Finite Group Databasebeta

← #10828672

Subgroups — #10828672

Order 768.

1130 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3494C211yesmaximal, characteristic
2#82914C211yesmaximal, characteristic, complemented
3#82664C211yesmaximal, characteristic, complemented
4C2×#4995C211yesmaximal, characteristic, complemented
5#71900C212yesmaximal, complemented
6#71852C212yesmaximal, complemented
7#71713C211yesmaximal, characteristic, complemented
8#71683C211yesmaximal, characteristic, complemented
9#71664C212yesmaximal, complemented
10#71647C212yesmaximal, complemented
11#71590C211yesmaximal, characteristic, complemented
12#24082—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
30C2×#935—22no—
31C22×C4⋊Dic3C2211yescharacteristic
32C22×#208C2211yescharacteristic
33C2×#837C2212yes—
34#5232C2212yes—
35#5221C2212yes—
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#4975C2212yes—
49C2×#808—21no—
50C2×#807—21no—
51#4949—22no—
52#4532C2212yes—
53#4532C2212yes—
54#4532C2212yes—
55#4530C2212yes—
56#3494D311yescharacteristic
57#3092—31no—
58C2×#428—31no—
59#2501—31no—
60#2060—32no—
61#2037—32no—
62#1900—31no—
63#1871—31no—
64#1859—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—
135#935—48no—
136C2×C4⋊Dic3—21no—
137C2×C4⋊Dic3—21no—
138C2×C4⋊Dic3—21no—
139C2×C4⋊Dic3—21no—
140C2×C4⋊Dic3—21no—
141C2×C4⋊Dic3C2312yes—
142C2×Dic3⋊C4—24no—
143C2×C4×Dic3C2312yes—
144C2×#208—21no—
145C2×#208—24no—
146C2×#208C2312yes—
147C2×#208C2312yes—
148C22×C3⋊C8—21no—
149#837—24no—
150#835—22no—
151#833—24no—
152#832—22no—
153C3⋊#95—21no—
154#830—21no—
155#830—21no—
156#808—22no—
157#808—22no—
158#808—22no—
159#807—22no—
160#807—22no—
161#807—22no—
162C22×M16—31no—
163#555—21no—
164#553—21no—
165#553—21no—
166#551—22no—
167#547—21no—
168#546—22no—
169#544C2×D311yescharacteristic
170#542—21no—
171#540—21no—
172#535C2×D311yescharacteristic
173#532—21no—
174#529—21no—
175#525C2×D311yescharacteristic
176C2×#123—62no—
177C2×#121C2×D311yescharacteristic
178#520C2×D311yescharacteristic
179#518C2×D311yescharacteristic
180C2×#117C2×D311yescharacteristic
181C22×C4⋊C4—31no—
182#484—61no—
183#483—61no—
184#481—61no—
185#480—61no—
186#428—32no—
187#428—32no—
188#428—32no—
189#428—61no—
190#428—61no—
191C2×#107—61no—
192C2×#106—61no—
193C2×#104—32no—
194#417—32no—
195C2×#100—32no—
196#407—31no—
197#399—62no—
198#390—32no—
199#384—32no—
200C4×C22⋊C4—31no—
201#343—32no—
202#339—32no—
203#339—32no—
204#339—32no—
205C24×C3C2×D411yescharacteristic
206C24×C3C2×D412yes—
207C3×#41—22no—
208C3×#41—22no—
209C3×#41—84no—
210C2×C3×Q8—22no—
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×D4—21no—
213C2×C3×D4—22no—
214C2×C3×D4—21no—
215C2×C3×D4—22no—
216C2×C3×D4—22no—
217C2×C3×D4—21no—
218C2×C3×D4—22no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4—44no—
224C2×C3×D4C2×D411yescharacteristic
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—21no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—22no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D412yes—
239C22×C3×C4C2411yescharacteristic
240C22×Dic3—22no—
241C22×Dic3—22no—
242C22×Dic3—44no—
243C22×Dic3—41no—
244C22×Dic3—41no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—22no—
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—21no—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—22no—
275C3×C22⋊C4—21no—
276C3×C22⋊C4—21no—
277C3×C42—22no—
278C3×C42—21no—
279C3×C42—21no—
280C3×C42—21no—
281C3×C42—21no—
282C3×C42—21no—
283#217—24no—
284#217—48no—
285C4⋊Dic3—22no—
286C4⋊Dic3—22no—
287C4⋊Dic3—41no—
288C4⋊Dic3—41no—
289Dic3⋊C4—48no—
290C4×Dic3—22no—
291C4×Dic3—22no—
292#208—24no—
293#208—22no—
294#208—22no—
295#208—22no—
296#208#4114yescomplemented
297C2×C3⋊C8—24no—
298C2×C3⋊C8—22no—
299C2×C3⋊C8—22no—
300C2×#41—22no—
301C2×#41C22×D311yescharacteristic
302C22×D4#6712yes—
303C22×D4C22×D311yescharacteristic
304C23×C4—61no—
305C23×C4C22×D311yescharacteristic
306C2×M16—32no—
307C2×M16—32no—
308C2×M16—64no—
309C2×M16—61no—
310C22×C8—61no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313#127—21no—
314#126—22no—
315#126—22no—
316#125—21no—
317#125—21no—
318#124—22no—
319#124—22no—
320#124—21no—
321#123—22no—
322#123—22no—
323#123—22no—
324#123—22no—
325#123—21no—
326#123—128no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8#6711yescharacteristic
330C22⋊Q8#6711yescharacteristic
331#121—22no—
332#121—22no—
333#121—22no—
334#121—21no—
335#121—21no—
336#121—21no—
337#121—21no—
338#121—21no—
339#121—21no—
340#121#6711yescharacteristic
341#121#6711yescharacteristic
342#120—21no—
343#120—22no—
344C4×Q8—21no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—21no—
348C4×D4—22no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—21no—
355#117—22no—
356#117—21no—
357#117—21no—
358#117—21no—
359#117#6711yescharacteristic
360#117#6711yescharacteristic
361#117C22×D311yescharacteristic
362#117C22×D311yescharacteristic
363#117C22×D311yescharacteristic
364C2×C4⋊C4—21no—
365C2×C4⋊C4—32no—
366C2×C4⋊C4—61no—
367C2×C4⋊C4—61no—
368C2×C4⋊C4—61no—
369C2×C4⋊C4—61no—
370C2×C4⋊C4—61no—
371C2×C4⋊C4—64no—
372C2×C4⋊C4#6711yescharacteristic
373C2×C4⋊C4#6711yescharacteristic
374C2×C4⋊C4C22×D311yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—61no—
378C2×C22⋊C4—64no—
379C2×C22⋊C4#6712yes—
380C2×C42—21no—
381C2×C42—32no—
382#107—62no—
383#107—62no—
384#107—62no—
385#106—62no—
386#106—62no—
387#106—62no—
388#104—64no—
389#103—62no—
390#102—62no—
391#100—64no—
392C22⋊C8—61no—
393C22⋊C8—61no—
394#95—61no—
395C23×C3—22no—
396C23×C3—22no—
397C23×C3—22no—
398C23×C3—21no—
399C23×C3—24no—
400C23×C3—41no—
401C23×C3—42no—
402C23×C3—42no—
403C23×C3#12011yescharacteristic
404C23×C3#12011yescharacteristic
405C23×C3C22×D411yescharacteristic
406C3×Q8—22no—
407C3×Q8—82no—
408C3×D4—22no—
409C3×D4—42no—
410C3×D4—44no—
411C3×D4—42no—
412C3×D4—41no—
413C3×D4—41no—
414C3×D4—42no—
415C3×D4—81no—
416C3×D4—82no—
417C3×D4—82no—
418C3×D4—82no—
419C2×C3×C4—22no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—21no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—21no—
430C2×C3×C4—21no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—22no—
434C2×C3×C4—42no—
435C2×C3×C4—41no—
436C2×C3×C4—41no—
437C2×C3×C4—42no—
438C2×C3×C4—81no—
439C2×C3×C4—82no—
440C2×C3×C4—82no—
441C2×C3×C4—82no—
442C2×C3×C4#12011yescharacteristic
443C2×C3×C4#12011yescharacteristic
444C2×C3×C4C22×D411yescharacteristic
445C2×C3×C4C22×D411yescharacteristic
446C2×C3×C4C2×#4112yes—
447C2×C3×C42−1+412yes—
448C2×Dic3—24no—
449C2×Dic3—42no—
450C2×Dic3—42no—
451C2×Dic3—44no—
452C2×Dic3—42no—
453C2×Dic3—82no—
454C2×Dic3—82no—
455C3⋊C8—24no—
456C3⋊C8—42no—
457C24C2×#6711yescharacteristic
458C24C2×#6712yes—
459#41—22no—
460#41—22no—
461#41—84no—
462C2×Q8—22no—
463C2×Q8C2×#6711yescharacteristic
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—44no—
476C2×D4C2×#6711yescharacteristic
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—124no—
488C22×C4—121no—
489C22×C4—121no—
490C22×C4C2×#6711yescharacteristic
491C22×C4C2×#6711yescharacteristic
492C22×C4C2×#6711yescharacteristic
493C22×C4C2×#6711yescharacteristic
494C22×C4C2×#6711yescharacteristic
495C22×C4C2×#6712yes—
496C22×C4C23×D311yescharacteristic
497M16—34no—
498M16—64no—
499M16—62no—
500M16—62no—
501M16—62no—
502C2×C8—64no—
503C2×C8—62no—
504C2×C8—62no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—62no—
516C4⋊C4—62no—
517C4⋊C4—128no—
518C4⋊C4—121no—
519C4⋊C4—121no—
520C4⋊C4C2×#6711yescharacteristic
521C4⋊C4C2×#6711yescharacteristic
522C4⋊C4C2×#6711yescharacteristic
523C4⋊C4C2×#6711yescharacteristic
524C4⋊C4C2×#6711yescharacteristic
525C4⋊C4C2×#6711yescharacteristic
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—64no—
543C22⋊C4—128no—
544C42—22no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—62no—
551C42—62no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3—21no—
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—21no—
559C22×C3—44no—
560C22×C3—41no—
561C22×C3—42no—
562C22×C3—41no—
563C22×C3—42no—
564C22×C3—42no—
565C22×C3—41no—
566C22×C3—84no—
567C22×C3#45312yes—
568C22×C3C2×#12011yescharacteristic
569C22×C3#54712yes—
570C3×C4—21no—
571C3×C4—21no—
572C3×C4—41no—
573C3×C4—41no—
574C3×C4—41no—
575C3×C4—42no—
576C3×C4—41no—
577C3×C4—41no—
578C3×C4—41no—
579C3×C4—41no—
580C3×C4#52511yescharacteristic
581C3×C4#53611yescharacteristic
582C3×C4#54712yes—
583Dic3—44no—
584Dic3—82no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—21no—
589C23—24no—
590C23—41no—
591C23—42no—
592C23—42no—
593C23#95311yescharacteristic
594C23#95311yescharacteristic
595C23C22×#6711yescharacteristic
596Q8—22no—
597Q8—82no—
598D4—22no—
599D4—44no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—41no—
604D4—41no—
605D4—82no—
606D4—81no—
607D4—82no—
608D4—82no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—64no—
629C2×C4—81no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—124no—
636C2×C4—122no—
637C2×C4—242no—
638C2×C4—242no—
639C2×C4#95311yescharacteristic
640C2×C4#95311yescharacteristic
641C2×C4D3×#4112yes—
642C2×C4#101012yes—
643C2×C4C22×#6711yescharacteristic
644C2×C4C22×#6711yescharacteristic
645C8—64no—
646C8—122no—
647C2×C3—22no—
648C2×C3—22no—
649C2×C3—41no—
650C2×C3—81no—
651C2×C3—82no—
652C2×C3#240611yescharacteristic
653C2×C3C2×#45311yescharacteristic
654C2×C3#302211yescharacteristic
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—44no—
663C22—41no—
664C22—42no—
665C22—41no—
666C22—42no—
667C22—42no—
668C22—41no—
669C22—84no—
670C22#524112yes—
671C22#582412yes—
672C22C2×#95311yescharacteristic
673C4—21no—
674C4—21no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—42no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—124no—
684C4—242no—
685C4#582412yes—
686C4#582511yescharacteristic
687C4#582811yescharacteristic
688C3#2408211yescharacteristic, complemented, Sylow
689C2—22no—
690C2—22no—
691C2—41no—
692C2—82no—
693C2—81no—
694C2#8409011yescharacteristic
695C2C2×#524111yescharacteristic
696C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3494C2
socleC22×C3C2×#120
hypercenter#3494D3
hypocenterC3#24082
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. #82664C2semidirect
4. C2×#4995C2semidirect
5. #71900C2semidirect
6. #71852C2semidirect
7. #71713C2semidirect
8. #71683C2semidirect
9. #71664C2semidirect
10. #71647C2semidirect
11. #71590C2semidirect
296. #208#41semidirect
688. C3#24082semidirect

Maximal subgroups

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

subgroupindex
C3×#34942
#829142
#826642
C2×#49952
#719002
#718522
#717132
#716832
#716642
#716472
#715902
#240823