Finite Group Databasebeta

← #11675284

Subgroups — #11675284

Order 768.

1798 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86224C211yesmaximal, characteristic, complemented
2#85766C211yesmaximal, characteristic, complemented
3C3×#3036C211yesmaximal, characteristic
4#83935C212yesmaximal, complemented
5#83930C212yesmaximal, complemented
6#83282C211yesmaximal, characteristic, complemented
7C2×#5018C211yesmaximal, characteristic, complemented
8#80523C214yesmaximal, complemented
9#80518C214yesmaximal, complemented
10#80418C214yesmaximal, complemented
11#80322C214yesmaximal, complemented
12#79068C211yesmaximal, characteristic, complemented
13#79045C211yesmaximal, characteristic, complemented
14#78877C212yesmaximal, complemented
15#78872C212yesmaximal, complemented
16#61333—31nomaximal, Sylow
17C2×C3×AGL(1,ℤ/8)C2212yes—
18C3×#532C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20#5827C2212yes—
21#5825C2212yes—
22#5798C2212yes—
23C22×#214C2211yescharacteristic
24C2×#924—22no—
25#5721C2212yescomplemented
26#5713C2212yescomplemented
27#5671C2214yescomplemented
28#5602—22no—
29#5600—22no—
30#5543—21no—
31#5543—21no—
32#5543C2212yes—
33#5542—21no—
34#5541—22no—
35#5539—22no—
36#5539C2214yes—
37D4×Dic6—21no—
38D4×Dic6C2212yes—
39#5500C2212yes—
40#5483—22no—
41#5470C2211yescharacteristic
42#5470C2211yescharacteristic
43C2×C4×Dic6C2211yescharacteristic
44C3×#463C2214yes—
45C3×#461C2212yes—
46C3×#459C2212yes—
47C3×#423C2211yescharacteristic
48C3×#417C2212yes—
49#5233C2212yes—
50#5233C2212yes—
51C2×#832C2211yescharacteristic
52#5164C2214yes—
53#5162C2214yes—
54#5162C2214yes—
55Dic3×SD16C2214yes—
56#5153C2214yes—
57#5153C2214yes—
58#5152C2214yes—
59Dic3×D8C2214yes—
60#5122C2212yes—
61#5121C2212yes—
62#5120C2212yes—
63#5119C2212yes—
64#5082C2212yescomplemented
65#5071C2212yescomplemented
66#5055C2212yescomplemented
67#5055C2212yescomplemented
68#5018C2214yes—
69#5018C2214yes—
70C2×#810C2211yescharacteristic
71C2×#807C2211yescharacteristic
72C2×#803C2211yescharacteristic
73#4851C2212yescomplemented
74#4850C2212yescomplemented
75#4829C2214yescomplemented
76#4828C2214yescomplemented
77#4828C2214yescomplemented
78#4799C2214yescomplemented
79#4797C2214yescomplemented
80#4793C2214yescomplemented
81#4792C2214yescomplemented
82#4768C2214yescomplemented
83#4765C2214yescomplemented
84#4763C2214yescomplemented
85#4759C2214yescomplemented
86#3507—31no—
87#3501—31no—
88#3376—34no—
89#3372—34no—
90#3370—34no—
91#3367—34no—
92#3137—32no—
93#3135—31no—
94#3083—32no—
95#3078—31no—
96#3042—32no—
97#3036D311yescharacteristic
98#2961—31no—
99#2946—32no—
100C2×#440—31no—
101C2×C3×#41C2312yes—
102C22×C3×D4C2311yescharacteristic
103C23×C3×C4—21no—
104C22×Dic6—22no—
105C22×Dic6C2311yescharacteristic
106C3×AGL(1,ℤ/8)—28no—
107C2×C3×SD16C2314yes—
108C2×C3×D8C2314yes—
109C2×C3×M16C2312yes—
110C3×C4⋊Q8C2311yescharacteristic
111C3×#127C2311yescharacteristic
112C3×#124C2312yes—
113C3×#121—21no—
114C3×#121—21no—
115C3×C4×D4—21no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4—21no—
119C3×C4×D4C2312yes—
120C3×C4×D4C2312yes—
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C22⋊C4—21no—
123C2×C3×C42C2311yescharacteristic
124C2×#217—24no—
125Q8×Dic3C2314yes—
126#944C2314yes—
127#936—22no—
128#936C2314yes—
129#935—24no—
130#935—24no—
131D4×Dic3—24no—
132D4×Dic3—22no—
133D4×Dic3—22no—
134D4×Dic3—22no—
135D4×Dic3C2314yes—
136D4×Dic3C2314yes—
137D4×Dic3C2314yes—
138C2×#214—24no—
139C2×#214C2314yes—
140#926—22no—
141#926C2312yes—
142#926C2312yes—
143C2×C4⋊Dic3—22no—
144C2×C4⋊Dic3—22no—
145C2×C4⋊Dic3C2311yescharacteristic
146#924—21no—
147#924—21no—
148#924—22no—
149#924—24no—
150#924—22no—
151#924—22no—
152#924—22no—
153#924—24no—
154#924D412yes—
155#924D412yes—
156C2×Dic3⋊C4—21no—
157C2×Dic3⋊C4—24no—
158C2×C4×Dic3—21no—
159C22×C3⋊C8C2311yescharacteristic
160C22×C3⋊C8C2311yescharacteristic
161#890—22no—
162#890—22no—
163#890C2314yes—
164#889—24no—
165#888—22no—
166#888C2314yes—
167#887—22no—
168#879—24no—
169#879—24no—
170#878—24no—
171#878—24no—
172#878—24no—
173#878—24no—
174#878—24no—
175#877—24no—
176#870D412yes—
177#870C2312yes—
178#869D411yescharacteristic
179#869D411yescharacteristic
180#869C2311yescharacteristic
181#869C2311yescharacteristic
182C4×Dic6—21no—
183C4×Dic6—21no—
184C4×Dic6—21no—
185C4×Dic6—21no—
186C4×Dic6C2312yes—
187C4×Dic6C2312yes—
188C4×Dic6C2312yes—
189C4×Dic6C2312yes—
190C3×C4⋊C8C2312yes—
191C3×C4⋊C8C2312yes—
192C3×#103C2314yes—
193C3×#102C2314yes—
194#835C2312yes—
195#835C2312yes—
196#835C2312yes—
197#835C2312yes—
198#832C2312yes—
199#832C2314yes—
200#832C2312yes—
201#832C2312yes—
202#832C2312yes—
203#818C2314yes—
204#817C2314yes—
205#816C2314yes—
206#816C2314yes—
207#815C2314yes—
208Dic3⋊C8C2314yes—
209Dic3⋊C8C2314yes—
210C8×Dic3C2314yes—
211#810C2314yes—
212#807C2314yes—
213#803C2312yes—
214#803C2312yes—
215C2×AGL(1,ℤ/8)C2×D312yes—
216C22×SD16—31no—
217#559—32no—
218#557—32no—
219#556—34no—
220#556—61no—
221#554—34no—
222#554—32no—
223#554—62no—
224#554—61no—
225#554—61no—
226#552—62no—
227D4×Q8—32no—
228D4×Q8—61no—
229#544—32no—
230#543—62no—
231#539—62no—
232#536—32no—
233#533—31no—
234#533—31no—
235#532C2×D311yescharacteristic
236#529—62no—
237C2×C22⋊Q8—62no—
238#520—32no—
239#518—32no—
240C2×C4×Q8—31no—
241C2×C4×D4C2×D311yescharacteristic
242#491—32no—
243#491—34no—
244#490—32no—
245#489—32no—
246#489—32no—
247#489—34no—
248#488—34no—
249#487—34no—
250#479—34no—
251#479—34no—
252#478—34no—
253#477—32no—
254#477—34no—
255#475—32no—
256#474—34no—
257#463—34no—
258#463—34no—
259#463C2×D314yes—
260#461—34no—
261#461—34no—
262#461C2×D312yes—
263#459C2×D312yes—
264#446—34no—
265#446—34no—
266#442—34no—
267#440—34no—
268#440—34no—
269#440—34no—
270C4×SD16—34no—
271C4×D8—34no—
272#428—32no—
273C2×#107—31no—
274#424—32no—
275#423C2×D311yescharacteristic
276#419—32no—
277#417C2×D312yes—
278#416—32no—
279#416—32no—
280C2×#103—31no—
281C2×#102—31no—
282#405—32no—
283C2×#97—31no—
284C24×C3—21no—
285C3×#41—24no—
286C3×#41—24no—
287C2×C3×Q8C2412yes—
288C2×C3×D4—22no—
289C2×C3×D4—21no—
290C2×C3×D4—21no—
291C2×C3×D4—21no—
292C2×C3×D4—22no—
293C2×C3×D4#4114yes—
294C2×C3×D4C2412yes—
295C2×C3×D4C2412yes—
296C22×C3×C4—21no—
297C22×C3×C4—22no—
298C22×C3×C4—22no—
299C22×C3×C4—22no—
300C22×C3×C4—82no—
301C22×C3×C4C2×D411yescharacteristic
302C22×C3×C4C2×D411yescharacteristic
303C22×C3×C4C2411yescharacteristic
304C22×Dic3—21no—
305C22×Dic3—21no—
306C22×Dic3—24no—
307C22×Dic3—24no—
308C22×Dic3—24no—
309C2×Dic6—22no—
310C2×Dic6—22no—
311C2×Dic6—21no—
312C2×Dic6—21no—
313C2×Dic6—21no—
314C2×Dic6—24no—
315C2×Dic6—84no—
316C2×Dic6C2×D412yes—
317C2×Dic6C2×D412yes—
318C2×Dic6C2412yes—
319C3×SD16—28no—
320C3×D8—28no—
321C3×M16—24no—
322C2×C3×C8C2414yes—
323C3×C4⋊C4—21no—
324C3×C4⋊C4—21no—
325C3×C4⋊C4—21no—
326C3×C4⋊C4C2412yes—
327C3×C22⋊C4—22no—
328C3×C22⋊C4—22no—
329C3×C22⋊C4—22no—
330C3×C22⋊C4—22no—
331C3×C42C2×D411yescharacteristic
332C3×C42C2×D411yescharacteristic
333C3×C42C2411yescharacteristic
334C3×C42C2411yescharacteristic
335#217—22no—
336#217—24no—
337#217—24no—
338#217—24no—
339#217—24no—
340#217—24no—
341#217C2×D414yes—
342#214—28no—
343C4⋊Dic3—21no—
344C4⋊Dic3—21no—
345C4⋊Dic3—21no—
346C4⋊Dic3—24no—
347C4⋊Dic3—24no—
348C4⋊Dic3C2×D412yes—
349C4⋊Dic3C2×D412yes—
350C4⋊Dic3C2×D412yes—
351C4⋊Dic3C2×D412yes—
352C4⋊Dic3C2412yes—
353C4⋊Dic3C2412yes—
354C4⋊Dic3C2412yes—
355Dic3⋊C4—22no—
356Dic3⋊C4—22no—
357Dic3⋊C4—22no—
358Dic3⋊C4—22no—
359Dic3⋊C4—24no—
360Dic3⋊C4—24no—
361Dic3⋊C4—24no—
362Dic3⋊C4—24no—
363Dic3⋊C4C2×D414yes—
364Dic3⋊C4C2×D414yes—
365C4×Dic3—22no—
366C4×Dic3—22no—
367C4×Dic3—22no—
368C4×Dic3—24no—
369C4×Dic3C2414yes—
370C2×C3⋊C8—22no—
371C2×C3⋊C8—22no—
372C2×C3⋊C8C2412yes—
373C2×C3⋊C8C2412yes—
374C2×#41C22×D312yes—
375C22×Q8—31no—
376C22×Q8—62no—
377C22×D4C22×D311yescharacteristic
378C23×C4—21no—
379AGL(1,ℤ/8)—28no—
380C2×SD16—34no—
381C2×SD16—64no—
382C2×SD16C22×D314yes—
383C2×D8C22×D314yes—
384C2×M16C22×D312yes—
385C22×C8—31no—
386C22×C8—31no—
387C4⋊Q8—34no—
388C4⋊Q8—34no—
389C4⋊Q8—31no—
390C4⋊Q8—31no—
391C4⋊Q8—31no—
392C4⋊Q8—31no—
393C4⋊Q8—62no—
394C4⋊Q8C22×D311yescharacteristic
395#127C22×D311yescharacteristic
396#126—64no—
397#126—64no—
398#125—34no—
399#125—32no—
400#125—32no—
401#125—62no—
402#125—64no—
403#125—62no—
404#124—34no—
405#124—62no—
406#124C22×D312yes—
407#123—64no—
408#123—64no—
409C22⋊Q8—32no—
410C22⋊Q8—32no—
411C22⋊Q8—64no—
412C22⋊Q8—64no—
413C22⋊Q8—64no—
414C22⋊Q8—64no—
415C22⋊Q8—64no—
416C22⋊Q8—64no—
417C22⋊Q8—62no—
418C22⋊Q8—64no—
419C22⋊Q8—62no—
420C22⋊Q8—62no—
421C22⋊Q8—62no—
422C22⋊Q8—61no—
423C22⋊Q8—61no—
424#121—21no—
425#121—21no—
426C4×Q8—32no—
427C4×Q8—32no—
428C4×Q8—34no—
429C4×Q8—32no—
430C4×Q8—32no—
431C4×Q8—62no—
432C4×Q8—61no—
433C4×Q8—61no—
434C4×Q8—61no—
435C4×Q8—61no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—34no—
441C4×D4—34no—
442C4×D4—34no—
443C4×D4—64no—
444C4×D4—62no—
445C4×D4—62no—
446C4×D4—62no—
447C4×D4C22×D312yes—
448C4×D4C22×D312yes—
449#117—32no—
450#117—32no—
451#117—64no—
452#117—62no—
453C2×C4⋊C4—31no—
454C2×C4⋊C4—62no—
455C2×C4⋊C4—64no—
456C2×C4⋊C4—62no—
457C2×C4⋊C4—61no—
458C2×C4⋊C4C22×D311yescharacteristic
459C2×C22⋊C4—21no—
460C2×C22⋊C4—64no—
461C2×C42—61no—
462C2×C42C22×D311yescharacteristic
463#107—34no—
464#107—34no—
465#106—34no—
466C4⋊C8—34no—
467C4⋊C8—34no—
468C4⋊C8C22×D312yes—
469C4⋊C8C22×D312yes—
470#103—32no—
471#103—32no—
472#103—32no—
473#103—32no—
474#103—34no—
475#103—34no—
476#103—34no—
477#103C22×D314yes—
478#102—34no—
479#102—32no—
480#102—32no—
481#102—32no—
482#102—32no—
483#102C22×D314yes—
484#97—34no—
485#97—32no—
486#97—32no—
487C4×C8—34no—
488C23×C3—22no—
489C23×C3—24no—
490C23×C3—22no—
491C23×C3—81no—
492C23×C3—81no—
493C23×C3C22×D411yescharacteristic
494C3×Q8—22no—
495C3×Q8—22no—
496C3×D4—24no—
497C3×D4—22no—
498C3×D4—22no—
499C3×D4—22no—
500C3×D4—42no—
501C3×D4—44no—
502C3×D4—81no—
503C3×D4C2×#4114yes—
504C2×C3×C4—22no—
505C2×C3×C4—22no—
506C2×C3×C4—41no—
507C2×C3×C4—41no—
508C2×C3×C4—44no—
509C2×C3×C4—44no—
510C2×C3×C4—82no—
511C2×C3×C4—81no—
512C2×C3×C4C22×D411yescharacteristic
513C2×C3×C4C22×D411yescharacteristic
514C2×C3×C4C22×D411yescharacteristic
515C2×C3×C4C22×D411yescharacteristic
516C2×C3×C4C22×D411yescharacteristic
517C2×C3×C4C2×#4112yes—
518C2×C3×C42−1+412yes—
519C2×C3×C4C2511yescharacteristic
520C2×Dic3—22no—
521C2×Dic3—22no—
522C2×Dic3—22no—
523C2×Dic3—22no—
524C2×Dic3—42no—
525C2×Dic3—84no—
526C2×Dic3—84no—
527C2×Dic3—84no—
528C2×Dic3—81no—
529C2×Dic3C22×D414yes—
530C2×Dic3C22×D414yes—
531Dic6—22no—
532Dic6—22no—
533Dic6—42no—
534Dic6—42no—
535Dic6—42no—
536Dic6—42no—
537Dic6—84no—
538Dic6—81no—
539C3×C8—24no—
540C3⋊C8—22no—
541C3⋊C8—22no—
542C24—21no—
543#41—24no—
544#41—24no—
545C2×Q8—32no—
546C2×Q8—32no—
547C2×Q8—32no—
548C2×Q8—64no—
549C2×Q8—61no—
550C2×Q8—61no—
551C2×Q8—61no—
552C2×Q8—62no—
553C2×Q8—62no—
554C2×Q8—244no—
555C2×Q8C23×D312yes—
556C2×D4—22no—
557C2×D4—21no—
558C2×D4—21no—
559C2×D4—21no—
560C2×D4—22no—
561C2×D4#23714yes—
562C2×D4C23×D312yes—
563C2×D4C23×D312yes—
564C22×C4—21no—
565C22×C4—22no—
566C22×C4—22no—
567C22×C4—22no—
568C22×C4—64no—
569C22×C4—64no—
570C22×C4—64no—
571C22×C4—61no—
572C22×C4—61no—
573C22×C4—82no—
574C22×C4D3×D411yescharacteristic
575C22×C4D3×D411yescharacteristic
576C22×C4C23×D311yescharacteristic
577SD16—28no—
578SD16—68no—
579D8—28no—
580M16—24no—
581C2×C8—32no—
582C2×C8—32no—
583C2×C8—62no—
584C2×C8—62no—
585C2×C8C23×D314yes—
586C4⋊C4—21no—
587C4⋊C4—21no—
588C4⋊C4—21no—
589C4⋊C4—32no—
590C4⋊C4—34no—
591C4⋊C4—32no—
592C4⋊C4—32no—
593C4⋊C4—34no—
594C4⋊C4—32no—
595C4⋊C4—32no—
596C4⋊C4—32no—
597C4⋊C4—32no—
598C4⋊C4—64no—
599C4⋊C4—64no—
600C4⋊C4—64no—
601C4⋊C4—64no—
602C4⋊C4—64no—
603C4⋊C4—64no—
604C4⋊C4—61no—
605C4⋊C4—62no—
606C4⋊C4—61no—
607C4⋊C4—62no—
608C4⋊C4—62no—
609C4⋊C4—62no—
610C4⋊C4—61no—
611C4⋊C4C23×D312yes—
612C22⋊C4—22no—
613C22⋊C4—22no—
614C22⋊C4—22no—
615C22⋊C4—22no—
616C22⋊C4—34no—
617C22⋊C4—64no—
618C22⋊C4—64no—
619C22⋊C4—64no—
620C22⋊C4—64no—
621C22⋊C4—64no—
622C22⋊C4—62no—
623C42—34no—
624C42—64no—
625C42—62no—
626C42—62no—
627C42—62no—
628C42D3×D411yescharacteristic
629C42D3×D411yescharacteristic
630C42C23×D311yescharacteristic
631C42C23×D311yescharacteristic
632C22×C3—21no—
633C22×C3—21no—
634C22×C3—24no—
635C22×C3—42no—
636C22×C3—42no—
637C22×C3—42no—
638C22×C3—42no—
639C22×C3—42no—
640C22×C3—84no—
641C22×C3#54812yes—
642C22×C3C23×D411yescharacteristic
643C3×C4—21no—
644C3×C4—21no—
645C3×C4—21no—
646C3×C4—21no—
647C3×C4—42no—
648C3×C4—42no—
649C3×C4#54812yes—
650C3×C4C22×#4111yescharacteristic
651C3×C4C2×2−1+411yescharacteristic
652Dic3—24no—
653Dic3—24no—
654Dic3—42no—
655Dic3—42no—
656Dic3—42no—
657Dic3—42no—
658C23—22no—
659C23—24no—
660C23—22no—
661C23—81no—
662C23—81no—
663C23C2×D3×D411yescharacteristic
664Q8—22no—
665Q8—22no—
666Q8—62no—
667Q8—62no—
668Q8—122no—
669Q8—122no—
670Q8—122no—
671Q8—122no—
672Q8—241no—
673Q8—244no—
674D4—22no—
675D4—22no—
676D4—22no—
677D4—24no—
678D4—42no—
679D4—44no—
680D4—81no—
681D4C2×#23714yes—
682C2×C4—22no—
683C2×C4—22no—
684C2×C4—34no—
685C2×C4—34no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—62no—
691C2×C4—62no—
692C2×C4—62no—
693C2×C4—62no—
694C2×C4—82no—
695C2×C4—81no—
696C2×C4—122no—
697C2×C4—244no—
698C2×C4—244no—
699C2×C4—244no—
700C2×C4—241no—
701C2×C4C2×D3×D411yescharacteristic
702C2×C4C2×D3×D411yescharacteristic
703C2×C4C2×D3×D411yescharacteristic
704C2×C4C2×D3×D411yescharacteristic
705C2×C4C2×D3×D411yescharacteristic
706C2×C4C2×#23712yes—
707C2×C4#101012yes—
708C2×C4C24×D311yescharacteristic
709C8—24no—
710C8—62no—
711C8—62no—
712C2×C3—22no—
713C2×C3—44no—
714C2×C3—82no—
715C2×C3C2×#54811yescharacteristic
716C2×C3#358511yescharacteristic
717C2×C3#358611yescharacteristic
718C22—21no—
719C22—21no—
720C22—24no—
721C22—42no—
722C22—42no—
723C22—42no—
724C22—42no—
725C22—42no—
726C22—84no—
727C22#559812yes—
728C22C22×D3×D411yescharacteristic
729C4—21no—
730C4—21no—
731C4—21no—
732C4—21no—
733C4—42no—
734C4—42no—
735C4—64no—
736C4—64no—
737C4—122no—
738C4—122no—
739C4—122no—
740C4—122no—
741C4#559812yes—
742C4C22×#23711yescharacteristic
743C4C2×#101011yescharacteristic
744C3#6133311yescharacteristic, complemented, Sylow
745C2—22no—
746C2—44no—
747C2—82no—
748C2C2×#559811yescharacteristic
749C2#8719211yescharacteristic
750C2#8720111yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
Fitting, prime radicalC3×#3036C2
socleC22×C3C23×D4
hypercenter#3036D3
hypocenterC3#61333
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #86224C2semidirect
2. #85766C2semidirect
4. #83935C2semidirect
5. #83930C2semidirect
6. #83282C2semidirect
7. C2×#5018C2semidirect
8. #80523C2semidirect
9. #80518C2semidirect
10. #80418C2semidirect
11. #80322C2semidirect
12. #79068C2semidirect
13. #79045C2semidirect
14. #78877C2semidirect
15. #78872C2semidirect
25. #5721C22semidirect
26. #5713C22semidirect
27. #5671C22semidirect
64. #5082C22semidirect
65. #5071C22semidirect
66. #5055C22semidirect
67. #5055C22semidirect
73. #4851C22semidirect
74. #4850C22semidirect
75. #4829C22semidirect
76. #4828C22semidirect
77. #4828C22semidirect
78. #4799C22semidirect
79. #4797C22semidirect
80. #4793C22semidirect
81. #4792C22semidirect
82. #4768C22semidirect
83. #4765C22semidirect
84. #4763C22semidirect
85. #4759C22semidirect
744. C3#61333semidirect

Maximal subgroups

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

subgroupindex
#862242
#857662
C3×#30362
#839352
#839302
#832822
C2×#50182
#805232
#805182
#804182
#803222
#790682
#790452
#788772
#788722
#613333