Finite Group Databasebeta

← #10770573

Subgroups — #10770573

Order 768.

936 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3538C211yesmaximal, characteristic, complemented
2#82931C211yesmaximal, characteristic
3#82833C211yesmaximal, characteristic, complemented
4#82736C211yesmaximal, characteristic, complemented
5#71902C212yesmaximal, complemented
6#71898C212yesmaximal, complemented
7#71888C212yesmaximal, complemented
8#71886C212yesmaximal, complemented
9#71710C211yesmaximal, characteristic, complemented
10#71691C211yesmaximal, characteristic, complemented
11#71689C211yesmaximal, characteristic, complemented
12#18454—31nomaximal, Sylow
13C3×#563—22no—
14C3×#552—21no—
15C3×#552—21no—
16C3×#551—22no—
17C3×D4×Q8—21no—
18C3×D4×Q8—21no—
19C3×#544—21no—
20C3×#544C2212yes—
21C3×#541—21no—
22C3×#539—21no—
23C3×#539C2212yes—
24C3×#536—21no—
25C3×#525C2211yescharacteristic
26C2×C3×C22⋊Q8C2211yescharacteristic
27C2×C3×C22⋊Q8C2211yescharacteristic
28C2×#930C2211yescharacteristic
29#5238—21no—
30#5231—21no—
31#5227—21no—
32#5211C2211yescharacteristic
33#5207C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5049—21no—
36#5048—22no—
37#5046C2212yes—
38#5046C2211yescharacteristic
39#5043C2212yes—
40#5043C2211yescharacteristic
41#5039—21no—
42#5036C2211yescharacteristic
43#5033C2211yescharacteristic
44#5003—21no—
45#4993—21no—
46#4991C2212yes—
47#4981C2212yes—
48#4968—21no—
49#4959—22no—
50#4538C2212yes—
51#4538C2212yes—
52#4538C2212yes—
53#4537C2212yes—
54#4528C2212yes—
55#4528C2212yes—
56#3538D311yescharacteristic, complemented
57#3194—31no—
58#3187—31no—
59#2816—31no—
60#1611—32no—
61#1609—32no—
62#1599—32no—
63#1597—32no—
64#1516—31no—
65#1511—31no—
66#1472—31no—
67C2×C3×#41D411yescharacteristic
68C22×C3×Q8—21no—
69C22×C3×Q8D411yescharacteristic
70C22×C3×Q8D411yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C22×#67—21no—
73C22×D12C2311yescharacteristic
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—22no—
78C3×#126—22no—
79C3×#126—22no—
80C3×#126—22no—
81C3×#126—22no—
82C3×#125—22no—
83C3×#125—21no—
84C3×#125—22no—
85C3×#125—22no—
86C3×#125—22no—
87C3×#124—22no—
88C3×#123—22no—
89C3×#123—22no—
90C3×#123—22no—
91C3×#123—21no—
92C3×#123—22no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8—22no—
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—21no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8—21no—
101C3×C22⋊Q8—21no—
102C3×C22⋊Q8—21no—
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8—22no—
105C3×C22⋊Q8D412yes—
106C3×C22⋊Q8D411yescharacteristic
107C3×C22⋊Q8C2312yes—
108C3×C22⋊Q8C2311yescharacteristic
109C3×#121D411yescharacteristic
110C3×#121C2311yescharacteristic
111C3×#120—21no—
112C3×C4×Q8—21no—
113C3×C4×Q8—21no—
114C3×C4×D4—22no—
115C3×C4×D4—21no—
116C3×C4×D4—21no—
117C3×C4×D4—22no—
118C3×#117—21no—
119C3×#117—22no—
120C3×#117—22no—
121C3×#117D411yescharacteristic
122C2×C3×C4⋊C4—21no—
123C2×C3×C4⋊C4D411yescharacteristic
124C2×C3×C4⋊C4D411yescharacteristic
125C2×C3×C4⋊C4C2312yes—
126C2×C3×C22⋊C4—21no—
127C2×C3×C22⋊C4D412yes—
128C2×#217—21no—
129C2×#215—22no—
130C2×#215—21no—
131C2×#213—21no—
132#930—21no—
133#930—21no—
134#930—21no—
135#930—21no—
136C2×(C2×D3)⋊C4—21no—
137C2×(C2×D3)⋊C4—22no—
138C2×(C2×D3)⋊C4—22no—
139C2×C4⋊Dic3C2311yescharacteristic
140C2×Dic3⋊C4—22no—
141C2×#208—21no—
142C22×C3⋊C8—21no—
143#835—22no—
144#835—21no—
145#832—21no—
146C3⋊#95—21no—
147C3⋊#95—22no—
148C3⋊#95C2312yes—
149#830—21no—
150#830—21no—
151#830C2312yes—
152#830C2311yescharacteristic
153#830C2311yescharacteristic
154#809—22no—
155#809—21no—
156#809—21no—
157#808—22no—
158#808—21no—
159#807—21no—
160#563—22no—
161#552—21no—
162#552—21no—
163#551—22no—
164D4×Q8—21no—
165D4×Q8—21no—
166#544—21no—
167#544C2×D312yes—
168#541—21no—
169#539—21no—
170#539C2×D312yes—
171#536—21no—
172#525C2×D311yescharacteristic
173C2×C22⋊Q8C2×D311yescharacteristic
174C2×C22⋊Q8C2×D311yescharacteristic
175C2×#121—31no—
176#483—32no—
177#483—31no—
178#482—61no—
179#481—61no—
180#480—31no—
181#469—61no—
182#468—61no—
183#465—62no—
184#450—32no—
185#450—31no—
186#449—61no—
187#448—61no—
188#447—31no—
189#407—31no—
190C2×C22⋊C8—31no—
191#399—61no—
192#398—32no—
193#394—32no—
194#394—62no—
195#393—61no—
196#381—31no—
197#328—32no—
198#328—32no—
199#328—32no—
200#327—32no—
201#324—32no—
202#324—32no—
203C24×C3C2×D411yescharacteristic
204C23×D3—21no—
205C3×#41—81no—
206C2×C3×Q8—22no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—81no—
211C2×C3×Q8—82no—
212C2×C3×Q8—81no—
213C2×C3×Q8C2×D412yes—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4C2×D411yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4—22no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—81no—
224C22×C3×C4C2×D412yes—
225C22×C3×C4C2×D412yes—
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C2×#67—41no—
233C2×#67—41no—
234C2×#67—82no—
235C22×Dic3—22no—
236C22×Dic3—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—81no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
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—22no—
254C3×C4⋊C4—22no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D412yes—
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—22no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—21no—
272C3×C42—22no—
273C3×C42—22no—
274C3×C42—22no—
275C3×C42—21no—
276#217—81no—
277#215—82no—
278#215—81no—
279#213—81no—
280(C2×D3)⋊C4—41no—
281(C2×D3)⋊C4—41no—
282(C2×D3)⋊C4—82no—
283(C2×D3)⋊C4—82no—
284C4⋊Dic3—21no—
285C4⋊Dic3—21no—
286Dic3⋊C4—82no—
287#208—81no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—22no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—81no—
292C2×#41#6711yescharacteristic
293C22×Q8—21no—
294C22×Q8#6711yescharacteristic
295C22×Q8#6711yescharacteristic
296C22×D4—31no—
297C22×D4—61no—
298C23×C4C22×D311yescharacteristic
299C2×SD16—62no—
300C2×SD16—61no—
301C2×D8—61no—
302C2×M16—61no—
303C22×C8—61no—
304C4⋊Q8—21no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308#126—22no—
309#126—22no—
310#126—22no—
311#126—22no—
312#125—21no—
313#125—22no—
314#125—22no—
315#125—22no—
316#125—22no—
317#124—22no—
318#123—22no—
319#123—22no—
320#123—22no—
321#123—21no—
322#123—22no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—22no—
335C22⋊Q8#6712yes—
336C22⋊Q8#6711yescharacteristic
337C22⋊Q8C22×D312yes—
338C22⋊Q8C22×D311yescharacteristic
339#121—61no—
340#121—61no—
341#121—61no—
342#121—61no—
343#121#6711yescharacteristic
344#121C22×D311yescharacteristic
345#120—21no—
346C4×Q8—21no—
347C4×Q8—21no—
348C4×D4—22no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—22no—
352#117—21no—
353#117—22no—
354#117—22no—
355#117#6711yescharacteristic
356C2×C4⋊C4—21no—
357C2×C4⋊C4—31no—
358C2×C4⋊C4—62no—
359C2×C4⋊C4#6711yescharacteristic
360C2×C4⋊C4#6711yescharacteristic
361C2×C4⋊C4C22×D312yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—61no—
364C2×C22⋊C4—61no—
365C2×C22⋊C4—62no—
366C2×C22⋊C4—62no—
367C2×C22⋊C4#6712yes—
368#107—61no—
369#106—62no—
370#106—61no—
371#103—62no—
372#103—61no—
373#102—61no—
374#102—61no—
375#102—62no—
376#102—61no—
377C22⋊C8—32no—
378C22⋊C8—31no—
379C22⋊C8—31no—
380C22⋊C8—61no—
381C22⋊C8—61no—
382#95—32no—
383#95—61no—
384#95—62no—
385C23×C3—22no—
386C23×C3—21no—
387C23×C3—41no—
388C23×C3—41no—
389C23×C3#12011yescharacteristic
390C23×C3#12011yescharacteristic
391C23×C3C22×D411yescharacteristic
392C22×D3—42no—
393C22×D3—42no—
394C22×D3—41no—
395C22×D3—42no—
396C3×Q8—42no—
397C3×Q8—41no—
398C3×Q8—42no—
399C3×Q8—42no—
400C3×Q8—81no—
401C3×Q8—82no—
402C3×D4—41no—
403C3×D4—42no—
404C3×D4—42no—
405C3×D4—81no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—21no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—42no—
422C2×C3×C4—42no—
423C2×C3×C4—42no—
424C2×C3×C4—81no—
425C2×C3×C4—82no—
426C2×C3×C4#12011yescharacteristic
427C2×C3×C4#12011yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C22×D411yescharacteristic
430#67—82no—
431#67—82no—
432C2×Dic3—42no—
433C2×Dic3—42no—
434C2×Dic3—41no—
435C2×Dic3—42no—
436C2×Dic3—42no—
437D12—41no—
438D12—41no—
439D12—42no—
440C3⋊C8—42no—
441C3⋊C8—41no—
442C3⋊C8—41no—
443C24—61no—
444C24C2×#6711yescharacteristic
445#41—81no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—81no—
451C2×Q8—82no—
452C2×Q8—81no—
453C2×Q8C2×#6712yes—
454C2×Q8C2×#6711yescharacteristic
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—121no—
461C2×D4—121no—
462C2×D4—241no—
463C2×D4—242no—
464C2×D4C2×#6711yescharacteristic
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—62no—
470C22×C4—61no—
471C22×C4—81no—
472C22×C4C2×#6712yes—
473C22×C4C2×#6712yes—
474C22×C4C2×#6711yescharacteristic
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C23×D311yescharacteristic
480SD16—242no—
481SD16—241no—
482D8—241no—
483M16—241no—
484C2×C8—61no—
485C2×C8—62no—
486C2×C8—61no—
487C2×C8—241no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—242no—
508C4⋊C4C2×#6712yes—
509C4⋊C4C2×#6711yescharacteristic
510C4⋊C4C2×#6711yescharacteristic
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—121no—
524C22⋊C4—121no—
525C22⋊C4—242no—
526C22⋊C4—242no—
527C22⋊C4—241no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—21no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—82no—
544C22×C3C2×#12011yescharacteristic
545C22×C3#57712yes—
546C2×D3—84no—
547C2×D3—81no—
548C2×D3—81no—
549C2×D3—81no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—42no—
555C3×C4—41no—
556C3×C4—42no—
557C3×C4—41no—
558C3×C4—41no—
559C3×C4—42no—
560C3×C4—42no—
561Dic3—82no—
562Dic3—81no—
563C23—22no—
564C23—21no—
565C23—41no—
566C23—41no—
567C23—122no—
568C23—122no—
569C23—122no—
570C23—121no—
571C23#95311yescharacteristic
572C23#95311yescharacteristic
573C23C22×#6711yescharacteristic
574Q8—42no—
575Q8—41no—
576Q8—42no—
577Q8—42no—
578Q8—81no—
579Q8—82no—
580D4—41no—
581D4—42no—
582D4—42no—
583D4—81no—
584D4—121no—
585D4—121no—
586D4—122no—
587D4—242no—
588D4—242no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—81no—
608C2×C4—82no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—121no—
614C2×C4#95311yescharacteristic
615C2×C4#95311yescharacteristic
616C2×C4C22×#6711yescharacteristic
617C2×C4C22×#6711yescharacteristic
618C8—122no—
619C8—121no—
620C8—121no—
621C2×C3—22no—
622C2×C3—41no—
623C2×C3—41no—
624C2×C3—81no—
625C2×C3#301011yescharacteristic
626C2×C3#302011yescharacteristic
627C2×C3#302611yescharacteristic
628D3—161no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—82no—
641C22—244no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22#583712yes—
646C22C2×#95311yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—242no—
659C4—241no—
660C3#1845411yescharacteristic, complemented, Sylow
661C2—22no—
662C2—41no—
663C2—41no—
664C2—81no—
665C2—481no—
666C2#8411011yescharacteristic
667C2#8411311yescharacteristic
668C2#8411611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3538C2
socleC22×C3C2×#120
prime radical#82931C2
hypercenter#3538D3
hypocenterC3#18454
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3538

Complements

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

NH ≅ G/Ntype
1. C3×#3538C2semidirect
3. #82833C2semidirect
4. #82736C2semidirect
5. #71902C2semidirect
6. #71898C2semidirect
7. #71888C2semidirect
8. #71886C2semidirect
9. #71710C2semidirect
10. #71691C2semidirect
11. #71689C2semidirect
56. #3538D3semidirect
660. C3#18454semidirect

Maximal subgroups

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

subgroupindex
C3×#35382
#829312
#828332
#827362
#719022
#718982
#718882
#718862
#717102
#716912
#716892
#184543