Finite Group Databasebeta

← #10828727

Subgroups — #10828727

Order 768.

1014 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85970C211yesmaximal, characteristic
2C3×#2502C211yesmaximal, characteristic, complemented
3#82139C211yesmaximal, characteristic
4C2×#4964C211yesmaximal, characteristic, complemented
5#71698C211yesmaximal, characteristic, complemented
6#69695C212yesmaximal, complemented
7#69672C212yesmaximal, complemented
8#69592C212yesmaximal, complemented
9#69573C212yesmaximal, complemented
10#69453C211yesmaximal, characteristic
11#69442C211yesmaximal, characteristic, complemented
12#24086—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15#5816C2211yescharacteristic
16#5812C2211yescharacteristic
17#5791C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×#208C2211yescharacteristic
20#5643—21no—
21#5642—21no—
22#5641—21no—
23#5639—21no—
24#5635—21no—
25#5625—21no—
26#5624—21no—
27#5622—21no—
28#5513—22no—
29#5504C2211yescharacteristic
30#5500C2211yescharacteristic
31#5495—22no—
32C3×#398—22no—
33C3×#391C2212yes—
34C3×#385—22no—
35C3×#384C2212yes—
36C3×C4×C22⋊C4C2211yescharacteristic
37C2×#836C2212yes—
38#5207C2211yescharacteristic
39#4971—21no—
40#4970—21no—
41#4968—21no—
42#4967—21no—
43#4966C2212yes—
44#4964—21no—
45#4964—21no—
46#4964C2212yes—
47#4964C2212yes—
48#4964C2212yes—
49C2×#808—21no—
50C2×#807—21no—
51C2×#805C2212yes—
52#4530C2212yes—
53#4523C2212yes—
54#4523C2212yes—
55#4523C2212yes—
56#3495—31no—
57#3092—31no—
58C2×#428—31no—
59#2502D311yescharacteristic, complemented
60#2060—32no—
61#2038—32no—
62#1901—31no—
63#1871—31no—
64#1859—32no—
65#1857—32no—
66#1826—31no—
67C22×C3×Q8C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C2×#235C2311yescharacteristic
71C3×C22⋊Q8—21no—
72C3×C22⋊Q8—21no—
73C3×C22⋊Q8—21no—
74C3×C22⋊Q8—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4D412yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4D412yes—
86C2×C3×C42C2312yes—
87#953—21no—
88C2×#217—21no—
89#947—21no—
90#946—21no—
91#946—21no—
92Q8×Dic3—21no—
93Q8×Dic3—21no—
94Q8×Dic3—21no—
95Q8×Dic3—21no—
96#944—21no—
97#930D411yescharacteristic
98#930D411yescharacteristic
99#929—22no—
100#929—21no—
101C4×#67—21no—
102C4×#67—21no—
103C4×#67—22no—
104#926—21no—
105#926—22no—
106#926—21no—
107#926D411yescharacteristic
108#926D411yescharacteristic
109#926C2311yescharacteristic
110#926C2311yescharacteristic
111#926C2311yescharacteristic
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3D411yescharacteristic
114#924—22no—
115#924D411yescharacteristic
116#924D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×#208—21no—
119C2×#208—24no—
120C2×#208C2312yes—
121C2×#208C2312yes—
122C22×C3⋊C8—21no—
123#898—22no—
124#898—22no—
125#897—22no—
126#893—21no—
127#892—22no—
128#892—21no—
129D3×C4⋊C4—21no—
130#890—22no—
131#890—21no—
132#890—21no—
133#889—22no—
134#889—22no—
135#888—21no—
136#888—21no—
137#887—22no—
138#887—21no—
139#886—22no—
140#885—22no—
141#883—22no—
142#67⋊C4—22no—
143#879—22no—
144#879—22no—
145#878—22no—
146#877—22no—
147C3×#95—21no—
148C3×#95—24no—
149#836—24no—
150#830—21no—
151#830—21no—
152#810—22no—
153#809—22no—
154#808—22no—
155#808—22no—
156#808—22no—
157#807—22no—
158#807—22no—
159#807—22no—
160#805—24no—
161C22×M16—31no—
162#556—61no—
163#554—61no—
164#552—61no—
165#552—61no—
166#552—62no—
167#551—62no—
168#544—31no—
169#542—61no—
170#541—61no—
171#536—31no—
172#533—61no—
173#529—61no—
174#525—31no—
175C2×C22⋊Q8C2×D311yescharacteristic
176#520—31no—
177#519—31no—
178C2×#117—31no—
179C22×C4⋊C4C2×D311yescharacteristic
180#484—61no—
181#483—61no—
182#481—61no—
183#480—61no—
184#428—32no—
185#428—32no—
186#428—32no—
187#428—61no—
188#428—61no—
189C2×#107—61no—
190C2×#106—61no—
191C2×#104—32no—
192#417—32no—
193C2×#101—32no—
194#407—31no—
195#398—22no—
196#391C2×D312yes—
197#385—22no—
198#384C2×D312yes—
199C4×C22⋊C4C2×D311yescharacteristic
200#343—32no—
201#339—32no—
202#339—32no—
203#339—32no—
204C24×C3C2×D411yescharacteristic
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—22no—
208C2×C3×Q8—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—41no—
214C22×C3×C4—41no—
215C22×C3×C4—82no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D412yes—
223C22×C3×C4C2411yescharacteristic
224C2×#67—21no—
225C2×#67—21no—
226C2×#67—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229#235—22no—
230#235—22no—
231C2×D12C2×D411yescharacteristic
232C2×C4×D3—21no—
233C2×C4×D3—22no—
234C2×Dic6—22no—
235C2×Dic6C2×D411yescharacteristic
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—84no—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C42—22no—
249C3×C42—22no—
250#217—21no—
251#217—21no—
252#217—21no—
253#217—21no—
254#217—21no—
255#217—21no—
256#217—21no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—21no—
261(C2×D3)⋊C4—21no—
262C4⋊Dic3—22no—
263C4⋊Dic3—22no—
264C4⋊Dic3—22no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3C2×D411yescharacteristic
268C4⋊Dic3C2×D411yescharacteristic
269C4⋊Dic3C2×D411yescharacteristic
270C4⋊Dic3C2×D411yescharacteristic
271C4⋊Dic3C2×D411yescharacteristic
272C4⋊Dic3C2×D411yescharacteristic
273Dic3⋊C4—22no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276Dic3⋊C4—22no—
277Dic3⋊C4—22no—
278Dic3⋊C4—21no—
279Dic3⋊C4—21no—
280C4×Dic3—22no—
281C4×Dic3—22no—
282C4×Dic3—22no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288#208—24no—
289#208—22no—
290#208—22no—
291#208—22no—
292#208#4114yes—
293C2×C3⋊C8—24no—
294C2×C3⋊C8—22no—
295C2×C3⋊C8—22no—
296C2×#41—31no—
297C22×Q8C22×D311yescharacteristic
298C23×C4—21no—
299C23×C4C22×D311yescharacteristic
300C2×M16—32no—
301C2×M16—32no—
302C2×M16—64no—
303C2×M16—61no—
304C22×C8—61no—
305C4⋊Q8—61no—
306C4⋊Q8—61no—
307C4⋊Q8—61no—
308#126—62no—
309#126—62no—
310#126—62no—
311#126—62no—
312#125—61no—
313#125—61no—
314#125—62no—
315#125—62no—
316#125—62no—
317#124—62no—
318#124—61no—
319#123—62no—
320#123—62no—
321#123—61no—
322#123—62no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327C22⋊Q8—31no—
328C22⋊Q8—31no—
329C22⋊Q8—62no—
330C22⋊Q8—62no—
331C22⋊Q8—61no—
332C22⋊Q8—61no—
333C22⋊Q8—62no—
334#121—31no—
335#121—31no—
336#120—61no—
337C4×Q8—61no—
338C4×Q8—62no—
339C4×Q8—61no—
340C4×Q8—61no—
341C4×Q8—61no—
342C4×Q8—61no—
343C4×D4—62no—
344C4×D4—61no—
345C4×D4—61no—
346C4×D4—61no—
347C4×D4—62no—
348#117—31no—
349#117—31no—
350#117—31no—
351#117—31no—
352#117—31no—
353#117—62no—
354#117—61no—
355#117—62no—
356#117—61no—
357#117—61no—
358#117—62no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—31no—
367C2×C4⋊C4—61no—
368C2×C4⋊C4#6712yes—
369C2×C4⋊C4C22×D312yes—
370C2×C4⋊C4C22×D311yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—61no—
373C2×C22⋊C4#6712yes—
374C2×C42—61no—
375C2×C42C22×D312yes—
376#107—62no—
377#107—62no—
378#107—62no—
379#106—62no—
380#106—62no—
381#106—62no—
382#104—64no—
383#103—62no—
384#102—62no—
385#101—64no—
386C22⋊C8—61no—
387C22⋊C8—61no—
388#95—21no—
389#95—24no—
390C23×C3—22no—
391C23×C3—22no—
392C23×C3—41no—
393C23×C3#12011yescharacteristic
394C23×C3#12011yescharacteristic
395C23×C3C22×D411yescharacteristic
396C22×D3—21no—
397C3×Q8—41no—
398C3×Q8—41no—
399C3×Q8—42no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—24no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—24no—
411C2×C3×C4—42no—
412C2×C3×C4—42no—
413C2×C3×C4—42no—
414C2×C3×C4—44no—
415C2×C3×C4—42no—
416C2×C3×C4—82no—
417C2×C3×C4—82no—
418C2×C3×C4#12011yescharacteristic
419C2×C3×C4#12011yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C2×#4112yes—
423C2×C3×C42−1+412yes—
424#67—42no—
425#67—42no—
426#67—81no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—42no—
435C2×Dic3—81no—
436D12—22no—
437C4×D3—41no—
438C4×D3—41no—
439C4×D3—82no—
440Dic6—22no—
441Dic6—82no—
442C3⋊C8—24no—
443C3⋊C8—42no—
444C24C2×#6711yescharacteristic
445#41—62no—
446#41—62no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—31no—
452C2×Q8—62no—
453C2×D4—31no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—62no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—82no—
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4D3×D411yescharacteristic
472C22×C4C2×#6712yes—
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6712yes—
475C22×C4C23×D311yescharacteristic
476M16—34no—
477M16—64no—
478M16—62no—
479M16—62no—
480M16—62no—
481C2×C8—64no—
482C2×C8—62no—
483C2×C8—62no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—31no—
491C4⋊C4—31no—
492C4⋊C4—31no—
493C4⋊C4—31no—
494C4⋊C4—31no—
495C4⋊C4—31no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—62no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—62no—
504C4⋊C4—62no—
505C4⋊C4—62no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—84no—
511C22⋊C4—24no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—62no—
515C22⋊C4—62no—
516C22⋊C4—62no—
517C22⋊C4—61no—
518C22⋊C4—61no—
519C22⋊C4—61no—
520C22⋊C4—61no—
521C22⋊C4—61no—
522C22⋊C4—61no—
523C22⋊C4—61no—
524C22⋊C4—61no—
525C22⋊C4—61no—
526C42—22no—
527C42—22no—
528C42—62no—
529C42—62no—
530C42—62no—
531C42—61no—
532C42—61no—
533C42—61no—
534C42—61no—
535C42—61no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—44no—
544C22×C3#45612yes—
545C22×C3C2×#12011yescharacteristic
546C22×C3#54712yes—
547C2×D3—41no—
548C2×D3—41no—
549C2×D3—41no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—44no—
553C3×C4—42no—
554C3×C4—42no—
555C3×C4—82no—
556C3×C4#52511yescharacteristic
557C3×C4#53611yescharacteristic
558C3×C4#54712yes—
559Dic3—41no—
560Dic3—41no—
561Dic3—41no—
562Dic3—41no—
563Dic3—41no—
564Dic3—41no—
565Dic3—41no—
566C23—22no—
567C23—22no—
568C23—41no—
569C23—61no—
570C23#93711yescharacteristic
571C23#93711yescharacteristic
572C23C22×#6711yescharacteristic
573Q8—41no—
574Q8—42no—
575Q8—41no—
576Q8—62no—
577Q8—242no—
578D4—62no—
579D4—122no—
580D4—122no—
581D4—241no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—122no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—242no—
611C2×C4—241no—
612C2×C4#93711yescharacteristic
613C2×C4#93711yescharacteristic
614C2×C4C2×D3×D411yescharacteristic
615C2×C4C2×D3×D411yescharacteristic
616C2×C4D3×#4112yes—
617C2×C4#101012yes—
618C8—64no—
619C8—122no—
620C2×C3—22no—
621C2×C3—22no—
622C2×C3—41no—
623C2×C3#240611yescharacteristic
624C2×C3C2×#45611yescharacteristic
625C2×C3#302111yescharacteristic
626D3—81no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—44no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22#509512yes—
639C22C2×#93711yescharacteristic
640C22#582412yes—
641C4—21no—
642C4—21no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—82no—
647C4—121no—
648C4—121no—
649C4—121no—
650C4—121no—
651C4—121no—
652C4—121no—
653C4—121no—
654C4#564811yescharacteristic
655C4#565512yes—
656C4#565611yescharacteristic
657C3#2408611yescharacteristic, complemented, Sylow
658C2—22no—
659C2—22no—
660C2—41no—
661C2—241no—
662C2#8284111yescharacteristic
663C2C2×#509511yescharacteristic
664C2#8339511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2502C2
socleC22×C3C2×#120
prime radical#5791C22
hypercenter#2502D3
hypocenterC3#24086
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
2. C3×#2502C2semidirect
4. C2×#4964C2semidirect
5. #71698C2semidirect
6. #69695C2semidirect
7. #69672C2semidirect
8. #69592C2semidirect
9. #69573C2semidirect
11. #69442C2semidirect
59. #2502D3semidirect
657. C3#24086semidirect

Maximal subgroups

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

subgroupindex
#859702
C3×#25022
#821392
C2×#49642
#716982
#696952
#696722
#695922
#695732
#694532
#694422
#240863