Finite Group Databasebeta

← #10828716

Subgroups — #10828716

Order 768.

998 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85970C211yesmaximal, characteristic
2C3×#2503C211yesmaximal, characteristic, complemented
3#82139C211yesmaximal, characteristic
4C2×#4964C211yesmaximal, characteristic, complemented
5#71698C211yesmaximal, characteristic, complemented
6#69695C212yesmaximal, complemented
7#69672C212yesmaximal, complemented
8#69562C212yesmaximal, complemented
9#69543C212yesmaximal, complemented
10#69453C211yesmaximal, characteristic
11#69442C211yesmaximal, characteristic, complemented
12#24085—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×#400—22no—
33C3×#391C2212yes—
34C3×#382C2212yes—
35C3×#381—22no—
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#2503D311yescharacteristic, complemented
60#2061—32no—
61#2038—32no—
62#1901—31no—
63#1871—31no—
64#1860—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⋊C4D412yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4D412yes—
85C2×C3×C42C2312yes—
86#953—21no—
87C2×#217—21no—
88#947—21no—
89#946—21no—
90#946—21no—
91Q8×Dic3—21no—
92Q8×Dic3—21no—
93Q8×Dic3—21no—
94Q8×Dic3—21no—
95#944—21no—
96#930D411yescharacteristic
97#930D411yescharacteristic
98#929—22no—
99#929—21no—
100C4×#67—21no—
101C4×#67—21no—
102C4×#67—22no—
103#926—21no—
104#926—22no—
105#926—21no—
106#926D411yescharacteristic
107#926D411yescharacteristic
108#926C2311yescharacteristic
109#926C2311yescharacteristic
110#926C2311yescharacteristic
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3D411yescharacteristic
113#924—22no—
114#924D411yescharacteristic
115#924D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×#208—21no—
118C2×#208—24no—
119C2×#208C2312yes—
120C2×#208C2312yes—
121C22×C3⋊C8—21no—
122#898—22no—
123#898—22no—
124#897—22no—
125#893—21no—
126#892—22no—
127#892—21no—
128D3×C4⋊C4—21no—
129#890—22no—
130#890—21no—
131#890—21no—
132#889—22no—
133#889—22no—
134#888—21no—
135#888—21no—
136#887—22no—
137#887—21no—
138#886—22no—
139#885—22no—
140#883—22no—
141#67⋊C4—22no—
142#879—22no—
143#879—22no—
144#878—22no—
145#877—22no—
146C3×#95—21no—
147C3×#95—24no—
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#400—22no—
196#391C2×D312yes—
197#382C2×D312yes—
198#381—22no—
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—44no—
214C22×C3×C4—41no—
215C22×C3×C4—41no—
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×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—42no—
247C3×C42—22no—
248C3×C42—22no—
249#217—21no—
250#217—21no—
251#217—21no—
252#217—21no—
253#217—21no—
254#217—21no—
255#217—21no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—21no—
261C4⋊Dic3—22no—
262C4⋊Dic3—22no—
263C4⋊Dic3—22no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3C2×D411yescharacteristic
267C4⋊Dic3C2×D411yescharacteristic
268C4⋊Dic3C2×D411yescharacteristic
269C4⋊Dic3C2×D411yescharacteristic
270C4⋊Dic3C2×D411yescharacteristic
271C4⋊Dic3C2×D411yescharacteristic
272Dic3⋊C4—22no—
273Dic3⋊C4—22no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276Dic3⋊C4—22no—
277Dic3⋊C4—21no—
278Dic3⋊C4—21no—
279C4×Dic3—22no—
280C4×Dic3—22no—
281C4×Dic3—22no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287#208—24no—
288#208—22no—
289#208—22no—
290#208—22no—
291#208#4114yes—
292C2×C3⋊C8—24no—
293C2×C3⋊C8—22no—
294C2×C3⋊C8—22no—
295C2×#41—31no—
296C22×Q8C22×D311yescharacteristic
297C23×C4—21no—
298C23×C4C22×D311yescharacteristic
299C2×M16—32no—
300C2×M16—32no—
301C2×M16—64no—
302C2×M16—61no—
303C22×C8—61no—
304C4⋊Q8—61no—
305C4⋊Q8—61no—
306C4⋊Q8—61no—
307#126—62no—
308#126—62no—
309#126—62no—
310#126—62no—
311#125—61no—
312#125—61no—
313#125—62no—
314#125—62no—
315#125—62no—
316#124—62no—
317#124—61no—
318#123—62no—
319#123—62no—
320#123—61no—
321#123—62no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—31no—
327C22⋊Q8—31no—
328C22⋊Q8—62no—
329C22⋊Q8—62no—
330C22⋊Q8—61no—
331C22⋊Q8—61no—
332C22⋊Q8—62no—
333#121—31no—
334#121—31no—
335#120—61no—
336C4×Q8—61no—
337C4×Q8—62no—
338C4×Q8—61no—
339C4×Q8—61no—
340C4×Q8—61no—
341C4×Q8—61no—
342C4×D4—62no—
343C4×D4—61no—
344C4×D4—61no—
345C4×D4—61no—
346C4×D4—62no—
347#117—31no—
348#117—31no—
349#117—31no—
350#117—31no—
351#117—31no—
352#117—62no—
353#117—61no—
354#117—62no—
355#117—61no—
356#117—61no—
357#117—62no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—31no—
364C2×C4⋊C4—31no—
365C2×C4⋊C4—61no—
366C2×C4⋊C4#6712yes—
367C2×C4⋊C4C22×D312yes—
368C2×C4⋊C4C22×D311yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—61no—
371C2×C22⋊C4#6712yes—
372C2×C42—61no—
373C2×C42C22×D312yes—
374#107—62no—
375#107—62no—
376#107—62no—
377#106—62no—
378#106—62no—
379#106—62no—
380#104—64no—
381#103—62no—
382#102—62no—
383#101—64no—
384C22⋊C8—61no—
385C22⋊C8—61no—
386#95—21no—
387#95—24no—
388#95—24no—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—41no—
392C23×C3#12011yescharacteristic
393C23×C3#12011yescharacteristic
394C23×C3C22×D411yescharacteristic
395C22×D3—21no—
396C3×Q8—41no—
397C3×Q8—41no—
398C3×Q8—42no—
399C2×C3×C4—22no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—24no—
409C2×C3×C4—42no—
410C2×C3×C4—42no—
411C2×C3×C4—42no—
412C2×C3×C4—44no—
413C2×C3×C4—42no—
414C2×C3×C4—42no—
415C2×C3×C4—82no—
416C2×C3×C4—82no—
417C2×C3×C4#12011yescharacteristic
418C2×C3×C4#12011yescharacteristic
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C2×#4112yes—
422C2×C3×C42−1+412yes—
423#67—42no—
424#67—42no—
425#67—81no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—42no—
434C2×Dic3—81no—
435D12—22no—
436C4×D3—41no—
437C4×D3—41no—
438C4×D3—82no—
439Dic6—22no—
440Dic6—82no—
441C3⋊C8—24no—
442C3⋊C8—42no—
443C24C2×#6711yescharacteristic
444#41—62no—
445#41—62no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—31no—
451C2×Q8—62no—
452C2×D4—31no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—44no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—62no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C2×#6712yes—
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6712yes—
474C22×C4C23×D311yescharacteristic
475M16—34no—
476M16—64no—
477M16—62no—
478M16—62no—
479M16—62no—
480C2×C8—64no—
481C2×C8—62no—
482C2×C8—62no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—31no—
490C4⋊C4—31no—
491C4⋊C4—31no—
492C4⋊C4—31no—
493C4⋊C4—31no—
494C4⋊C4—31no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—62no—
498C4⋊C4—62no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—62no—
504C4⋊C4—62no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—42no—
512C22⋊C4—62no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—61no—
516C22⋊C4—61no—
517C22⋊C4—61no—
518C22⋊C4—61no—
519C22⋊C4—61no—
520C22⋊C4—61no—
521C22⋊C4—61no—
522C22⋊C4—61no—
523C22⋊C4—61no—
524C42—22no—
525C42—22no—
526C42—62no—
527C42—62no—
528C42—62no—
529C42—61no—
530C42—61no—
531C42—61no—
532C42—61no—
533C42—61no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—44no—
542C22×C3#45412yes—
543C22×C3C2×#12011yescharacteristic
544C22×C3#54712yes—
545C2×D3—41no—
546C2×D3—41no—
547C2×D3—41no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—42no—
551C3×C4—42no—
552C3×C4—82no—
553C3×C4—82no—
554C3×C4#52511yescharacteristic
555C3×C4#53611yescharacteristic
556C3×C4#54712yes—
557Dic3—41no—
558Dic3—41no—
559Dic3—41no—
560Dic3—41no—
561Dic3—41no—
562Dic3—41no—
563Dic3—41no—
564C23—22no—
565C23—22no—
566C23—41no—
567C23—61no—
568C23#93711yescharacteristic
569C23#93711yescharacteristic
570C23C22×#6711yescharacteristic
571Q8—41no—
572Q8—42no—
573Q8—41no—
574Q8—62no—
575Q8—242no—
576D4—62no—
577D4—122no—
578D4—122no—
579D4—241no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—122no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—242no—
609C2×C4—241no—
610C2×C4#93711yescharacteristic
611C2×C4#93711yescharacteristic
612C2×C4C2×D3×D411yescharacteristic
613C2×C4C2×D3×D411yescharacteristic
614C2×C4D3×#4112yes—
615C2×C4#101012yes—
616C8—64no—
617C8—122no—
618C2×C3—22no—
619C2×C3—22no—
620C2×C3—41no—
621C2×C3#240611yescharacteristic
622C2×C3C2×#45411yescharacteristic
623C2×C3#302211yescharacteristic
624D3—81no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—44no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22#506212yes—
637C22C2×#93711yescharacteristic
638C22#582412yes—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—82no—
645C4—121no—
646C4—121no—
647C4—121no—
648C4—121no—
649C4—121no—
650C4—121no—
651C4—121no—
652C4#564811yescharacteristic
653C4#565512yes—
654C4#565611yescharacteristic
655C3#2408511yescharacteristic, complemented, Sylow
656C2—22no—
657C2—22no—
658C2—41no—
659C2—241no—
660C2#8284111yescharacteristic
661C2C2×#506211yescharacteristic
662C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2503C2
socleC22×C3C2×#120
prime radical#5791C22
hypercenter#2503D3
hypocenterC3#24085
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
2. C3×#2503C2semidirect
4. C2×#4964C2semidirect
5. #71698C2semidirect
6. #69695C2semidirect
7. #69672C2semidirect
8. #69562C2semidirect
9. #69543C2semidirect
11. #69442C2semidirect
59. #2503D3semidirect
655. C3#24085semidirect

Maximal subgroups

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

subgroupindex
#859702
C3×#25032
#821392
C2×#49642
#716982
#696952
#696722
#695622
#695432
#694532
#694422
#240853