Finite Group Databasebeta

← #10800836

Subgroups — #10800836

Order 768.

1868 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic3C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic
3C2×#4757C212yesmaximal, complemented
4#73866C211yesmaximal, characteristic, complemented
5C3×#1786C211yesmaximal, characteristic
6#70171C212yesmaximal, complemented
7#70167C212yesmaximal, complemented
8#70152C212yesmaximal, complemented
9#69930C211yesmaximal, characteristic, complemented
10#69906C211yesmaximal, characteristic, complemented
11#69504C211yesmaximal, characteristic, complemented
12#21453—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×Dic3—21no—
15C22×#217—21no—
16C2×D4×Dic3—28no—
17C2×D4×Dic3—21no—
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2212yes—
20C2×D4×Dic3C2212yes—
21C22×#214C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C2×#924C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C2×#887C2211yescharacteristic, complemented
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30#5218C418yescomplemented
31C2×#832C2212yes—
32C2×#830C2211yescharacteristic
33C2×#816C2212yescomplemented
34C2×#815C2212yescomplemented
35Dic3×C4⋊C4C2211yescharacteristic, complemented
36#4973C2211yescharacteristic, complemented
37#4972C2211yescharacteristic, complemented
38C2×#807C2212yescomplemented
39#4946—21no—
40Dic3×C22⋊C4—21no—
41#4757—28no—
42#4652—21no—
43#4649—21no—
44#4551C2212yescomplemented
45#4550C2212yescomplemented
46#4529C2212yescomplemented
47C22×C4×D4—31no—
48C2×#450—31no—
49C2×#440—32no—
50C4×C22⋊Q8—31no—
51#1877—32no—
52#1872—31no—
53#1862—32no—
54#1852—31no—
55#1786D311yescharacteristic
56#1762—32no—
57#1749—31no—
58C25×C3—21no—
59C22×C3×D4—24no—
60C22×C3×D4—24no—
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4C2×C414yes—
64C22×C3×D4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—28no—
67C23×Dic3—21no—
68C23×Dic3—22no—
69C23×Dic3—22no—
70C23×Dic3—44no—
71C23×Dic3—44no—
72C23×Dic3D411yescharacteristic
73C23×Dic3D411yescharacteristic
74C22×Dic6C2311yescharacteristic
75C22×C3×C8C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×#217—28no—
80C2×#217—21no—
81C2×#217—22no—
82C2×#217—22no—
83D4×Dic3—28no—
84D4×Dic3—28no—
85D4×Dic3—24no—
86D4×Dic3—24no—
87D4×Dic3—24no—
88D4×Dic3D418yes—
89C2×#214—24no—
90C2×#214—24no—
91C2×#214C2×C418yescomplemented
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
96#924—22no—
97#924C2×C414yescomplemented
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—24no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3D411yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2312yes—
109#887—24no—
110C3×#102—24no—
111C3×#102—24no—
112C3×C22⋊C8—22no—
113C3×#95—21no—
114#832C2×C418yescomplemented
115C3⋊#95—21no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119#830C2×C414yescomplemented
120#816—24no—
121#815—24no—
122#807—24no—
123C23×D4C2×D311yescharacteristic
124C24×C4—61no—
125C22×SD16—32no—
126C2×C22⋊Q8—31no—
127C2×C4×Q8—31no—
128C2×C4×D4—32no—
129C2×C4×D4—32no—
130C2×C4×D4—61no—
131C2×C4×D4—61no—
132C2×C4×D4—68no—
133C22×C4⋊C4—31no—
134C22×C22⋊C4—61no—
135C22×C42—31no—
136#450—38no—
137#440—68no—
138C2×#107—32no—
139C2×#103—32no—
140C2×#102—32no—
141C2×#102C2×D312yes—
142C2×#102C2×D312yes—
143C2×C22⋊C8—31no—
144C2×C22⋊C8C2×D311yescharacteristic
145C2×#97—32no—
146#391—31no—
147#389—31no—
148#387—61no—
149#385—61no—
150#380C2×D311yescharacteristic
151C4×C4⋊C4—31no—
152C4×C4⋊C4—61no—
153C4×C22⋊C4—61no—
154#340—32no—
155#340—32no—
156#336—32no—
157C24×C3—24no—
158C24×C3—24no—
159C24×C3—24no—
160C24×C3—22no—
161C24×C3—42no—
162C24×C3—42no—
163C24×C3—42no—
164C24×C3—42no—
165C24×C3C2×D411yescharacteristic
166C2×C3×D4—24no—
167C2×C3×D4—24no—
168C2×C3×D4—216no—
169C2×C3×D4—216no—
170C2×C3×D4—24no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4C22×C414yes—
175C2×C3×D4C2×D414yes—
176C2×C3×D4#4114yes—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4C22×C412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4#4112yes—
184C22×C3×C4C2411yescharacteristic
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—24no—
189C22×Dic3—28no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—28no—
196C22×Dic3—41no—
197C22×Dic3—432no—
198C22×Dic3—44no—
199C22×Dic3—44no—
200C22×Dic3—44no—
201C22×Dic3—42no—
202C22×Dic3—42no—
203C22×Dic3—44no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×Dic6—22no—
209C2×Dic6—22no—
210C2×Dic6—21no—
211C2×Dic6C22×C412yescomplemented
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C22⋊C4—42no—
218#217—28no—
219#217—24no—
220#217—28no—
221#214—216no—
222C4⋊Dic3—24no—
223C4⋊Dic3—22no—
224C4⋊Dic3—21no—
225C4⋊Dic3C22×C412yescomplemented
226C4⋊Dic3C2×D414yes—
227Dic3⋊C4—24no—
228Dic3⋊C4—42no—
229Dic3⋊C4—42no—
230C4×Dic3—22no—
231C4×Dic3—22no—
232C4×Dic3—22no—
233C4×Dic3—42no—
234C4×Dic3—42no—
235C4×Dic3C2×D414yes—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8C22×C414yescomplemented
239C25—21no—
240C22×Q8—31no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4C4×D314yes—
246C22×D4C22×D312yes—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—62no—
250C23×C4—62no—
251C23×C4—68no—
252C23×C4—61no—
253C23×C4—124no—
254C23×C4—124no—
255C23×C4C22×D311yescharacteristic
256C2×SD16—38no—
257C2×SD16—64no—
258C2×SD16—64no—
259C22×C8—32no—
260C22×C8C22×D312yes—
261C22⋊Q8—34no—
262C22⋊Q8—62no—
263C4×Q8—64no—
264C4×D4—38no—
265C4×D4—64no—
266C4×D4—64no—
267C4×D4—68no—
268C4×D4—68no—
269C4×D4—64no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—68no—
283C2×C22⋊C4—61no—
284C2×C42—31no—
285C2×C42—31no—
286C2×C42—31no—
287C2×C42—31no—
288C2×C42—64no—
289C2×C42—61no—
290C2×C42—61no—
291#107—64no—
292#103—64no—
293#102—24no—
294#102—24no—
295#102—38no—
296C22⋊C8—22no—
297C22⋊C8—34no—
298#97—64no—
299#95—21no—
300#95—61no—
301#95—61no—
302#95—61no—
303#95—61no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—216no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—48no—
313C23×C3—48no—
314C23×C3—48no—
315C23×C3—48no—
316C23×C3—48no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—48no—
322C23×C3C4×D412yes—
323C23×C3AGL(1,ℤ/8)12yes—
324C23×C3#13712yes—
325C23×C3C22×D411yescharacteristic
326C3×D4—216no—
327C3×D4—24no—
328C3×D4—24no—
329C3×D4—24no—
330C3×D4C4×D418yes—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4C4×D412yes—
342C2×C3×C4C23×C411yescharacteristic
343C2×C3×C4C22×D412yes—
344C2×C3×C4C2×#4112yes—
345C2×C3×C4C2×#4111yescharacteristic
346C2×Dic3—28no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—21no—
356C2×Dic3—24no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—44no—
360C2×Dic3—416no—
361C2×Dic3—416no—
362C2×Dic3—48no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—41no—
369C2×Dic3#12012yes—
370C2×Dic3#12012yes—
371C2×Dic3AGL(1,ℤ/8)12yes—
372Dic6—24no—
373Dic6—42no—
374C3×C8—42no—
375C3⋊C8—24no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24D3×D411yescharacteristic
385C2×Q8—32no—
386C2×Q8—61no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—216no—
392C2×D4—216no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4C2×C4×D314yes—
398C2×D4D3×D414yes—
399C2×D4#23714yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—68no—
412C22×C4—62no—
413C22×C4—64no—
414C22×C4—68no—
415C22×C4—62no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4—124no—
420C22×C4—121no—
421C22×C4—1232no—
422C22×C4—124no—
423C22×C4—124no—
424C22×C4—124no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4C2×C4×D312yes—
428C22×C4D3×D411yescharacteristic
429C22×C4#23712yes—
430C22×C4C23×D311yescharacteristic
431SD16—616no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—34no—
436C2×C8—62no—
437C2×C8—62no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—32no—
441C4⋊C4—34no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4—62no—
445C4⋊C4—61no—
446C4⋊C4—122no—
447C4⋊C4—122no—
448C22⋊C4—42no—
449C22⋊C4—68no—
450C22⋊C4—68no—
451C22⋊C4—64no—
452C42—34no—
453C42—62no—
454C42—62no—
455C42—62no—
456C42—122no—
457C42—122no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—28no—
465C22×C3—432no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3#44014yescomplemented
473C22×C3C2×C4×D411yescharacteristic
474C22×C3C2×#12011yescharacteristic
475C22×C3#52511yescharacteristic
476C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
477C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
478C22×C3C2×#13711yescharacteristic
479C22×C3#57511yescharacteristic
480C3×C4—22no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4C2×C4×D412yes—
484C3×C4#52512yes—
485Dic3—22no—
486Dic3—22no—
487Dic3—22no—
488Dic3—42no—
489Dic3—44no—
490Dic3—41no—
491Dic3—42no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—216no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—48no—
510C23#67⋊C412yes—
511C23#91112yes—
512C23#91512yes—
513C23C2×D3×D411yescharacteristic
514Q8—64no—
515Q8—122no—
516D4—216no—
517D4—24no—
518D4—24no—
519D4—24no—
520D4#67⋊C418yes—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
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—68no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—64no—
547C2×C4—124no—
548C2×C4—1216no—
549C2×C4—1216no—
550C2×C4—128no—
551C2×C4—122no—
552C2×C4—121no—
553C2×C4—122no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4#67⋊C412yes—
558C2×C4C22×C4×D311yescharacteristic
559C2×C4C2×D3×D412yes—
560C2×C4C2×#23712yes—
561C2×C4C2×#23711yescharacteristic
562C8—42no—
563C8—64no—
564C2×C3—24no—
565C2×C3—48no—
566C2×C3C4×#12011yescharacteristic
567C2×C3C2×#44011yescharacteristic
568C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
569C2×C3#300311yescharacteristic
570C2×C3#300711yescharacteristic
571C2×C3#311011yescharacteristic
572C2×C3#311411yescharacteristic
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—28no—
580C22—432no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22#475714yescomplemented
588C22C2×#67⋊C411yescharacteristic
589C22#558611yescharacteristic
590C22D3×#12011yescharacteristic
591C22C2×#91111yescharacteristic
592C22C2×#91511yescharacteristic
593C22D3×AGL(1,ℤ/8)11yescharacteristic
594C22#577311yescharacteristic
595C4—22no—
596C4—62no—
597C4—62no—
598C4—62no—
599C4—81no—
600C4—81no—
601C4—121no—
602C4—122no—
603C4—122no—
604C4—124no—
605C4C2×#67⋊C412yes—
606C4#558612yes—
607C3#2145311yescharacteristic, complemented, Sylow
608C2—24no—
609C2—48no—
610C2#7381811yescharacteristic
611C2C2×#475711yescharacteristic
612C2#7891411yescharacteristic
613C2#7999111yescharacteristic
614C2#8000211yescharacteristic
615C2#8013411yescharacteristic
616C2#8014711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1786C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
cosocleC22×C3×C4C24
hypercenter#1786D3
hypocenterC3#21453
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
3. C2×#4757C2semidirect
4. #73866C2semidirect
6. #70171C2semidirect
7. #70167C2semidirect
8. #70152C2semidirect
9. #69930C2semidirect
10. #69906C2semidirect
11. #69504C2semidirect
25. C2×#887C22semidirect
30. #5218C4semidirect
33. C2×#816C22semidirect
34. C2×#815C22semidirect
35. Dic3×C4⋊C4C22semidirect
36. #4973C22semidirect
37. #4972C22semidirect
38. C2×#807C22semidirect
44. #4551C22semidirect
45. #4550C22semidirect
46. #4529C22semidirect
91. C2×#214C2×C4semidirect
97. #924C2×C4semidirect
114. #832C2×C4semidirect
119. #830C2×C4semidirect
211. C2×Dic6C22×C4semidirect
225. C4⋊Dic3C22×C4semidirect
238. C2×C3⋊C8C22×C4semidirect
472. C22×C3#440semidirect
587. C22#4757semidirect
607. C3#21453semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic32
C2×#52182
C2×#47572
#738662
C3×#17862
#701712
#701672
#701522
#699302
#699062
#695042
#214533