Finite Group Databasebeta

← #16796035

Subgroups — #16796035

Order 3328.

680 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131631C211yesmaximal, characteristic, complemented
2D4×#6122C211yesmaximal, characteristic
3#13130473C211yesmaximal, characteristic, complemented
4#13130472C211yesmaximal, characteristic, complemented
5#13130446C211yesmaximal, characteristic, complemented
6#13130445C211yesmaximal, characteristic, complemented
7#13130441C211yesmaximal, characteristic, complemented
8#13130439C211yesmaximal, characteristic, complemented
9#13130419C211yesmaximal, characteristic, complemented
10SD16×#1057C211yesmaximal, characteristic, complemented
11#13130403C211yesmaximal, characteristic, complemented
12#13130184C211yesmaximal, characteristic, complemented
13C2×D4⋊#1057C211yesmaximal, characteristic
14#13130037C211yesmaximal, characteristic, complemented
15#13124839C211yesmaximal, characteristic
16C22×#6122—21no—
17C2×D4×Dic13C2211yescharacteristic
18#11719327C2211yescharacteristic
19#11719169—21no—
20#11719158C2211yescharacteristic, complemented
21Q8×#1057C2211yescharacteristic, complemented
22#11719145C2211yescharacteristic
23#11719145C2211yescharacteristic
24#11719144—21no—
25#11719144—21no—
26D4×#1057C2211yescharacteristic
27D4×#1057C2211yescharacteristic
28#11719129C2211yescharacteristic
29C4×#6122C2211yescharacteristic
30C2×C4⋊#1057C2211yescharacteristic
31C2×C4×#1057C2211yescharacteristic
32#11719080—21no—
33#11719080C2211yescharacteristic, complemented
34#11719078—21no—
35#11719074C2211yescharacteristic, complemented
36#11719067—21no—
37#11719062—21no—
38C13×#450C411yescharacteristic
39#11718879C411yescharacteristic
40#11718823C411yescharacteristic
41#11718823C411yescharacteristic
42SD16×Dic13C2211yescharacteristic
43SD16×Dic13C2211yescharacteristic
44#11718704C411yescharacteristic
45#11718422C411yescharacteristic
46#11718422C411yescharacteristic
47#11718419C2211yescharacteristic
48#11718419C2211yescharacteristic
49#11718382C411yescharacteristic
50#11718379C2211yescharacteristic
51Q8⋊#1057C2211yescharacteristic, complemented
52Q8⋊#1057C2211yescharacteristic, complemented
53D4⋊#1057C2211yescharacteristic
54D4⋊#1057C2211yescharacteristic
55D4⋊#1057C2211yescharacteristic
56D4⋊#1057C2211yescharacteristic
57#11718255C2211yescharacteristic, complemented
58#11718255C2211yescharacteristic, complemented
59#11718253C2211yescharacteristic, complemented
60#11718253C2211yescharacteristic, complemented
61#1057⋊C8C2211yescharacteristic, complemented
62#1057⋊C8C2211yescharacteristic, complemented
63C8×#1057C2211yescharacteristic, complemented
64C8×#1057C2211yescharacteristic, complemented
65#11718244C2211yescharacteristic, complemented
66#11718244C2211yescharacteristic, complemented
67C22×D4×C13C2×C411yescharacteristic
68C23×Dic13—21no—
69C2×#6122—21no—
70C2×#6122—21no—
71C2×#6122—21no—
72C2×#6122—44no—
73C2×#6122D411yescharacteristic
74C2×#6122D411yescharacteristic
75C22×#1057—21no—
76C22×#1057—21no—
77C22×#1057—21no—
78C22×#1057—21no—
79C2×C13×SD16C2×C411yescharacteristic
80C2×C13×SD16C2×C411yescharacteristic
81C13×C22⋊Q8C2×C411yescharacteristic
82C2×C22⋊Dic13—21no—
83Q8×Dic13C2311yescharacteristic
84#88198C2×C411yescharacteristic
85D4×Dic13—21no—
86D4×Dic13—21no—
87D4×Dic13C2×C411yescharacteristic
88D4×Dic13C2×C411yescharacteristic
89D4×Dic13C2311yescharacteristic
90D4×Dic13C2311yescharacteristic
91C2×#6105C2×C411yescharacteristic
92C2×#6105C2×C411yescharacteristic
93C2×C4⋊Dic13C2×C411yescharacteristic
94#88178C2×C411yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96#88143—21no—
97#88142C2×C411yescharacteristic
98#88141—21no—
99#88141C2311yescharacteristic
100#88132—21no—
101#88131—21no—
102C22⋊#1057—21no—
103C22⋊#1057—21no—
104C22⋊#1057—21no—
105C22⋊#1057—21no—
106#88112—21no—
107#88112—21no—
108#88112—21no—
109#88110—21no—
110#88110—21no—
111#88110D411yescharacteristic
112#88110D411yescharacteristic
113C4⋊#1057—21no—
114C4⋊#1057—21no—
115C4⋊#1057—21no—
116C4⋊#1057C2311yescharacteristic
117C4⋊#1057C2311yescharacteristic
118C4⋊#1057C2311yescharacteristic
119C4⋊#1057C2311yescharacteristic
120C4×#1057—21no—
121C4×#1057—21no—
122C4×#1057—21no—
123C4×#1057C2311yescharacteristic
124C4×#1057C2311yescharacteristic
125C4×#1057C2311yescharacteristic
126C4×#1057C2311yescharacteristic
127C13×#102C2×C411yescharacteristic
128C13×#102C2×C411yescharacteristic
129C13×C22⋊C8C2×C411yescharacteristic
130#88075C2×C411yescharacteristic
131#88075C2×C411yescharacteristic
132#88072C2×C411yescharacteristic
133#88072C2×C411yescharacteristic
134C22⋊#1055C2×C411yescharacteristic
135#88057C2×C411yescharacteristic
136#88057C2×C411yescharacteristic
137#88056C2×C411yescharacteristic
138#88056C2×C411yescharacteristic
139#88054C2×C411yescharacteristic
140#88054C2×C411yescharacteristic
141C8×Dic13C2311yescharacteristic
142C8×Dic13C2311yescharacteristic
143#88048C2×C411yescharacteristic
144#88048C2×C411yescharacteristic
145#17995—131nomaximal, Sylow
146C24×C13—21no—
147C2×Q8×C13C22×C411yescharacteristic
148C2×D4×C13—22no—
149C2×D4×C13—21no—
150C2×D4×C13—41no—
151C2×D4×C13C22×C411yescharacteristic
152C2×D4×C13C22×C411yescharacteristic
153C22×C4×C13C22×C411yescharacteristic
154C22×Dic13—21no—
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C22×Dic13—21no—
159C22×Dic13—42no—
160C22×Dic13—42no—
161C22×Dic13C2×D411yescharacteristic
162C22×Dic13#4111yescharacteristic
163C2×Dic26—21no—
164C2×Dic26C22×C411yescharacteristic
165#6122—22no—
166#6122—42no—
167#6122—42no—
168#6122—42no—
169#6122SD1614yescomplemented
170C2×#1057—21no—
171C2×#1057—21no—
172C2×#1057—21no—
173C2×#1057—21no—
174C2×#1057—21no—
175C2×#1057—21no—
176C2×#1057—42no—
177C2×#1057—42no—
178C2×#1057—42no—
179C2×#1057—42no—
180C2×#1057C2×D411yescharacteristic
181C2×#1057C2×D411yescharacteristic
182C2×#1057C2×D411yescharacteristic
183C2×#1057C2×D411yescharacteristic
184C13×SD16—22no—
185C13×SD16—41no—
186C2×C8×C13C22×C411yescharacteristic
187C2×C8×C13C22×C411yescharacteristic
188C13×C4⋊C4—21no—
189C13×C4⋊C4C22×C411yescharacteristic
190C13×C22⋊C4—21no—
191C22⋊Dic13—21no—
192C22⋊Dic13—21no—
193C22⋊Dic13—21no—
194C22⋊Dic13—21no—
195C22⋊Dic13—21no—
196#6105—22no—
197#6105—41no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13C22×C411yescharacteristic
201C4⋊Dic13C22×C411yescharacteristic
202C4⋊Dic13C22×C411yescharacteristic
203#6101—21no—
204#6101—21no—
205#6101—21no—
206#6101—21no—
207#6101—21no—
208C4×Dic13—21no—
209C4×Dic13—21no—
210C4×Dic13C22×C411yescharacteristic
211C4×Dic13C2×D411yescharacteristic
212C4×Dic13#4111yescharacteristic
213C4×Dic13C2411yescharacteristic
214C2×#1055C22×C411yescharacteristic
215C2×#1055C22×C411yescharacteristic
216#3101—131no—
217#2962—131no—
218D4×M16—131no—
219#1590—131no—
220#1590—131no—
221#1583—131no—
222#1583—131no—
223#1582—131no—
224#1582—131no—
225C8×SD16—131no—
226C8×SD16—131no—
227#1570—131no—
228#1493—131no—
229C2×D4⋊C8—131no—
230C8×M16—131no—
231C23×C13—21no—
232C23×C13—21no—
233C23×C13—42no—
234C23×C13—42no—
235C23×C13—81no—
236C23×C13C4×D411yescharacteristic
237Q8×C13—21no—
238Q8×C13—21no—
239D4×C13—21no—
240D4×C13—21no—
241D4×C13—21no—
242D4×C13—42no—
243D4×C13#13112yes—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13C4×D411yescharacteristic
249C2×C4×C13C23×C411yescharacteristic
250C2×Dic13—21no—
251C2×Dic13—21no—
252C2×Dic13—21no—
253C2×Dic13—21no—
254C2×Dic13—41no—
255C2×Dic13—42no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—42no—
261C2×Dic13—42no—
262C2×Dic13C4×D411yescharacteristic
263C2×Dic13C4×D411yescharacteristic
264C2×Dic13C2×SD1612yes—
265C2×Dic13#13512yes—
266C2×Dic13C22×D411yescharacteristic
267C2×Dic13C2×#4111yescharacteristic
268Dic26—21no—
269Dic26—21no—
270Dic26—81no—
271#1057—22no—
272#1057—21no—
273#1057—21no—
274#1057—41no—
275#1057—41no—
276#1057—41no—
277#1057—41no—
278#1057—41no—
279#1057C2×SD1612yes—
280#1057C2×SD1612yes—
281C8×C13—21no—
282C8×C13—21no—
283#1055—21no—
284#1055—21no—
285C22×M16—261no—
286#533—131no—
287C2×C4×D4—131no—
288#483—131no—
289#483—131no—
290#476—131no—
291#476—131no—
292#461—131no—
293#461—131no—
294#450—131no—
295#450#26111yescharacteristic
296#446—131no—
297#446—131no—
298#446—261no—
299#446—261no—
300C8×Q8—131no—
301C8×Q8—131no—
302C4×SD16—131no—
303C4×SD16—131no—
304C4×SD16—131no—
305C4×SD16—131no—
306#436—131no—
307#436—131no—
308#435—261no—
309#435—261no—
310C8×D4—131no—
311C8×D4—131no—
312#431—131no—
313#424—261no—
314#423—131no—
315C2×C4⋊C8—131no—
316#407—261no—
317#405—261no—
318C4×M16—131no—
319C2×C4×C8—131no—
320#334—131no—
321#334—131no—
322#334—131no—
323#334—131no—
324Q8⋊C8—131no—
325Q8⋊C8—131no—
326Q8⋊C8—131no—
327Q8⋊C8—131no—
328D4⋊C8—131no—
329D4⋊C8—131no—
330D4⋊C8—131no—
331D4⋊C8—131no—
332#322—131no—
333#322—131no—
334C82—131no—
335C82—131no—
336C22×C13—21no—
337C22×C13—21no—
338C22×C13—22no—
339C22×C13—41no—
340C22×C13—41no—
341C22×C13—41no—
342C22×C13—41no—
343C22×C13—41no—
344C22×C13—84no—
345C22×C13C4×SD1612yes—
346C22×C13C2×C4×D411yescharacteristic
347C4×C13—21no—
348C4×C13—41no—
349C4×C13—41no—
350C4×C13C2×#13111yescharacteristic
351C4×C13#56811yescharacteristic
352Dic13—21no—
353Dic13—21no—
354Dic13—41no—
355Dic13—41no—
356Dic13—41no—
357Dic13—41no—
358Dic13C4×SD1612yes—
359Dic13C22×SD1611yescharacteristic
360Dic13C2×#13511yescharacteristic
361C22×D4C2×#26111yescharacteristic
362C23×C4—261no—
363C2×SD16—131no—
364C2×SD16—131no—
365C2×SD16C2×#26111yescharacteristic
366C2×SD16C2×#26111yescharacteristic
367C2×M16—131no—
368C2×M16—131no—
369C2×M16—261no—
370C2×M16—261no—
371C2×M16—261no—
372C2×M16—524no—
373C22×C8—261no—
374C22×C8—261no—
375C22×C8—261no—
376C22×C8—261no—
377C4⋊Q8—131no—
378C4⋊Q8—131no—
379#125—261no—
380C22⋊Q8—131no—
381C22⋊Q8—261no—
382C22⋊Q8C2×#26111yescharacteristic
383C4×Q8—131no—
384C4×Q8—131no—
385C4×Q8—261no—
386C4×D4—131no—
387C4×D4—131no—
388C4×D4—131no—
389C4×D4—131no—
390C4×D4—261no—
391C4×D4—261no—
392#117—261no—
393C2×C4⋊C4—131no—
394C2×C22⋊C4—261no—
395C2×C42—131no—
396#106—131no—
397#106—131no—
398#106—131no—
399#106—131no—
400C4⋊C8—131no—
401C4⋊C8—131no—
402C4⋊C8—131no—
403C4⋊C8—131no—
404C4⋊C8—131no—
405C4⋊C8—131no—
406C4⋊C8—261no—
407C4⋊C8—261no—
408C4⋊C8—261no—
409C4⋊C8—261no—
410C4⋊C8—261no—
411C4⋊C8—261no—
412#103—131no—
413#103—131no—
414#103—131no—
415#103—131no—
416#102—131no—
417#102—131no—
418#102C2×#26111yescharacteristic
419#102C2×#26111yescharacteristic
420C22⋊C8—131no—
421C22⋊C8—261no—
422C22⋊C8—261no—
423C22⋊C8—261no—
424C22⋊C8—261no—
425C22⋊C8C2×#26111yescharacteristic
426#97—131no—
427#97—131no—
428#97—261no—
429#97—261no—
430C4×C8—131no—
431C4×C8—131no—
432C4×C8—131no—
433C4×C8—131no—
434C4×C8—131no—
435C4×C8—131no—
436C4×C8—261no—
437C4×C8—261no—
438C4×C8—261no—
439C2×C13—22no—
440C2×C13—42no—
441C2×C13—81no—
442C2×C13#293311yescharacteristic
443C2×C13C2×C4×SD1611yescharacteristic
444C2×C13#295611yescharacteristic
445C24—21no—
446C2×Q8—131no—
447C2×Q8—261no—
448C2×Q8C22×#26111yescharacteristic
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—41no—
452C2×D4C22×#26111yescharacteristic
453C2×D4C22×#26111yescharacteristic
454C22×C4—131no—
455C22×C4—131no—
456C22×C4—261no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—261no—
461C22×C4—522no—
462C22×C4—522no—
463C22×C4C22×#26111yescharacteristic
464SD16—22no—
465SD16—41no—
466SD16—262no—
467SD16—521no—
468M16—134no—
469M16—262no—
470M16—522no—
471M16—522no—
472M16—522no—
473C2×C8—131no—
474C2×C8—131no—
475C2×C8—131no—
476C2×C8—131no—
477C2×C8—131no—
478C2×C8—131no—
479C2×C8—261no—
480C2×C8—261no—
481C2×C8—261no—
482C2×C8—261no—
483C2×C8—261no—
484C2×C8—261no—
485C2×C8—522no—
486C2×C8—522no—
487C2×C8—522no—
488C2×C8—522no—
489C2×C8C22×#26111yescharacteristic
490C2×C8C22×#26111yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—131no—
493C4⋊C4—131no—
494C4⋊C4—131no—
495C4⋊C4—261no—
496C4⋊C4—261no—
497C4⋊C4—261no—
498C4⋊C4—261no—
499C4⋊C4—261no—
500C4⋊C4—261no—
501C4⋊C4—261no—
502C4⋊C4C22×#26111yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—261no—
505C22⋊C4—261no—
506C22⋊C4—261no—
507C22⋊C4—261no—
508C22⋊C4—261no—
509C42—131no—
510C42—131no—
511C42—131no—
512C42—131no—
513C42—261no—
514C42—261no—
515C13#1799511yescharacteristic, complemented, Sylow
516C23—21no—
517C23—21no—
518C23—42no—
519C23—42no—
520C23—81no—
521C23D4×#26111yescharacteristic
522Q8—21no—
523Q8—21no—
524Q8—261no—
525Q8—261no—
526Q8—1041no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—42no—
531D4#8823812yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—132no—
537C2×C4—132no—
538C2×C4—131no—
539C2×C4—131no—
540C2×C4—131no—
541C2×C4—131no—
542C2×C4—261no—
543C2×C4—261no—
544C2×C4—261no—
545C2×C4—261no—
546C2×C4—521no—
547C2×C4—521no—
548C2×C4—521no—
549C2×C4—521no—
550C2×C4—522no—
551C2×C4—522no—
552C2×C4—521no—
553C2×C4—522no—
554C2×C4D4×#26111yescharacteristic
555C2×C4C23×#26111yescharacteristic
556C8—21no—
557C8—21no—
558C8—132no—
559C8—132no—
560C8—261no—
561C8—261no—
562C8—262no—
563C8—261no—
564C8—261no—
565C8—521no—
566C8—521no—
567C8—521no—
568C8—521no—
569C8—521no—
570C22—21no—
571C22—21no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—84no—
579C22SD16×#26112yes—
580C22C2×D4×#26111yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—131no—
585C4—131no—
586C4—132no—
587C4—261no—
588C4—261no—
589C4—521no—
590C4—521no—
591C4—521no—
592C4—521no—
593C4C2×#8823811yescharacteristic
594C4#1171962111yescharacteristic
595C2—22no—
596C2—42no—
597C2—81no—
598C2#1313156811yescharacteristic
599C2C2×SD16×#26111yescharacteristic
600C2#1313161011yescharacteristic
subgroupG/H
centerC22C2×D4×#261
commutatorC2×C4×C13C23×C4
FrattiniC2×C4C23×#261
FittingC13×#450C4
socleC22×C13C2×C4×D4
prime radicalC22×D4×C13C2×C4
cosocleC4×Dic13C24
hypercenter#450#261
hypocenterC13#17995
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #13131631C2semidirect
3. #13130473C2semidirect
4. #13130472C2semidirect
5. #13130446C2semidirect
6. #13130445C2semidirect
7. #13130441C2semidirect
8. #13130439C2semidirect
9. #13130419C2semidirect
10. SD16×#1057C2semidirect
11. #13130403C2semidirect
12. #13130184C2semidirect
14. #13130037C2semidirect
20. #11719158C22semidirect
21. Q8×#1057C22semidirect
33. #11719080C22semidirect
35. #11719074C22semidirect
51. Q8⋊#1057C22semidirect
52. Q8⋊#1057C22semidirect
57. #11718255C22semidirect
58. #11718255C22semidirect
59. #11718253C22semidirect
60. #11718253C22semidirect
61. #1057⋊C8C22semidirect
62. #1057⋊C8C22semidirect
63. C8×#1057C22semidirect
64. C8×#1057C22semidirect
65. #11718244C22semidirect
66. #11718244C22semidirect
169. #6122SD16semidirect
515. C13#17995semidirect

Maximal subgroups

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

subgroupindex
#131316312
D4×#61222
#131304732
#131304722
#131304462
#131304452
#131304412
#131304392
#131304192
SD16×#10572
#131304032
#131301842
C2×D4⋊#10572
#131300372
#131248392
#1799513