Finite Group Databasebeta

← #10877416

Subgroups — #10877416

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83233C211yesmaximal, characteristic
2#76519C211yesmaximal, characteristic
3C3×#2024C211yesmaximal, characteristic, complemented
4#71787C211yesmaximal, characteristic, complemented
5#70270C211yesmaximal, characteristic
6#70268C211yesmaximal, characteristic
7#70253C211yesmaximal, characteristic, complemented
8#70233C211yesmaximal, characteristic, complemented
9#69674C211yesmaximal, characteristic, complemented
10#68872C211yesmaximal, characteristic
11D3×#340C211yesmaximal, characteristic
12#68847C211yesmaximal, characteristic, complemented
13#68843C211yesmaximal, characteristic, complemented
14#68647C211yesmaximal, characteristic
15#68643C211yesmaximal, characteristic, complemented
16#27697—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#946—21no—
19C2×#946C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#391C2211yescharacteristic
26#5230—21no—
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4989—21no—
33#4987C2211yescharacteristic
34#4984C2211yescharacteristic
35#4982—21no—
36#4975C2211yescharacteristic
37#4975C2211yescharacteristic
38C4×(C2×D3)⋊C4C2211yescharacteristic
39#4935C2211yescharacteristic
40#4928C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4668—21no—
43#4668—21no—
44#4657—21no—
45#4657—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#336C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2502—31no—
61#2440—31no—
62#2100—31no—
63#2100—31no—
64#2072—31no—
65#2024D311yescharacteristic, complemented
66#1990—31no—
67#1933—31no—
68#1933—31no—
69#1892—31no—
70#1872—31no—
71#1834—31no—
72#1824—31no—
73C8×C22⋊C4—31no—
74C2×#340—31no—
75C22×C3×Q8—21no—
76C22×C3×Q8C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#946—22no—
90#946—22no—
91#946—44no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C2×D3×C8—22no—
115C2×D3×C8—22no—
116C2×D3×C8—41no—
117D3×C4⋊C4—22no—
118D3×C4⋊C4—22no—
119D3×C4⋊C4—22no—
120D3×C4⋊C4—41no—
121C3×#103—22no—
122C3×#103—41no—
123C3×#95—21no—
124C3×#95—21no—
125#835—22no—
126#835—41no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134(C2×D3)⋊C8—22no—
135C8×Dic3—22no—
136C4×C3⋊C8—41no—
137C23×C8—31no—
138C2×C4⋊Q8C2×D311yescharacteristic
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—61no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#103C2×D311yescharacteristic
146C2×#103C2×D311yescharacteristic
147C2×C22⋊C8—31no—
148C2×C4×C8—31no—
149C2×C4×C8—31no—
150#398—61no—
151#398—61no—
152#398—61no—
153#395—61no—
154#391—31no—
155#391C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#385—61no—
159#385—61no—
160#385—61no—
161#384—31no—
162#384—31no—
163#384—31no—
164#380—31no—
165C4×C22⋊C4—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#340C2×D311yescharacteristic
178#336—31no—
179#336C2×D311yescharacteristic
180C23×D3C2×D411yescharacteristic
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—42no—
188C2×C3×Q8—42no—
189C2×C3×Q8—81no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—81no—
212C2×C4×D3—81no—
213C2×C4×D3—81no—
214C2×C4×D3#4112yes—
215C2×C4×D3#4112yes—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4#4112yes—
229C3×C42—21no—
230C3×C42—21no—
231(C2×D3)⋊C4—42no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4SD1614yes—
234(C2×D3)⋊C4Q1614yes—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3—81no—
245C4⋊Dic3—81no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—81no—
252Dic3⋊C4—81no—
253Dic3⋊C4—81no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8#4112yes—
262D3×C8—44no—
263C22×Q8—21no—
264C22×Q8C22×D311yescharacteristic
265C23×C4—31no—
266C23×C4—61no—
267C23×C4—61no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8—62no—
271C22×C8—62no—
272C22×C8—121no—
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C4⋊Q8—22no—
276C4⋊Q8—41no—
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8—124no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—121no—
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C42—31no—
308C2×C42C22×D311yescharacteristic
309#103—22no—
310#103—41no—
311#103—62no—
312#103—121no—
313C22⋊C8—62no—
314C4×C8—62no—
315C4×C8—121no—
316#95—21no—
317#95—21no—
318#95—31no—
319#95—31no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3C22×D411yescharacteristic
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3#12412yes—
331C22×D3C2×SD1612yes—
332C22×D3C2×Q1612yes—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×Q8—84no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4C2×SD1612yes—
355C2×C3×C4C2×Q1612yes—
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C42−1+411yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3#12412yes—
381C2×Dic3C2×SD1612yes—
382C2×Dic3C2×Q1612yes—
383C4×D3—22no—
384C4×D3—22no—
385C4×D3—22no—
386C4×D3—22no—
387C4×D3—84no—
388C4×D3—84no—
389C3×C8—41no—
390C3×C8—41no—
391C3⋊C8—22no—
392C3⋊C8—41no—
393C24—31no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—81no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—241no—
424C22×C4—241no—
425C22×C4—241no—
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4C23×D311yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—32no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—124no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—42no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—241no—
463C4⋊C4—241no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C4⋊C4—241no—
467C4⋊C4#23712yes—
468C22⋊C4—34no—
469C22⋊C4—34no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C42—21no—
473C42—21no—
474C42—61no—
475C42—61no—
476C22×C3C2×#12411yescharacteristic
477C22×C3#54611yescharacteristic
478C22×C3#54711yescharacteristic
479C22×C3C22×SD1611yescharacteristic
480C22×C3C22×Q1611yescharacteristic
481C22×C3#57611yescharacteristic
482C22×C3#57711yescharacteristic
483C2×D3—22no—
484C2×D3—22no—
485C2×D3—22no—
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—22no—
489C2×D3#48714yes—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—42no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4#52911yescharacteristic
497C3×C4#53911yescharacteristic
498C3×C4#55111yescharacteristic
499C3×C4#55211yescharacteristic
500Dic3—22no—
501Dic3—22no—
502Dic3—22no—
503Dic3—81no—
504Dic3—81no—
505Dic3—81no—
506Dic3—81no—
507C23—32no—
508C23—32no—
509C23—32no—
510C23—61no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23C2×D3×D411yescharacteristic
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—84no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—32no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—244no—
562C2×C4—244no—
563C2×C4D3×SD1612yes—
564C2×C4D3×Q1612yes—
565C2×C4C2×#23711yescharacteristic
566C2×C4#100411yescharacteristic
567C2×C4#100711yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C8—41no—
572C8—41no—
573C8—62no—
574C8—121no—
575C2×C3#248811yescharacteristic
576C2×C3C2×#48711yescharacteristic
577C2×C3#313611yescharacteristic
578C2×C3#329511yescharacteristic
579C2×C3#329911yescharacteristic
580C2×C3#333511yescharacteristic
581C2×C3#333811yescharacteristic
582D3—24no—
583C22—34no—
584C22—62no—
585C22—62no—
586C22—62no—
587C22—62no—
588C22—62no—
589C22—62no—
590C22#559111yescharacteristic
591C22#563111yescharacteristic
592C22D3×#12411yescharacteristic
593C22C2×D3×SD1611yescharacteristic
594C22C2×D3×Q1611yescharacteristic
595C22#577411yescharacteristic
596C22#577811yescharacteristic
597C4—22no—
598C4—22no—
599C4—42no—
600C4—62no—
601C4—62no—
602C4—62no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—241no—
607C4—241no—
608C4—241no—
609C4—241no—
610C4#563811yescharacteristic
611C4#563911yescharacteristic
612C4#567911yescharacteristic
613C4#568011yescharacteristic
614C3#2769711yescharacteristic, complemented, Sylow
615C2—64no—
616C2#7686711yescharacteristic
617C2#8015811yescharacteristic
618C2#8019311yescharacteristic
619C2#8102811yescharacteristic
620C2#8112511yescharacteristic
621C2D3×#48711yescharacteristic
622C2#8168011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2024C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2024D3
hypocenterC3#27697
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
3. C3×#2024C2semidirect
4. #71787C2semidirect
7. #70253C2semidirect
8. #70233C2semidirect
9. #69674C2semidirect
12. #68847C2semidirect
13. #68843C2semidirect
15. #68643C2semidirect
65. #2024D3semidirect
614. C3#27697semidirect

Maximal subgroups

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

subgroupindex
#832332
#765192
C3×#20242
#717872
#702702
#702682
#702532
#702332
#696742
#688722
D3×#3402
#688472
#688432
#686472
#686432
#276973