Finite Group Databasebeta

← #10800831

Subgroups — #10800831

Order 768.

1952 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#447C211yesmaximal, characteristic
3C2×#5152C212yesmaximal, complemented
4Dic3×#121C211yesmaximal, characteristic, complemented
5#72128C211yesmaximal, characteristic, complemented
6#72037C212yesmaximal, complemented
7#71677C211yesmaximal, characteristic, complemented
8#70170C212yesmaximal, complemented
9#70152C212yesmaximal, complemented
10#69928C211yesmaximal, characteristic, complemented
11#69906C211yesmaximal, characteristic, complemented
12#21452—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×Dic3—21no—
15C22×C3×D8C2212yes—
16C2×C3×#121C2211yescharacteristic
17C22×#217—21no—
18C2×D4×Dic3—28no—
19C2×D4×Dic3—21no—
20C2×D4×Dic3—21no—
21C2×D4×Dic3—21no—
22C2×D4×Dic3C2212yes—
23C2×D4×Dic3C2212yes—
24C2×D4×Dic3C2211yescharacteristic, complemented
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C3×#447C418yescomplemented
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30#5221C2211yescharacteristic, complemented
31#5220—21no—
32C2×#832C2212yes—
33C2×#832C2212yes—
34C2×#832C2212yescomplemented
35#5212C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5152—28no—
38C2×#817C2212yescomplemented
39C2×#815C2212yescomplemented
40#4984C2211yescharacteristic, complemented
41Dic3×C4⋊C4C2211yescharacteristic, complemented
42#4951—21no—
43Dic3×C22⋊C4—21no—
44#4551C2212yescomplemented
45#4550C2212yescomplemented
46#4529C2212yescomplemented
47C22×C4×D4—31no—
48C2×#447D311yescharacteristic
49C2×#442—32no—
50C4×#121—31no—
51#1879—32no—
52#1874—31no—
53#1864—32no—
54#1853—31no—
55#1786—31no—
56#1762—32no—
57#1749—31no—
58C25×C3—21no—
59C22×C3×D4—24no—
60C22×C3×D4—24no—
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4—22no—
64C22×C3×D4C2×C414yescomplemented
65C22×C3×D4C2311yescharacteristic
66C22×C3×D4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3—28no—
69C23×Dic3—21no—
70C23×Dic3—22no—
71C23×Dic3—22no—
72C23×Dic3—21no—
73C23×Dic3—44no—
74C23×Dic3—44no—
75C23×Dic3D411yescharacteristic
76C23×Dic3D411yescharacteristic
77C2×C3×D8—24no—
78C2×C3×D8—24no—
79C2×C3×D8C2×C418yescomplemented
80C22×C3×C8C2312yes—
81C3×#121—22no—
82C3×#121C2×C414yescomplemented
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×#217—28no—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92D4×Dic3—28no—
93D4×Dic3—28no—
94D4×Dic3—24no—
95D4×Dic3—24no—
96D4×Dic3—24no—
97D4×Dic3—24no—
98D4×Dic3—44no—
99D4×Dic3D418yescomplemented
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—24no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2312yes—
106C2×C4×Dic3—24no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2312yes—
113C3×#102C2×C418yescomplemented
114C3×C22⋊C8C2×C414yescomplemented
115#832—24no—
116#832—24no—
117#832—24no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121#830—22no—
122#817—24no—
123#815—24no—
124C23×D4C2×D311yescharacteristic
125C24×C4—61no—
126C22×D8C2×D312yes—
127C2×#121C2×D311yescharacteristic
128C2×C4×D4—32no—
129C2×C4×D4—32no—
130C2×C4×D4—31no—
131C2×C4×D4—61no—
132C2×C4×D4—61no—
133C2×C4×D4—68no—
134C2×C4×D4—61no—
135C22×C4⋊C4—31no—
136C22×C22⋊C4—61no—
137C22×C42—31no—
138#447Dic318yescomplemented
139#442—68no—
140C2×#106—32no—
141C2×#102—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102C2×D312yes—
145C2×C22⋊C8—31no—
146C2×C22⋊C8C2×D311yescharacteristic
147C2×#97—32no—
148#390—31no—
149#389—31no—
150#388—61no—
151#386—61no—
152#380—31no—
153C4×C4⋊C4—31no—
154C4×C22⋊C4—61no—
155#340—32no—
156#340—32no—
157#336—32no—
158C24×C3—24no—
159C24×C3—24no—
160C24×C3—24no—
161C24×C3—22no—
162C24×C3—21no—
163C24×C3—42no—
164C24×C3—42no—
165C24×C3—42no—
166C24×C3—42no—
167C24×C3C2×D411yescharacteristic
168C2×C3×D4—24no—
169C2×C3×D4—24no—
170C2×C3×D4—216no—
171C2×C3×D4—216no—
172C2×C3×D4—24no—
173C2×C3×D4—24no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4C22×C414yes—
183C2×C3×D4C22×C412yescomplemented
184C2×C3×D4C2×D414yes—
185C2×C3×D4#4114yes—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4C22×C412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4#4112yes—
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—24no—
197C22×Dic3—28no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—28no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—41no—
205C22×Dic3—432no—
206C22×Dic3—44no—
207C22×Dic3—44no—
208C22×Dic3—44no—
209C22×Dic3—42no—
210C22×Dic3—42no—
211C22×Dic3—42no—
212C22×Dic3—42no—
213C22×Dic3—44no—
214C22×Dic3—42no—
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C3×D8—216no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8C22×C414yescomplemented
223C3×C4⋊C4—21no—
224C3×C4⋊C4C22×C412yescomplemented
225C3×C22⋊C4—24no—
226#217—28no—
227#217—28no—
228#217—42no—
229#217—42no—
230#217—42no—
231#217—42no—
232C4⋊Dic3—24no—
233C4⋊Dic3—22no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C4⋊Dic3—42no—
237C4⋊Dic3C2×D414yes—
238C4×Dic3—22no—
239C4×Dic3—22no—
240C4×Dic3—22no—
241C4×Dic3—42no—
242C4×Dic3C2×D414yes—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C25—21no—
247C22×D4—24no—
248C22×D4—24no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4C2×Dic314yescomplemented
253C22×D4C22×D311yescharacteristic
254C22×D4C22×D312yes—
255C23×C4—31no—
256C23×C4—31no—
257C23×C4—62no—
258C23×C4—62no—
259C23×C4—61no—
260C23×C4—68no—
261C23×C4—61no—
262C23×C4—124no—
263C23×C4—124no—
264C23×C4C22×D311yescharacteristic
265C2×D8—24no—
266C2×D8—24no—
267C2×D8C2×Dic318yescomplemented
268C22×C8—32no—
269C22×C8C22×D312yes—
270#121—22no—
271#121C2×Dic314yescomplemented
272C4×D4—38no—
273C4×D4—64no—
274C4×D4—64no—
275C4×D4—64no—
276C4×D4—68no—
277C4×D4—68no—
278C4×D4—64no—
279C4×D4—124no—
280C2×C4⋊C4—32no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—68no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—64no—
300C2×C42—61no—
301#106—64no—
302#102—64no—
303#102—64no—
304#102—64no—
305#102C2×Dic318yescomplemented
306C22⋊C8—62no—
307C22⋊C8C2×Dic314yescomplemented
308#97—64no—
309#95—61no—
310#95—61no—
311#95—61no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—216no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—22no—
321C23×C3—48no—
322C23×C3—48no—
323C23×C3—48no—
324C23×C3—48no—
325C23×C3—48no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—48no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C4×D412yes—
338C23×C3AGL(1,ℤ/8)12yes—
339C23×C3AGL(1,ℤ/8)12yes—
340C23×C3C22×D411yescharacteristic
341C3×D4—216no—
342C3×D4—24no—
343C3×D4—24no—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—48no—
347C3×D4—42no—
348C3×D4C4×D418yescomplemented
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4C4×D412yes—
357C2×C3×C4C23×C411yescharacteristic
358C2×C3×C4C22×D412yes—
359C2×C3×C4C2×#4112yes—
360C2×C3×C4C2×#4111yescharacteristic
361C2×Dic3—28no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—24no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—44no—
373C2×Dic3—416no—
374C2×Dic3—416no—
375C2×Dic3—48no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—84no—
385C2×Dic3#12012yes—
386C2×Dic3#12012yes—
387C2×Dic3AGL(1,ℤ/8)12yes—
388C3×C8—24no—
389C3⋊C8—42no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—22no—
394C24—21no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24D3×D411yescharacteristic
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4D3×D414yes—
415C2×D4#23714yes—
416C2×D4C22×Dic314yes—
417C2×D4C22×Dic312yescomplemented
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—68no—
429C22×C4—62no—
430C22×C4—64no—
431C22×C4—68no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—124no—
437C22×C4—122no—
438C22×C4—121no—
439C22×C4—1232no—
440C22×C4—124no—
441C22×C4—124no—
442C22×C4—124no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4D3×D411yescharacteristic
448C22×C4#23712yes—
449C22×C4C22×Dic312yes—
450C22×C4C23×D311yescharacteristic
451D8—216no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8C22×Dic314yescomplemented
458C4⋊C4—21no—
459C4⋊C4—34no—
460C4⋊C4—64no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—122no—
465C4⋊C4C22×Dic312yescomplemented
466C22⋊C4—24no—
467C22⋊C4—68no—
468C22⋊C4—68no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C42—34no—
474C42—62no—
475C42—62no—
476C42—62no—
477C42—122no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—28no—
485C22×C3—432no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3#44214yescomplemented
500C22×C3C2×C4×D411yescharacteristic
501C22×C3C2×#12011yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
505C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
506C22×C3#57511yescharacteristic
507C3×C4—22no—
508C3×C4—42no—
509C3×C4C2×C4×D412yes—
510C3×C4#52512yes—
511Dic3—22no—
512Dic3—22no—
513Dic3—22no—
514Dic3—44no—
515Dic3—41no—
516Dic3—81no—
517Dic3—81no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—216no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—48no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23#91112yes—
544C23#91112yes—
545C23D4×Dic312yes—
546C23C2×D3×D411yescharacteristic
547D4—216no—
548D4—24no—
549D4—24no—
550D4—24no—
551D4—24no—
552D4—48no—
553D4—42no—
554D4D4×Dic318yescomplemented
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—32no—
561C2×C4—32no—
562C2×C4—32no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—68no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—64no—
576C2×C4—124no—
577C2×C4—1216no—
578C2×C4—1216no—
579C2×C4—128no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—244no—
589C2×C4D4×Dic312yes—
590C2×C4C2×D3×D412yes—
591C2×C4C2×#23712yes—
592C2×C4C2×#23711yescharacteristic
593C2×C4C23×Dic311yescharacteristic
594C8—24no—
595C8—122no—
596C2×C3—24no—
597C2×C3—48no—
598C2×C3—82no—
599C2×C3C4×#12011yescharacteristic
600C2×C3C2×#44211yescharacteristic
601C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
602C2×C3#300311yescharacteristic
603C2×C3#300611yescharacteristic
604C2×C3#310911yescharacteristic
605C2×C3#311411yescharacteristic
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—28no—
613C22—432no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22#515214yescomplemented
628C22#558611yescharacteristic
629C22D3×#12011yescharacteristic
630C22C2×#91111yescharacteristic
631C22C2×#91111yescharacteristic
632C22D3×AGL(1,ℤ/8)11yescharacteristic
633C22#577311yescharacteristic
634C22C2×D4×Dic311yescharacteristic
635C4—22no—
636C4—42no—
637C4—62no—
638C4—62no—
639C4—62no—
640C4—124no—
641C4—121no—
642C4—241no—
643C4—241no—
644C4#558612yes—
645C4C2×D4×Dic312yes—
646C3#2145211yescharacteristic, complemented, Sylow
647C2—24no—
648C2—48no—
649C2—82no—
650C2#7999011yescharacteristic
651C2#7999111yescharacteristic
652C2#8000211yescharacteristic
653C2#8000311yescharacteristic
654C2Dic3×#12011yescharacteristic
655C2C2×#515211yescharacteristic
656C2Dic3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#447C2
socleC23×C3C22×D4
cosocleC22×C3×C4C24
hypercenterC2×#447D3
hypocenterC3#21452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C22×D4×Dic3C2semidirect
3. C2×#5152C2semidirect
4. Dic3×#121C2semidirect
5. #72128C2semidirect
6. #72037C2semidirect
7. #71677C2semidirect
8. #70170C2semidirect
9. #70152C2semidirect
10. #69928C2semidirect
11. #69906C2semidirect
24. C2×D4×Dic3C22semidirect
27. C3×#447C4semidirect
30. #5221C22semidirect
34. C2×#832C22semidirect
38. C2×#817C22semidirect
39. C2×#815C22semidirect
40. #4984C22semidirect
41. Dic3×C4⋊C4C22semidirect
44. #4551C22semidirect
45. #4550C22semidirect
46. #4529C22semidirect
64. C22×C3×D4C2×C4semidirect
79. C2×C3×D8C2×C4semidirect
82. C3×#121C2×C4semidirect
99. D4×Dic3D4semidirect
113. C3×#102C2×C4semidirect
114. C3×C22⋊C8C2×C4semidirect
138. #447Dic3semidirect
183. C2×C3×D4C22×C4semidirect
222. C2×C3×C8C22×C4semidirect
224. C3×C4⋊C4C22×C4semidirect
252. C22×D4C2×Dic3semidirect
267. C2×D8C2×Dic3semidirect
271. #121C2×Dic3semidirect
305. #102C2×Dic3semidirect
307. C22⋊C8C2×Dic3semidirect
348. C3×D4C4×D4semidirect
417. C2×D4C22×Dic3semidirect
457. C2×C8C22×Dic3semidirect
465. C4⋊C4C22×Dic3semidirect
499. C22×C3#442semidirect
554. D4D4×Dic3semidirect
627. C22#5152semidirect
646. C3#21452semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic32
C2×C3×#4472
C2×#51522
Dic3×#1212
#721282
#720372
#716772
#701702
#701522
#699282
#699062
#214523