no—
318C8×D4—131no—
319#432—131no—
320#432—261no—
321#428—131no—
322#423—131no—
323#418—131no—
324#417—131no—
325#407—261no—
326C4×M16—131no—
327C4×M16—131no—
328C4×M16—131no—
329C2×#97—261no—
330#335—131no—
331#335—131no—
332#334—131no—
333#334—131no—
334Q8⋊C8—131no—
335Q8⋊C8—131no—
336D4⋊C8—131no—
337D4⋊C8—131no—
338D4⋊C8—131no—
339D4⋊C8—131no—
340D4⋊C8—131no—
341D4⋊C8—131no—
342#322—131no—
343#322—131no—
344#322—131no—
345C82—131no—
346C22×C13—21no—
347C22×C13—22no—
348C22×C13—22no—
349C22×C13—21no—
350C22×C13—21no—
351C22×C13—41no—
352C22×C13—41no—
353C22×C13—41no—
354C22×C13—41no—
355C22×C13—41no—
356C22×C13—41no—
357C22×C13—41no—
358C22×C13—84no—
359C22×C13C2×C4×D411yescharacteristic
360C4×C13—21no—
361C4×C13—41no—
362C4×C13C22×M1611yescharacteristic
363C4×C13#56811yescharacteristic
364Dic13—21no—
365Dic13—21no—
366Dic13—21no—
367Dic13—41no—
368Dic13—41no—
369Dic13—41no—
370Dic13—41no—
371Dic13C2×AGL(1,ℤ/8)11yescharacteristic
372Dic13#57511yescharacteristic
373C2×#41C2×#26111yescharacteristic
374C22×D4C2×#26111yescharacteristic
375C23×C4—261no—
376AGL(1,ℤ/8)—24no—
377C2×SD16C2×#26111yescharacteristic
378C2×SD16C2×#26111yescharacteristic
379C2×D8C2×#26111yescharacteristic
380C2×D8C2×#26111yescharacteristic
381C2×M16—131no—
382C2×M16—131no—
383C2×M16—131no—
384C2×M16—261no—
385C2×M16—261no—
386C2×M16—261no—
387C2×M16—261no—
388C2×M16—524no—
389C2×M16C2×#26111yescharacteristic
390C22×C8—261no—
391C22×C8—261no—
392C22×C8—261no—
393C22×C8—261no—
394C4×Q8—131no—
395C4×Q8—131no—
396C4×D4—131no—
397C4×D4—131no—
398C4×D4—131no—
399C4×D4—131no—
400C4×D4—131no—
401C4×D4—131no—
402C4×D4—261no—
403C4×D4—261no—
404C4×D4—261no—
405C4×D4—261no—
406#117—131no—
407#117—261no—
408C2×C4⋊C4—131no—
409C2×C22⋊C4—261no—
410C2×C42—131no—
411C2×C42—261no—
412#107—131no—
413#107—131no—
414#106—131no—
415#106—131no—
416C4⋊C8—131no—
417C4⋊C8—131no—
418C4⋊C8—131no—
419C4⋊C8—131no—
420C4⋊C8—261no—
421C4⋊C8—261no—
422C4⋊C8—261no—
423#103—131no—
424#103—131no—
425#102—131no—
426#102—131no—
427#102—131no—
428#102—131no—
429#102—131no—
430#102—131no—
431C22⋊C8—261no—
432C22⋊C8—261no—
433C22⋊C8—261no—
434C22⋊C8—261no—
435C22⋊C8—261no—
436C22⋊C8—261no—
437C22⋊C8—261no—
438#97—131no—
439#97—131no—
440#97—131no—
441#97—131no—
442#97—261no—
443#97—261no—
444#97—261no—
445#97—261no—
446C4×C8—131no—
447C4×C8—131no—
448C4×C8—131no—
449C4×C8—131no—
450C4×C8—131no—
451C4×C8—131no—
452C2×C13—41no—
453C2×C13—42no—
454C2×C13—42no—
455C2×C13—81no—
456C2×C13D4×M1611yescharacteristic
457C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
458C2×C13#295611yescharacteristic
459C24—21no—
460#41—22no—
461#41—41no—
462C2×Q8C22×#26111yescharacteristic
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—41no—
467C2×D4C22×#26111yescharacteristic
468C2×D4C22×#26111yescharacteristic
469C2×D4C22×#26111yescharacteristic
470C22×C4—21no—
471C22×C4—131no—
472C22×C4—131no—
473C22×C4—261no—
474C22×C4—261no—
475C22×C4—261no—
476C22×C4—261no—
477C22×C4—261no—
478C22×C4—261no—
479C22×C4—261no—
480C22×C4—522no—
481C22×C4—522no—
482C22×C4C22×#26111yescharacteristic
483SD16—22no—
484SD16—22no—
485D8—22no—
486D8#612214yescomplemented
487M16—22no—
488M16—262no—
489M16—522no—
490M16—521no—
491M16—521no—
492M16—522no—
493M16—522no—
494M16—522no—
495C2×C8—131no—
496C2×C8—131no—
497C2×C8—131no—
498C2×C8—131no—
499C2×C8—131no—
500C2×C8—131no—
501C2×C8—261no—
502C2×C8—261no—
503C2×C8—261no—
504C2×C8—261no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—522no—
508C2×C8—522no—
509C2×C8—522no—
510C2×C8—522no—
511C2×C8C22×#26111yescharacteristic
512C2×C8C22×#26111yescharacteristic
513C4⋊C4—131no—
514C4⋊C4—131no—
515C4⋊C4—131no—
516C4⋊C4—131no—
517C4⋊C4—261no—
518C4⋊C4—261no—
519C4⋊C4—261no—
520C22⋊C4—261no—
521C22⋊C4—261no—
522C22⋊C4—261no—
523C22⋊C4—261no—
524C22⋊C4—261no—
525C22⋊C4—261no—
526C22⋊C4—261no—
527C42—131no—
528C42—131no—
529C42—131no—
530C42—131no—
531C42—261no—
532C42—261no—
533C42—261no—
534C13#1804611yescharacteristic, complemented, Sylow
535C23—21no—
536C23—21no—
537C23—21no—
538C23—42no—
539C23—42no—
540C23—81no—
541C23D4×#26111yescharacteristic
542Q8—21no—
543Q8—21no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—42no—
549D4—42no—
550D4C2×#612212yes—
551D4C2×#612212yescomplemented
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—42no—
556C2×C4—131no—
557C2×C4—131no—
558C2×C4—131no—
559C2×C4—131no—
560C2×C4—261no—
561C2×C4—261no—
562C2×C4—261no—
563C2×C4—261no—
564C2×C4—261no—
565C2×C4—261no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—521no—
569C2×C4—521no—
570C2×C4—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—522no—
574C2×C4—522no—
575C2×C4—522no—
576C2×C4—1041no—
577C2×C4D4×#26111yescharacteristic
578C2×C4C23×#26111yescharacteristic
579C8—21no—
580C8—132no—
581C8—261no—
582C8—261no—
583C8—261no—
584C8—262no—
585C8—261no—
586C8—261no—
587C8—521no—
588C8—521no—
589C8—521no—
590C8—521no—
591C8—521no—
592C8C2×#612212yescomplemented
593C22—21no—
594C22—22no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—84no—
606C22C2×D4×#26111yescharacteristic
607C4—21no—
608C4—41no—
609C4—131no—
610C4—131no—
611C4—261no—
612C4—261no—
613C4—261no—
614C4—521no—
615C4—521no—
616C4—521no—
617C4—521no—
618C4#1171962111yescharacteristic
619C4C22×#612211yescharacteristic
620C2—41no—
621C2—42no—
622C2—42no—
623C2—81no—
624C2D4×#612211yescharacteristic
625C2AGL(1,ℤ/8)×#26111yescharacteristic
626C2#1313161011yescharacteristic
subgroupG/H
centerC22C2×D4×#261
commutatorC2×C4×C13C23×C4
FrattiniC2×C4C23×#261
Fitting, prime radicalC2×C13×AGL(1,ℤ/8)C4
socleC22×C13C2×C4×D4
cosocleC4×Dic13C24
hypercenterC2×AGL(1,ℤ/8)#261
hypocenterC13#18046
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #13131673C2semidirect
2. D4×#6122C2semidirect
3. #13130479C2semidirect
4. #13130472C2semidirect
5. #13130457C2semidirect
6. #13130453C2semidirect
7. #13130441C2semidirect
8. #13130437C2semidirect
9. #13130432C2semidirect
10. D8×#1057C2semidirect
11. #13130400C2semidirect
12. #13130175C2semidirect
13. #13130171C2semidirect
14. #13130059C2semidirect
21. #11719158C22semidirect
22. #11719158C22semidirect
31. D4×#1057C22semidirect
32. D4×#1057C22semidirect
51. Q8⋊#1057C22semidirect
52. Q8⋊#1057C22semidirect
57. D4⋊#1057C22semidirect
58. D4⋊#1057C22semidirect
59. #11718256C22semidirect
60. #11718255C22semidirect
61. #11718254C22semidirect
62. #11718253C22semidirect
63. #1057⋊C8C22semidirect
64. #1057⋊C8C22semidirect
65. #11718251C22semidirect
66. C8×#1057C22semidirect
194. C13×D8M16semidirect
253. D4×C13C2×M16semidirect
290. #1057AGL(1,ℤ/8)semidirect
292. C8×C13C2×M16semidirect
486. D8#6122semidirect
534. C13#18046semidirect
551. D4C2×#6122semidirect
592. C8C2×#6122semidirect

Maximal subgroups

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

subgroupindex
#131316732
D4×#61222
#131304792
#131304722
#131304572
#131304532
#131304412
#131304372
#131304322
D8×#10572
#131304002
#131301752
#131301712
#131300592
AGL(1,ℤ/8)×Dic132
#1804613