Finite Group Databasebeta

← #10828709

Subgroups — #10828709

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
1C3×#3495C211yesmaximal, characteristic
2#82914C211yesmaximal, characteristic
3#82776C211yesmaximal, characteristic, complemented
4C2×#4995C211yesmaximal, characteristic, complemented
5#71901C212yesmaximal, complemented
6#71863C212yesmaximal, complemented
7#71713C211yesmaximal, characteristic
8#71698C211yesmaximal, characteristic, complemented
9#71665C212yesmaximal, complemented
10#71658C212yesmaximal, complemented
11#71590C211yesmaximal, characteristic, complemented
12#24085—31nomaximal, Sylow
13C3×#556—21no—
14C3×#554—21no—
15C3×#552—21no—
16C3×#552—22no—
17C3×#552—21no—
18C3×#551—22no—
19C3×#544C2211yescharacteristic
20C3×#542—21no—
21C3×#541—21no—
22C3×#536C2211yescharacteristic
23C3×#533—21no—
24C3×#529—21no—
25C3×#525C2211yescharacteristic
26C2×C3×C22⋊Q8C2211yescharacteristic
27C3×#520C2211yescharacteristic
28C3×#519C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C22×C4⋊Dic3C2211yescharacteristic
31C22×#208C2211yescharacteristic
32C2×#837C2212yes—
33#5232C2212yes—
34#5229C2212yes—
35C2×#836C2212yes—
36C4×#217C2211yescharacteristic
37#5207C2211yescharacteristic
38#5044—21no—
39#5043—21no—
40#5034—21no—
41#5033—21no—
42#4995—21no—
43#4995—21no—
44#4995C2212yes—
45#4995C2212yes—
46#4995C2212yes—
47#4986—22no—
48#4977C2212yes—
49C2×#808—21no—
50C2×#807—21no—
51#4943—22no—
52#4532C2212yes—
53#4532C2212yes—
54#4532C2212yes—
55#4530C2212yes—
56#3495D311yescharacteristic
57#3092—31no—
58C2×#428—31no—
59#2503—31no—
60#2061—32no—
61#2038—32no—
62#1901—31no—
63#1871—31no—
64#1860—32no—
65#1857—32no—
66#1826—31no—
67C2×C3×#41C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×Dic3—21no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×#126—22no—
75C3×#126—22no—
76C3×#126—22no—
77C3×#126—22no—
78C3×#125—22no—
79C3×#125—22no—
80C3×#125—22no—
81C3×#125—21no—
82C3×#125—21no—
83C3×#124—22no—
84C3×#124—21no—
85C3×#123—22no—
86C3×#123—22no—
87C3×#123—21no—
88C3×#123—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—21no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—22no—
96C3×C22⋊Q8—21no—
97C3×C22⋊Q8—21no—
98C3×C22⋊Q8D411yescharacteristic
99C3×C22⋊Q8D411yescharacteristic
100C3×#121D411yescharacteristic
101C3×#121D411yescharacteristic
102C3×#120—21no—
103C3×C4×Q8—22no—
104C3×C4×Q8—21no—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8—21no—
109C3×C4×D4—22no—
110C3×C4×D4—21no—
111C3×C4×D4—22no—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×#117—22no—
115C3×#117—22no—
116C3×#117—21no—
117C3×#117—21no—
118C3×#117—22no—
119C3×#117—21no—
120C3×#117D411yescharacteristic
121C3×#117D411yescharacteristic
122C3×#117C2311yescharacteristic
123C3×#117C2311yescharacteristic
124C3×#117C2311yescharacteristic
125C2×C3×C4⋊C4—21no—
126C2×C3×C4⋊C4D412yes—
127C2×C3×C4⋊C4D411yescharacteristic
128C2×C3×C4⋊C4D411yescharacteristic
129C2×C3×C4⋊C4C2311yescharacteristic
130C2×C3×C22⋊C4—21no—
131C2×C3×C22⋊C4D412yes—
132C2×C3×C42—21no—
133C2×#217—21no—
134C2×C4⋊Dic3—21no—
135C2×C4⋊Dic3—21no—
136C2×C4⋊Dic3—21no—
137C2×C4⋊Dic3—21no—
138C2×C4⋊Dic3—21no—
139C2×C4⋊Dic3C2312yes—
140C2×C4×Dic3C2312yes—
141C2×#208—21no—
142C2×#208—24no—
143C2×#208C2312yes—
144C2×#208C2312yes—
145C22×C3⋊C8—21no—
146#837—24no—
147#836—24no—
148#835—22no—
149#832—22no—
150C3⋊#95—24no—
151C3⋊#95—24no—
152C3⋊#95—21no—
153#830—21no—
154#830—21no—
155#808—22no—
156#808—22no—
157#808—22no—
158#807—22no—
159#807—22no—
160#807—22no—
161C22×M16—31no—
162#556—21no—
163#554—21no—
164#552—21no—
165#552—22no—
166#552—21no—
167#551—22no—
168#544C2×D311yescharacteristic
169#542—21no—
170#541—21no—
171#536C2×D311yescharacteristic
172#533—21no—
173#529—21no—
174#525C2×D311yescharacteristic
175C2×C22⋊Q8C2×D311yescharacteristic
176#520C2×D311yescharacteristic
177#519C2×D311yescharacteristic
178C2×#117C2×D311yescharacteristic
179C22×C4⋊C4—31no—
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—62no—
196#391—32no—
197#382—32no—
198#381—62no—
199C4×C22⋊C4—31no—
200#343—32no—
201#339—32no—
202#339—32no—
203#339—32no—
204C24×C3C2×D411yescharacteristic
205C3×#41—22no—
206C3×#41—22no—
207C2×C3×Q8—22no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—22no—
211C2×C3×Q8—22no—
212C2×C3×Q8C2×D411yescharacteristic
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4C2×D411yescharacteristic
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—21no—
220C22×C3×C4—22no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D412yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D412yes—
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—22no—
232C22×Dic3—22no—
233C22×Dic3—44no—
234C22×Dic3—41no—
235C22×Dic3—41no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4C2×D411yescharacteristic
253C3×C4⋊C4C2×D411yescharacteristic
254C3×C4⋊C4C2×D411yescharacteristic
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—21no—
262C3×C22⋊C4—21no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—21no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—21no—
272C3×C42—22no—
273C3×C42—22no—
274C3×C42—22no—
275C3×C42—21no—
276C3×C42—21no—
277C3×C42—21no—
278C3×C42—21no—
279C3×C42—21no—
280#217—42no—
281C4⋊Dic3—22no—
282C4⋊Dic3—22no—
283C4⋊Dic3—41no—
284C4⋊Dic3—41no—
285C4×Dic3—22no—
286C4×Dic3—22no—
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×#41C22×D311yescharacteristic
296C22×Q8C22×D311yescharacteristic
297C23×C4—61no—
298C23×C4C22×D311yescharacteristic
299C2×M16—32no—
300C2×M16—32no—
301C2×M16—64no—
302C2×M16—61no—
303C22×C8—61no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307#126—22no—
308#126—22no—
309#126—22no—
310#126—22no—
311#125—22no—
312#125—22no—
313#125—22no—
314#125—21no—
315#125—21no—
316#124—22no—
317#124—21no—
318#123—22no—
319#123—22no—
320#123—21no—
321#123—22no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C22⋊Q8—22no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8#6711yescharacteristic
332C22⋊Q8#6711yescharacteristic
333#121#6711yescharacteristic
334#121#6711yescharacteristic
335#120—21no—
336C4×Q8—22no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×D4—22no—
343C4×D4—21no—
344C4×D4—22no—
345C4×D4—21no—
346C4×D4—21no—
347#117—22no—
348#117—22no—
349#117—21no—
350#117—21no—
351#117—22no—
352#117—21no—
353#117#6711yescharacteristic
354#117#6711yescharacteristic
355#117C22×D311yescharacteristic
356#117C22×D311yescharacteristic
357#117C22×D311yescharacteristic
358C2×C4⋊C4—21no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—61no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4—61no—
364C2×C4⋊C4—61no—
365C2×C4⋊C4#6712yes—
366C2×C4⋊C4#6711yescharacteristic
367C2×C4⋊C4#6711yescharacteristic
368C2×C4⋊C4C22×D311yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—61no—
371C2×C22⋊C4#6712yes—
372C2×C42—21no—
373C2×C42—32no—
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—61no—
387#95—64no—
388#95—64no—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—21no—
392C23×C3—41no—
393C23×C3#12011yescharacteristic
394C23×C3#12011yescharacteristic
395C23×C3C22×D411yescharacteristic
396C3×Q8—22no—
397C3×Q8—41no—
398C3×Q8—41no—
399C3×Q8—42no—
400C3×Q8—82no—
401C3×D4—22no—
402C3×D4—42no—
403C3×D4—42no—
404C3×D4—81no—
405C2×C3×C4—22no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—22no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—22no—
421C2×C3×C4—24no—
422C2×C3×C4—42no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—42no—
426C2×C3×C4—81no—
427C2×C3×C4—82no—
428C2×C3×C4#12011yescharacteristic
429C2×C3×C4#12011yescharacteristic
430C2×C3×C4C22×D411yescharacteristic
431C2×C3×C4C22×D411yescharacteristic
432C2×C3×C4C2×#4112yes—
433C2×C3×C42−1+412yes—
434C2×Dic3—42no—
435C2×Dic3—42no—
436C2×Dic3—44no—
437C2×Dic3—42no—
438C2×Dic3—42no—
439C2×Dic3—82no—
440C2×Dic3—82no—
441C3⋊C8—24no—
442C3⋊C8—42no—
443C24C2×#6711yescharacteristic
444#41—22no—
445#41—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×#6711yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4C2×#6711yescharacteristic
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—124no—
465C22×C4—121no—
466C22×C4—121no—
467C22×C4C2×#6712yes—
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6711yescharacteristic
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—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—121no—
502C4⋊C4—121no—
503C4⋊C4C2×#6711yescharacteristic
504C4⋊C4C2×#6711yescharacteristic
505C4⋊C4C2×#6711yescharacteristic
506C4⋊C4C2×#6711yescharacteristic
507C4⋊C4C2×#6711yescharacteristic
508C4⋊C4C2×#6711yescharacteristic
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—122no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—62no—
533C42—62no—
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—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3#45412yes—
546C22×C3C2×#12011yescharacteristic
547C22×C3#54712yes—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—42no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—41no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—41no—
559C3×C4#52511yescharacteristic
560C3×C4#53611yescharacteristic
561C3×C4#54712yes—
562Dic3—82no—
563Dic3—82no—
564C23—22no—
565C23—22no—
566C23—21no—
567C23—41no—
568C23#95311yescharacteristic
569C23#95311yescharacteristic
570C23C22×#6711yescharacteristic
571Q8—22no—
572Q8—42no—
573Q8—41no—
574Q8—41no—
575Q8—82no—
576D4—22no—
577D4—42no—
578D4—42no—
579D4—81no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—122no—
604C2×C4—124no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—242no—
609C2×C4—242no—
610C2×C4#95311yescharacteristic
611C2×C4#95311yescharacteristic
612C2×C4D3×#4112yes—
613C2×C4#101012yes—
614C2×C4C22×#6711yescharacteristic
615C2×C4C22×#6711yescharacteristic
616C8—64no—
617C8—122no—
618C2×C3—22no—
619C2×C3—22no—
620C2×C3—41no—
621C2×C3—81no—
622C2×C3#240611yescharacteristic
623C2×C3C2×#45411yescharacteristic
624C2×C3#302211yescharacteristic
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—44no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22#524512yes—
637C22#582412yes—
638C22C2×#95311yescharacteristic
639C4—21no—
640C4—21no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—42no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—242no—
651C4—242no—
652C4#582412yes—
653C4#582511yescharacteristic
654C4#582811yescharacteristic
655C3#2408511yescharacteristic, complemented, Sylow
656C2—22no—
657C2—22no—
658C2—41no—
659C2—81no—
660C2#8409011yescharacteristic
661C2C2×#524511yescharacteristic
662C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3495C2
socleC22×C3C2×#120
prime radicalC3×#525C22
hypercenter#3495D3
hypocenterC3#24085
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3495

Complements

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

NH ≅ G/Ntype
3. #82776C2semidirect
4. C2×#4995C2semidirect
5. #71901C2semidirect
6. #71863C2semidirect
8. #71698C2semidirect
9. #71665C2semidirect
10. #71658C2semidirect
11. #71590C2semidirect
655. C3#24085semidirect

Maximal subgroups

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

subgroupindex
C3×#34952
#829142
#827762
C2×#49952
#719012
#718632
#717132
#716982
#716652
#716582
#715902
#240853