Finite Group Databasebeta

← #10770617

Subgroups — #10770617

Order 768.

1036 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3558C211yesmaximal, characteristic, complemented
2#83306C211yesmaximal, characteristic, complemented
3#83144C211yesmaximal, characteristic, complemented
4#83007C211yesmaximal, characteristic, complemented
5#71885C212yesmaximal, complemented
6#71884C212yesmaximal, complemented
7#71846C212yesmaximal, complemented
8#71845C212yesmaximal, complemented
9#71814C211yesmaximal, characteristic, complemented
10#71756C211yesmaximal, characteristic, complemented
11#71720C211yesmaximal, characteristic, complemented
12#18458—31nomaximal, Sylow
13C3×#562—21no—
14C3×H(3,𝔽4)—21no—
15C3×#559—21no—
16C3×#548—21no—
17C3×#546—21no—
18C3×#546—22no—
19C3×D42—21no—
20C3×#542—21no—
21C3×#538—22no—
22C3×#538C2212yes—
23C3×#535—21no—
24C3×#534C2212yes—
25C3×#532C2211yescharacteristic
26C3×#529C2211yescharacteristic
27C2×C3×#124C2211yescharacteristic
28C2×#875C2211yescharacteristic, complemented
29#5231—21no—
30#5223—21no—
31#5089—21no—
32#5074—21no—
33#5068—21no—
34#5059—21no—
35#5050C2211yescharacteristic, complemented
36#5046C2211yescharacteristic, complemented
37#5040C2211yescharacteristic
38#5040C2212yescomplemented
39#5036C2211yescharacteristic
40#5036C2212yescomplemented
41#5028—21no—
42#5016—22no—
43#5016—21no—
44#4961—22no—
45#4960C2212yescomplemented
46#4947C2212yescomplemented
47#4936C2211yescharacteristic, complemented
48#4926C2211yescharacteristic, complemented
49#4925C2211yescharacteristic, complemented
50#4538C2212yescomplemented
51#4537C2212yes—
52#4537C2212yescomplemented
53#4537C2212yescomplemented
54#4525C2212yescomplemented
55#4525C2212yescomplemented
56#3558D311yescharacteristic, complemented
57#3213—31no—
58#3211—31no—
59#2819—31no—
60#1606—32no—
61#1601—32no—
62#1600—32no—
63#1595—32no—
64#1517—31no—
65#1512—31no—
66#1469—31no—
67C2×C3×#41—21no—
68C2×C3×#41—21no—
69C22×C3×Q8D411yescharacteristic
70C22×C3×D4—21no—
71C22×C3×D4—22no—
72C22×C3×D4D411yescharacteristic
73C22×#67—22no—
74C22×D12C2311yescharacteristic
75C22×Dic6C2311yescharacteristic
76C3×C4⋊Q8D411yescharacteristic, complemented
77C3×#127—21no—
78C3×#127D411yescharacteristic, complemented
79C3×#126—22no—
80C3×#126—21no—
81C3×#125D411yescharacteristic, complemented
82C3×#124—21no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124—22no—
86C3×#124—21no—
87C3×#124—21no—
88C3×#124—21no—
89C3×#124—21no—
90C3×#124—21no—
91C3×#124D411yescharacteristic, complemented
92C3×#123—22no—
93C3×#123—21no—
94C3×#123—22no—
95C3×#123—22no—
96C3×#123—22no—
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8C2311yescharacteristic
101C3×#121—22no—
102C3×#121—22no—
103C3×#121—22no—
104C3×#121—21no—
105C3×#121—22no—
106C3×#121—21no—
107C3×#121—22no—
108C3×#121C2311yescharacteristic
109C3×#121C2312yes—
110C3×#120—22no—
111C3×#120—22no—
112C3×#120—22no—
113C3×#120—22no—
114C3×#120—22no—
115C3×C4×Q8—21no—
116C3×C4×Q8D411yescharacteristic, complemented
117C3×C4×D4—22no—
118C3×C4×D4—21no—
119C3×C4×D4—21no—
120C3×C4×D4—22no—
121C3×C4×D4—21no—
122C3×C4×D4D411yescharacteristic, complemented
123C3×C4×D4D412yescomplemented
124C3×#117—21no—
125C2×C3×C22⋊C4—22no—
126C2×C3×C22⋊C4D412yes—
127C2×C3×C22⋊C4C2312yes—
128C2×C3×C42C2311yescharacteristic
129C2×#217—22no—
130C2×#216—21no—
131C2×#215—21no—
132C2×#214—21no—
133C2×#214—22no—
134C2×#213—21no—
135C2×#213—22no—
136C2×(C2×D3)⋊C4—21no—
137C2×(C2×D3)⋊C4—21no—
138C2×(C2×D3)⋊C4—22no—
139C2×C4⋊Dic3—21no—
140#875—21no—
141#875—21no—
142#875—21no—
143#875—21no—
144C3⋊#95—21no—
145C3⋊#95—22no—
146C3⋊#95C2312yes—
147#830C2311yescharacteristic
148#830C2311yescharacteristic
149#830C2312yes—
150#810—22no—
151#810—21no—
152#810—21no—
153#809—22no—
154#809—21no—
155#809—21no—
156C4⋊(C3⋊C8)—21no—
157C4⋊(C3⋊C8)—21no—
158#803—21no—
159C4×C3⋊C8—21no—
160#562—21no—
161H(3,𝔽4)—21no—
162#559—21no—
163#548—21no—
164#546—21no—
165#546—22no—
166D42—21no—
167#542—21no—
168#538—22no—
169#538C2×D312yes—
170#535—21no—
171#534C2×D312yes—
172#532C2×D311yescharacteristic
173#529C2×D311yescharacteristic
174C2×#124—31no—
175C2×#124C2×D311yescharacteristic
176#496—61no—
177#495—61no—
178#489—61no—
179#488—61no—
180#464—61no—
181#463—61no—
182#463—62no—
183#451—31no—
184#450—31no—
185#448—31no—
186#448—32no—
187#447—31no—
188#447—32no—
189#432—31no—
190#431—31no—
191#394—62no—
192#393—32no—
193#393—61no—
194#392—32no—
195#392—61no—
196#382—31no—
197#328—32no—
198#327—32no—
199#327—32no—
200#327—32no—
201#323—32no—
202#323—32no—
203C24×C3—22no—
204C24×C3C2×D412yes—
205C23×D3—21no—
206C3×#41—81no—
207C3×#41—81no—
208C3×#41—82no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—81no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4—22no—
215C2×C3×D4—21no—
216C2×C3×D4—22no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—82no—
226C2×C3×D4—81no—
227C2×C3×D4—82no—
228C2×C3×D4—82no—
229C2×C3×D4C2×D411yescharacteristic
230C2×C3×D4C2×D412yes—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4—21no—
234C22×C3×C4—22no—
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D412yes—
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2411yescharacteristic
239C2×#67—82no—
240C2×#67—82no—
241C2×#67—82no—
242C22×Dic3—22no—
243C22×Dic3—21no—
244C2×D12—21no—
245C2×D12—21no—
246C2×D12—81no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—81no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4C2×D412yes—
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—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—22no—
272C3×C22⋊C4—22no—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—22no—
276C3×C42—22no—
277C3×C42—21no—
278C3×C42C2×D411yescharacteristic
279C3×C42C2×D411yescharacteristic
280C3×C42C2×D411yescharacteristic
281C3×C42C2×D411yescharacteristic
282#217—82no—
283#216—81no—
284#215—81no—
285#214—82no—
286#214—81no—
287#213—82no—
288#213—81no—
289(C2×D3)⋊C4—41no—
290(C2×D3)⋊C4—41no—
291(C2×D3)⋊C4—41no—
292(C2×D3)⋊C4—41no—
293(C2×D3)⋊C4—82no—
294C4⋊Dic3—81no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—22no—
298C2×#41—21no—
299C2×#41—21no—
300C22×Q8—31no—
301C22×Q8#6711yescharacteristic
302C22×D4—21no—
303C22×D4—22no—
304C22×D4—31no—
305C22×D4—62no—
306C22×D4#6711yescharacteristic
307C2×Q16—61no—
308C2×SD16—61no—
309C2×SD16—61no—
310C2×SD16—62no—
311C2×D8—61no—
312C2×D8—62no—
313C4⋊Q8#6711yescharacteristic, complemented
314#127—21no—
315#127#6711yescharacteristic, complemented
316#126—22no—
317#126—21no—
318#125#6711yescharacteristic, complemented
319#124—21no—
320#124—22no—
321#124—22no—
322#124—22no—
323#124—21no—
324#124—21no—
325#124—21no—
326#124—21no—
327#124—21no—
328#124—61no—
329#124—61no—
330#124—61no—
331#124—61no—
332#124#6711yescharacteristic, complemented
333#123—22no—
334#123—21no—
335#123—22no—
336#123—22no—
337#123—22no—
338C22⋊Q8—22no—
339C22⋊Q8—21no—
340C22⋊Q8—22no—
341C22⋊Q8C22×D311yescharacteristic
342#121—22no—
343#121—22no—
344#121—22no—
345#121—21no—
346#121—22no—
347#121—21no—
348#121—22no—
349#121C22×D311yescharacteristic
350#121C22×D312yes—
351#120—22no—
352#120—22no—
353#120—22no—
354#120—22no—
355#120—22no—
356C4×Q8—21no—
357C4×Q8#6711yescharacteristic, complemented
358C4×D4—22no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—22no—
362C4×D4—21no—
363C4×D4#6711yescharacteristic, complemented
364C4×D4#6712yescomplemented
365#117—21no—
366C2×C4⋊C4—61no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—61no—
369C2×C22⋊C4—61no—
370C2×C22⋊C4—62no—
371C2×C22⋊C4—62no—
372C2×C22⋊C4#6712yes—
373C2×C22⋊C4C22×D312yes—
374C2×C42C22×D311yescharacteristic
375C4⋊C8—61no—
376C4⋊C8—61no—
377#103—62no—
378#103—61no—
379#103—61no—
380#102—62no—
381#102—61no—
382#102—61no—
383C22⋊C8—31no—
384C22⋊C8—31no—
385C22⋊C8—32no—
386#97—61no—
387C4×C8—61no—
388#95—32no—
389#95—61no—
390#95—62no—
391C23×C3—22no—
392C23×C3—21no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—22no—
396C23×C3—42no—
397C23×C3—42no—
398C23×C3—82no—
399C23×C3—82no—
400C23×C3—82no—
401C23×C3C22×D411yescharacteristic
402C22×D3—42no—
403C22×D3—42no—
404C22×D3—41no—
405C22×D3—41no—
406C22×D3—41no—
407C3×Q8—41no—
408C3×Q8—42no—
409C3×Q8—81no—
410C3×Q8—81no—
411C3×D4—42no—
412C3×D4—41no—
413C3×D4—42no—
414C3×D4—42no—
415C3×D4—41no—
416C3×D4—42no—
417C3×D4—41no—
418C3×D4—42no—
419C3×D4—81no—
420C3×D4—81no—
421C3×D4—82no—
422C3×D4—82no—
423C3×D4—82no—
424C3×D4—82no—
425C3×D4—82no—
426C2×C3×C4—21no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—41no—
434C2×C3×C4—41no—
435C2×C3×C4—42no—
436C2×C3×C4—42no—
437C2×C3×C4—82no—
438C2×C3×C4—82no—
439C2×C3×C4—82no—
440C2×C3×C4#12011yescharacteristic
441C2×C3×C4#12011yescharacteristic
442C2×C3×C4#12011yescharacteristic
443C2×C3×C4#12011yescharacteristic
444C2×C3×C4C22×D411yescharacteristic
445C2×C3×C4C22×D411yescharacteristic
446#67—88no—
447C2×Dic3—42no—
448C2×Dic3—42no—
449C2×Dic3—42no—
450C2×Dic3—41no—
451C2×Dic3—42no—
452D12—42no—
453D12—41no—
454D12—41no—
455Dic6—42no—
456Dic6—41no—
457Dic6—41no—
458C3⋊C8—42no—
459C3⋊C8—41no—
460C3⋊C8—41no—
461C24—22no—
462C24—61no—
463C24C2×#6712yes—
464#41—81no—
465#41—81no—
466#41—82no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—81no—
473C2×Q8—241no—
474C2×Q8C2×#6711yescharacteristic
475C2×D4—22no—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—61no—
487C2×D4—61no—
488C2×D4—82no—
489C2×D4—81no—
490C2×D4—82no—
491C2×D4—82no—
492C2×D4—241no—
493C2×D4—242no—
494C2×D4—242no—
495C2×D4—242no—
496C2×D4C2×#6711yescharacteristic
497C2×D4C2×#6712yes—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—62no—
503C22×C4—61no—
504C22×C4C2×#6711yescharacteristic
505C22×C4C2×#6712yes—
506C22×C4C2×#6711yescharacteristic
507C22×C4C23×D311yescharacteristic
508Q16—241no—
509SD16—242no—
510SD16—241no—
511SD16—241no—
512D8—242no—
513D8—241no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—62no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—241no—
525C4⋊C4C2×#6712yes—
526C4⋊C4C2×#6711yescharacteristic
527C4⋊C4C2×#6711yescharacteristic
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—121no—
545C22⋊C4—121no—
546C22⋊C4—121no—
547C22⋊C4—121no—
548C22⋊C4—242no—
549C22⋊C4—242no—
550C42—22no—
551C42—21no—
552C42C2×#6711yescharacteristic
553C42C2×#6711yescharacteristic
554C42C2×#6711yescharacteristic
555C42C2×#6711yescharacteristic
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—41no—
561C22×C3—41no—
562C22×C3—42no—
563C22×C3—42no—
564C22×C3—42no—
565C22×C3—41no—
566C22×C3—42no—
567C22×C3—82no—
568C22×C3—82no—
569C22×C3—164no—
570C22×C3C2×#12011yescharacteristic
571C22×C3#57611yescharacteristic
572C22×C3#57711yescharacteristic
573C2×D3—82no—
574C2×D3—82no—
575C2×D3—81no—
576C2×D3—81no—
577C2×D3—81no—
578C3×C4—21no—
579C3×C4—21no—
580C3×C4—21no—
581C3×C4—21no—
582C3×C4—21no—
583C3×C4—21no—
584C3×C4—41no—
585C3×C4—42no—
586C3×C4—41no—
587C3×C4—42no—
588C3×C4—41no—
589Dic3—82no—
590Dic3—81no—
591C23—22no—
592C23—21no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—42no—
597C23—42no—
598C23—82no—
599C23—82no—
600C23—82no—
601C23—122no—
602C23—122no—
603C23—121no—
604C23—121no—
605C23—121no—
606C23C22×#6711yescharacteristic
607Q8—41no—
608Q8—42no—
609Q8—81no—
610Q8—81no—
611Q8—122no—
612Q8—121no—
613Q8—121no—
614D4—42no—
615D4—41no—
616D4—42no—
617D4—42no—
618D4—41no—
619D4—42no—
620D4—41no—
621D4—42no—
622D4—81no—
623D4—81no—
624D4—82no—
625D4—82no—
626D4—82no—
627D4—82no—
628D4—82no—
629D4—122no—
630D4—121no—
631D4—121no—
632D4—248no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—121no—
652C2×C4#95311yescharacteristic
653C2×C4#95311yescharacteristic
654C2×C4#95311yescharacteristic
655C2×C4#95311yescharacteristic
656C2×C4C22×#6711yescharacteristic
657C2×C4C22×#6711yescharacteristic
658C8—122no—
659C8—121no—
660C8—121no—
661C2×C3—21no—
662C2×C3—21no—
663C2×C3—82no—
664C2×C3—82no—
665C2×C3—81no—
666C2×C3#301211yescharacteristic
667C2×C3#301911yescharacteristic
668C2×C3#302711yescharacteristic
669D3—161no—
670C22—21no—
671C22—21no—
672C22—42no—
673C22—42no—
674C22—41no—
675C22—41no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—41no—
680C22—42no—
681C22—82no—
682C22—82no—
683C22—164no—
684C22—242no—
685C22—242no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22#583511yescharacteristic
690C22#583611yescharacteristic
691C22C2×#95311yescharacteristic
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—41no—
699C4—42no—
700C4—41no—
701C4—42no—
702C4—41no—
703C4—242no—
704C4—241no—
705C3#1845811yescharacteristic, complemented, Sylow
706C2—21no—
707C2—21no—
708C2—82no—
709C2—81no—
710C2—82no—
711C2—481no—
712C2#8410211yescharacteristic
713C2#8410411yescharacteristic
714C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3558C2
socleC22×C3C2×#120
hypercenter#3558D3
hypocenterC3#18458
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3558

Complements

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

NH ≅ G/Ntype
1. C3×#3558C2semidirect
2. #83306C2semidirect
3. #83144C2semidirect
4. #83007C2semidirect
5. #71885C2semidirect
6. #71884C2semidirect
7. #71846C2semidirect
8. #71845C2semidirect
9. #71814C2semidirect
10. #71756C2semidirect
11. #71720C2semidirect
28. C2×#875C22semidirect
35. #5050C22semidirect
36. #5046C22semidirect
38. #5040C22semidirect
40. #5036C22semidirect
45. #4960C22semidirect
46. #4947C22semidirect
47. #4936C22semidirect
48. #4926C22semidirect
49. #4925C22semidirect
50. #4538C22semidirect
52. #4537C22semidirect
53. #4537C22semidirect
54. #4525C22semidirect
55. #4525C22semidirect
56. #3558D3semidirect
76. C3×C4⋊Q8D4semidirect
78. C3×#127D4semidirect
81. C3×#125D4semidirect
91. C3×#124D4semidirect
116. C3×C4×Q8D4semidirect
122. C3×C4×D4D4semidirect
123. C3×C4×D4D4semidirect
313. C4⋊Q8#67semidirect
315. #127#67semidirect
318. #125#67semidirect
332. #124#67semidirect
357. C4×Q8#67semidirect
363. C4×D4#67semidirect
364. C4×D4#67semidirect
705. C3#18458semidirect

Maximal subgroups

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

subgroupindex
C3×#35582
#833062
#831442
#830072
#718852
#718842
#718462
#718452
#718142
#717562
#717202
#184583