Finite Group Databasebeta

← #10828683

Subgroups — #10828683

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3494C211yesmaximal, characteristic
2#82914C211yesmaximal, characteristic, complemented
3#82665C211yesmaximal, characteristic, complemented
4C2×#4995C211yesmaximal, characteristic, complemented
5#71901C212yesmaximal, complemented
6#71852C212yesmaximal, complemented
7#71713C211yesmaximal, characteristic, complemented
8#71683C211yesmaximal, characteristic, complemented
9#71665C212yesmaximal, complemented
10#71647C212yesmaximal, complemented
11#71590C211yesmaximal, characteristic, complemented
12#24083—31nomaximal, Sylow
13C3×#555—21no—
14C3×#553—21no—
15C3×#553—21no—
16C3×#551—22no—
17C3×#547—21no—
18C3×#546—22no—
19C3×#544C2211yescharacteristic
20C3×#542—21no—
21C3×#540—21no—
22C3×#535C2211yescharacteristic
23C3×#532—21no—
24C3×#529—21no—
25C3×#525C2211yescharacteristic
26C2×C3×#121C2211yescharacteristic
27C3×#520C2211yescharacteristic
28C3×#518C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C22×C4⋊Dic3C2211yescharacteristic
31C22×#208C2211yescharacteristic
32C2×#837C2212yes—
33#5232C2212yes—
34#5221C2212yes—
35#5220—22no—
36C2×#833C2212yes—
37C4×#217C2211yescharacteristic
38#5207C2211yescharacteristic
39#5044—21no—
40#5043—21no—
41#5034—21no—
42#5033—21no—
43#4995—21no—
44#4995—21no—
45#4995C2212yes—
46#4995C2212yes—
47#4995C2212yes—
48#4977C2212yes—
49C2×#808—21no—
50C2×#807—21no—
51#4953—22no—
52#4532C2212yes—
53#4532C2212yes—
54#4532C2212yes—
55#4530C2212yes—
56#3494D311yescharacteristic
57#3092—31no—
58C2×#428—31no—
59#2504—31no—
60#2061—32no—
61#2037—32no—
62#1900—31no—
63#1871—31no—
64#1860—32no—
65#1856—32no—
66#1826—31no—
67C2×C3×#41—22no—
68C2×C3×#41C2311yescharacteristic
69C22×C3×D4D412yes—
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×Dic3—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×#127—21no—
76C3×#126—22no—
77C3×#126—22no—
78C3×#125—21no—
79C3×#125—21no—
80C3×#124—22no—
81C3×#124—22no—
82C3×#124—21no—
83C3×#123—22no—
84C3×#123—22no—
85C3×#123—22no—
86C3×#123—22no—
87C3×#123—21no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8D411yescharacteristic
91C3×C22⋊Q8D411yescharacteristic
92C3×#121—22no—
93C3×#121—22no—
94C3×#121—22no—
95C3×#121—21no—
96C3×#121—21no—
97C3×#121—21no—
98C3×#121—21no—
99C3×#121—21no—
100C3×#121—21no—
101C3×#121D411yescharacteristic
102C3×#121D411yescharacteristic
103C3×#120—21no—
104C3×#120—22no—
105C3×C4×Q8—21no—
106C3×C4×D4—22no—
107C3×C4×D4—22no—
108C3×C4×D4—21no—
109C3×C4×D4—22no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×#117—22no—
117C3×#117—21no—
118C3×#117—21no—
119C3×#117—21no—
120C3×#117D411yescharacteristic
121C3×#117D411yescharacteristic
122C3×#117C2311yescharacteristic
123C3×#117C2311yescharacteristic
124C3×#117C2311yescharacteristic
125C2×C3×C4⋊C4—21no—
126C2×C3×C4⋊C4D411yescharacteristic
127C2×C3×C4⋊C4D411yescharacteristic
128C2×C3×C4⋊C4C2311yescharacteristic
129C2×C3×C22⋊C4—21no—
130C2×C3×C22⋊C4—22no—
131C2×C3×C22⋊C4D412yes—
132C2×C3×C42—21no—
133C2×#217—21no—
134C2×#217—24no—
135C2×C4⋊Dic3—21no—
136C2×C4⋊Dic3—21no—
137C2×C4⋊Dic3—21no—
138C2×C4⋊Dic3—21no—
139C2×C4⋊Dic3—21no—
140C2×C4⋊Dic3C2312yes—
141C2×C4×Dic3C2312yes—
142C2×#208—21no—
143C2×#208—24no—
144C2×#208C2312yes—
145C2×#208C2312yes—
146C22×C3⋊C8—21no—
147#837—24no—
148#835—22no—
149#833—24no—
150#832—22no—
151C3⋊#95—21no—
152C3⋊#95—24no—
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#555—21no—
163#553—21no—
164#553—21no—
165#551—22no—
166#547—21no—
167#546—22no—
168#544C2×D311yescharacteristic
169#542—21no—
170#540—21no—
171#535C2×D311yescharacteristic
172#532—21no—
173#529—21no—
174#525C2×D311yescharacteristic
175C2×#121C2×D311yescharacteristic
176#520C2×D311yescharacteristic
177#518C2×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×#100—32no—
194#407—31no—
195#397—62no—
196#390—32no—
197#386—62no—
198#382—32no—
199C4×C22⋊C4—31no—
200#343—32no—
201#339—32no—
202#339—32no—
203#339—32no—
204C24×C3C2×D411yescharacteristic
205C24×C3C2×D412yes—
206C3×#41—22no—
207C3×#41—22no—
208C3×#41—84no—
209C2×C3×Q8—22no—
210C2×C3×Q8C2×D411yescharacteristic
211C2×C3×D4—21no—
212C2×C3×D4—22no—
213C2×C3×D4—21no—
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4—21no—
217C2×C3×D4—22no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—44no—
223C2×C3×D4C2×D411yescharacteristic
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—21no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D412yes—
238C22×C3×C4C2411yescharacteristic
239C22×Dic3—22no—
240C22×Dic3—22no—
241C22×Dic3—41no—
242C22×Dic3—41no—
243C22×Dic3—82no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
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⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4C2×D411yescharacteristic
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C22⋊C4—22no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
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×C22⋊C4—22no—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—21no—
276C3×C42—22no—
277C3×C42—21no—
278C3×C42—21no—
279C3×C42—21no—
280C3×C42—21no—
281C3×C42—21no—
282#217—42no—
283#217—84no—
284C4⋊Dic3—22no—
285C4⋊Dic3—22no—
286C4⋊Dic3—41no—
287C4⋊Dic3—41no—
288C4×Dic3—22no—
289C4×Dic3—22no—
290#208—24no—
291#208—22no—
292#208—22no—
293#208—22no—
294#208#4114yescomplemented
295C2×C3⋊C8—24no—
296C2×C3⋊C8—22no—
297C2×C3⋊C8—22no—
298C2×#41—22no—
299C2×#41C22×D311yescharacteristic
300C22×D4#6712yes—
301C22×D4C22×D311yescharacteristic
302C23×C4—61no—
303C23×C4C22×D311yescharacteristic
304C2×M16—32no—
305C2×M16—32no—
306C2×M16—64no—
307C2×M16—61no—
308C22×C8—61no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311#127—21no—
312#126—22no—
313#126—22no—
314#125—21no—
315#125—21no—
316#124—22no—
317#124—22no—
318#124—21no—
319#123—22no—
320#123—22no—
321#123—22no—
322#123—22no—
323#123—21no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8#6711yescharacteristic
327C22⋊Q8#6711yescharacteristic
328#121—22no—
329#121—22no—
330#121—22no—
331#121—21no—
332#121—21no—
333#121—21no—
334#121—21no—
335#121—21no—
336#121—21no—
337#121#6711yescharacteristic
338#121#6711yescharacteristic
339#120—21no—
340#120—22no—
341C4×Q8—21no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—21no—
345C4×D4—22no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352#117—22no—
353#117—21no—
354#117—21no—
355#117—21no—
356#117#6711yescharacteristic
357#117#6711yescharacteristic
358#117C22×D311yescharacteristic
359#117C22×D311yescharacteristic
360#117C22×D311yescharacteristic
361C2×C4⋊C4—21no—
362C2×C4⋊C4—32no—
363C2×C4⋊C4—61no—
364C2×C4⋊C4—61no—
365C2×C4⋊C4—61no—
366C2×C4⋊C4—61no—
367C2×C4⋊C4—61no—
368C2×C4⋊C4#6711yescharacteristic
369C2×C4⋊C4#6711yescharacteristic
370C2×C4⋊C4C22×D311yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—61no—
374C2×C22⋊C4—64no—
375C2×C22⋊C4#6712yes—
376C2×C42—21no—
377C2×C42—32no—
378#107—62no—
379#107—62no—
380#107—62no—
381#106—62no—
382#106—62no—
383#106—62no—
384#104—64no—
385#103—62no—
386#102—62no—
387#100—64no—
388C22⋊C8—61no—
389C22⋊C8—61no—
390#95—61no—
391#95—64no—
392C23×C3—22no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—21no—
396C23×C3—24no—
397C23×C3—41no—
398C23×C3—42no—
399C23×C3—42no—
400C23×C3#12011yescharacteristic
401C23×C3#12011yescharacteristic
402C23×C3C22×D411yescharacteristic
403C3×Q8—22no—
404C3×Q8—82no—
405C3×D4—22no—
406C3×D4—42no—
407C3×D4—44no—
408C3×D4—42no—
409C3×D4—41no—
410C3×D4—41no—
411C3×D4—42no—
412C3×D4—81no—
413C3×D4—82no—
414C3×D4—82no—
415C3×D4—82no—
416C2×C3×C4—22no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—21no—
424C2×C3×C4—21no—
425C2×C3×C4—21no—
426C2×C3×C4—21no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—21no—
430C2×C3×C4—22no—
431C2×C3×C4—42no—
432C2×C3×C4—41no—
433C2×C3×C4—41no—
434C2×C3×C4—42no—
435C2×C3×C4—81no—
436C2×C3×C4—82no—
437C2×C3×C4—82no—
438C2×C3×C4—82no—
439C2×C3×C4#12011yescharacteristic
440C2×C3×C4#12011yescharacteristic
441C2×C3×C4C22×D411yescharacteristic
442C2×C3×C4C22×D411yescharacteristic
443C2×C3×C4C2×#4112yes—
444C2×C3×C42−1+412yes—
445C2×Dic3—42no—
446C2×Dic3—42no—
447C2×Dic3—44no—
448C2×Dic3—42no—
449C2×Dic3—42no—
450C2×Dic3—82no—
451C2×Dic3—82no—
452C3⋊C8—24no—
453C3⋊C8—42no—
454C24C2×#6711yescharacteristic
455C24C2×#6712yes—
456#41—22no—
457#41—22no—
458#41—84no—
459C2×Q8—22no—
460C2×Q8C2×#6711yescharacteristic
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—44no—
473C2×D4C2×#6711yescharacteristic
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—121no—
485C22×C4—121no—
486C22×C4—242no—
487C22×C4C2×#6711yescharacteristic
488C22×C4C2×#6711yescharacteristic
489C22×C4C2×#6711yescharacteristic
490C22×C4C2×#6711yescharacteristic
491C22×C4C2×#6711yescharacteristic
492C22×C4C2×#6712yes—
493C22×C4C23×D311yescharacteristic
494M16—34no—
495M16—64no—
496M16—62no—
497M16—62no—
498M16—62no—
499C2×C8—64no—
500C2×C8—62no—
501C2×C8—62no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—62no—
513C4⋊C4—62no—
514C4⋊C4—121no—
515C4⋊C4—121no—
516C4⋊C4C2×#6711yescharacteristic
517C4⋊C4C2×#6711yescharacteristic
518C4⋊C4C2×#6711yescharacteristic
519C4⋊C4C2×#6711yescharacteristic
520C4⋊C4C2×#6711yescharacteristic
521C4⋊C4C2×#6711yescharacteristic
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—122no—
539C22⋊C4—244no—
540C42—22no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—62no—
547C42—62no—
548C22×C3—21no—
549C22×C3—21no—
550C22×C3—21no—
551C22×C3—21no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3—21no—
555C22×C3—44no—
556C22×C3—41no—
557C22×C3—42no—
558C22×C3—41no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3—41no—
562C22×C3—84no—
563C22×C3#45512yes—
564C22×C3C2×#12011yescharacteristic
565C22×C3#54712yes—
566C3×C4—21no—
567C3×C4—21no—
568C3×C4—41no—
569C3×C4—41no—
570C3×C4—41no—
571C3×C4—42no—
572C3×C4—41no—
573C3×C4—41no—
574C3×C4—41no—
575C3×C4—41no—
576C3×C4#52511yescharacteristic
577C3×C4#53611yescharacteristic
578C3×C4#54712yes—
579Dic3—82no—
580Dic3—82no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—21no—
585C23—24no—
586C23—41no—
587C23—42no—
588C23—42no—
589C23#95311yescharacteristic
590C23#95311yescharacteristic
591C23C22×#6711yescharacteristic
592Q8—22no—
593Q8—82no—
594D4—22no—
595D4—44no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—41no—
600D4—41no—
601D4—82no—
602D4—81no—
603D4—82no—
604D4—82no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—81no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—124no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—242no—
634C2×C4—242no—
635C2×C4#95311yescharacteristic
636C2×C4#95311yescharacteristic
637C2×C4D3×#4112yes—
638C2×C4#101012yes—
639C2×C4C22×#6711yescharacteristic
640C2×C4C22×#6711yescharacteristic
641C8—64no—
642C8—122no—
643C2×C3—22no—
644C2×C3—22no—
645C2×C3—41no—
646C2×C3—81no—
647C2×C3—82no—
648C2×C3#240611yescharacteristic
649C2×C3C2×#45511yescharacteristic
650C2×C3#302111yescharacteristic
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—44no—
659C22—41no—
660C22—42no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—41no—
665C22—84no—
666C22#524212yes—
667C22#582412yes—
668C22C2×#95311yescharacteristic
669C4—21no—
670C4—21no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—42no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—242no—
680C4—242no—
681C4#582412yes—
682C4#582511yescharacteristic
683C4#582811yescharacteristic
684C3#2408311yescharacteristic, complemented, Sylow
685C2—22no—
686C2—22no—
687C2—41no—
688C2—82no—
689C2—81no—
690C2#8409011yescharacteristic
691C2C2×#524211yescharacteristic
692C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3494C2
socleC22×C3C2×#120
hypercenter#3494D3
hypocenterC3#24083
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3494

Complements

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

NH ≅ G/Ntype
2. #82914C2semidirect
3. #82665C2semidirect
4. C2×#4995C2semidirect
5. #71901C2semidirect
6. #71852C2semidirect
7. #71713C2semidirect
8. #71683C2semidirect
9. #71665C2semidirect
10. #71647C2semidirect
11. #71590C2semidirect
294. #208#41semidirect
684. C3#24083semidirect

Maximal subgroups

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

subgroupindex
C3×#34942
#829142
#826652
C2×#49952
#719012
#718522
#717132
#716832
#716652
#716472
#715902
#240833