Finite Group Databasebeta

← #10800829

Subgroups — #10800829

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×#893C211yesmaximal, characteristic, complemented
2C2×#4869C212yesmaximal, complemented
3C2×#4732C211yesmaximal, characteristic
4#73865C211yesmaximal, characteristic, complemented
5C3×#1853C211yesmaximal, characteristic, complemented
6#71560C211yesmaximal, characteristic, complemented
7#70584C212yesmaximal, complemented
8#69906C211yesmaximal, characteristic, complemented
9#69521C211yesmaximal, characteristic, complemented
10#68897C212yesmaximal, complemented
11#68782C212yesmaximal, complemented
12#21452—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3—21no—
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C22×(C2×D3)⋊C4—21no—
18C22×C4×Dic3C2211yescharacteristic
19C22×D24C2212yes—
20C2×#893—21no—
21C2×#893—21no—
22C2×#893—28no—
23C2×#893C2212yes—
24C2×#893C2212yes—
25C2×#893C2211yescharacteristic, complemented
26C2×#67⋊C4—21no—
27C2×C3×#106C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#821C2212yes—
32C2×#815C2212yescomplemented
33#4988C2211yescharacteristic, complemented
34Dic3×C4⋊C4C2211yescharacteristic, complemented
35#4972C2211yescharacteristic, complemented
36C2×#809C2212yes—
37C2×#809C2212yes—
38C2×#809C2212yescomplemented
39#4958—21no—
40#4869—28no—
41#4732C418yescomplemented
42#4669—21no—
43#4666—21no—
44C3×#340C2212yescomplemented
45#4550C2212yescomplemented
46#4529C2212yescomplemented
47C22×C4×D4—31no—
48C2×#447—31no—
49C2×#442—32no—
50C4×#121—31no—
51#1879—32no—
52#1874—31no—
53#1864—32no—
54#1853D311yescharacteristic, complemented
55#1786—31no—
56#1762—32no—
57#1749—31no—
58C24×D3—21no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3D411yescharacteristic
63C23×Dic3D411yescharacteristic
64C22×D12—21no—
65C22×D12—22no—
66C22×D12—24no—
67C22×D12—24no—
68C22×D12C2×C414yescomplemented
69C22×D12C2311yescharacteristic
70C22×D12C2312yes—
71C22×C4×D3—28no—
72C22×C4×D3—22no—
73C22×C4×D3—22no—
74C22×C4×D3—21no—
75C22×C4×D3—44no—
76C22×C4×D3—44no—
77C22×C3×C8C2312yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—21no—
84#930—22no—
85#930C2×C414yescomplemented
86C2×(C2×D3)⋊C4—28no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×Dic3⋊C4—21no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3D411yescharacteristic
98C2×C4×Dic3D411yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100C2×C4×Dic3C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C2×D24—24no—
103C2×D24—24no—
104C2×D24C2×C418yescomplemented
105#893—28no—
106#893—28no—
107#893—24no—
108#893—24no—
109#893—24no—
110#893—24no—
111#893D418yescomplemented
112#67⋊C4—44no—
113C3×#106—24no—
114C3×C22⋊C8C2×C414yescomplemented
115C3×#95—21no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118#830—22no—
119#821C2×C418yescomplemented
120#815—24no—
121#809—24no—
122#809—24no—
123#809—24no—
124C23×D4—31no—
125C24×C4—61no—
126C22×D8—32no—
127C2×#121—31no—
128C2×C4×D4—32no—
129C2×C4×D4—32no—
130C2×C4×D4—31no—
131C2×C4×D4—61no—
132C2×C4×D4—61no—
133C2×C4×D4—61no—
134C2×C4×D4—68no—
135C22×C4⋊C4C2×D311yescharacteristic
136C22×C22⋊C4—61no—
137C22×C42—31no—
138#447—38no—
139#442—68no—
140C2×#106C2×D312yescomplemented
141C2×#102—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8C2×D311yescharacteristic
147C2×#97—32no—
148#390—31no—
149#389—31no—
150#388—61no—
151#386—61no—
152#380C2×D311yescharacteristic
153C4×C4⋊C4—31no—
154C4×C22⋊C4—61no—
155#340—32no—
156#340C2×D312yescomplemented
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C23×D3—24no—
160C23×D3—24no—
161C23×D3—24no—
162C23×D3—22no—
163C23×D3—21no—
164C23×D3—42no—
165C23×D3—42no—
166C23×D3—42no—
167C23×D3—42no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4C22×C412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—24no—
181C2×#67—42no—
182C2×#67—42no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—41no—
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×D12—24no—
195C2×D12—24no—
196C2×D12—216no—
197C2×D12—24no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—216no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—21no—
205C2×D12C22×C414yes—
206C2×D12C22×C412yescomplemented
207C2×D12C2×D414yes—
208C2×D12#4114yes—
209C2×C4×D3—28no—
210C2×C4×D3—28no—
211C2×C4×D3—432no—
212C2×C4×D3—44no—
213C2×C4×D3—44no—
214C2×C4×D3—44no—
215C2×C4×D3—44no—
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×C4×D3—42no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8C22×C414yescomplemented
222C3×C4⋊C4—24no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4C2×D414yes—
227C3×C22⋊C4—42no—
228(C2×D3)⋊C4—28no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—28no—
231(C2×D3)⋊C4—42no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234C4⋊Dic3—21no—
235C4⋊Dic3C22×C412yescomplemented
236Dic3⋊C4—42no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239C4×Dic3—22no—
240C4×Dic3—42no—
241C4×Dic3C2×D414yes—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245D24—216no—
246C25—61no—
247C22×D4—34no—
248C22×D4—32no—
249C22×D4—31no—
250C22×D4—64no—
251C22×D4—64no—
252C22×D4—61no—
253C22×D4—61no—
254C22×D4—62no—
255C23×C4—21no—
256C23×C4—31no—
257C23×C4—31no—
258C23×C4—62no—
259C23×C4—62no—
260C23×C4—61no—
261C23×C4—68no—
262C23×C4—124no—
263C23×C4—124no—
264C23×C4C22×D311yescharacteristic
265C2×D8—38no—
266C2×D8—64no—
267C2×D8—64no—
268C22×C8—32no—
269C22×C8C22×D312yes—
270#121—34no—
271#121—62no—
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—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D312yes—
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—24no—
302#102—38no—
303#102—64no—
304#102—64no—
305#102—64no—
306C22⋊C8—62no—
307C22⋊C8C4×D314yescomplemented
308#97—64no—
309#95—21no—
310#95—61no—
311#95—61no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3C4×D412yes—
317C23×C3AGL(1,ℤ/8)12yes—
318C23×C3AGL(1,ℤ/8)12yes—
319C23×C3C22×D411yescharacteristic
320C22×D3—216no—
321C22×D3—24no—
322C22×D3—24no—
323C22×D3—24no—
324C22×D3—22no—
325C22×D3—48no—
326C22×D3—48no—
327C22×D3—48no—
328C22×D3—48no—
329C22×D3—48no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—48no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—48no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4C4×D412yes—
356C2×C3×C4C23×C411yescharacteristic
357C2×C3×C4C22×D412yes—
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4112yes—
360#67—48no—
361C2×Dic3—28no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—22no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—44no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3#12012yes—
376C2×Dic3#12012yes—
377C2×Dic3AGL(1,ℤ/8)12yes—
378D12—216no—
379D12—24no—
380D12—24no—
381D12—24no—
382D12—24no—
383D12—42no—
384D12C4×D418yescomplemented
385C4×D3—416no—
386C4×D3—416no—
387C4×D3—84no—
388C3×C8—24no—
389C3⋊C8—42no—
390C24—64no—
391C24—64no—
392C24—64no—
393C24—62no—
394C24—61no—
395C24—122no—
396C24—122no—
397C24—122no—
398C24—122no—
399C24D3×D411yescharacteristic
400C2×D4—34no—
401C2×D4—32no—
402C2×D4—34no—
403C2×D4—34no—
404C2×D4—616no—
405C2×D4—64no—
406C2×D4—64no—
407C2×D4—616no—
408C2×D4—64no—
409C2×D4—64no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—61no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—122no—
417C2×D4—122no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—68no—
431C22×C4—68no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—124no—
439C22×C4—122no—
440C22×C4—1232no—
441C22×C4—124no—
442C22×C4—124no—
443C22×C4—124no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—121no—
447C22×C4C2×C4×D312yes—
448C22×C4D3×D411yescharacteristic
449C22×C4#23912yes—
450C22×C4C23×D311yescharacteristic
451D8—616no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8C2×C4×D314yescomplemented
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—32no—
463C4⋊C4—61no—
464C4⋊C4—122no—
465C4⋊C4D3×D414yes—
466C22⋊C4—42no—
467C22⋊C4—68no—
468C22⋊C4—68no—
469C22⋊C4—64no—
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#44214yescomplemented
485C22×C3C2×C4×D411yescharacteristic
486C22×C3C2×#12011yescharacteristic
487C22×C3#52511yescharacteristic
488C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
489C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
490C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
491C22×C3#57511yescharacteristic
492C2×D3—28no—
493C2×D3—432no—
494C2×D3—44no—
495C2×D3—44no—
496C2×D3—44no—
497C2×D3—44no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C3×C4—22no—
508C3×C4—44no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4C2×C4×D412yes—
512C3×C4#52512yes—
513Dic3—22no—
514Dic3—22no—
515Dic3—22no—
516Dic3—42no—
517Dic3—41no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—616no—
523C23—64no—
524C23—64no—
525C23—64no—
526C23—62no—
527C23—128no—
528C23—128no—
529C23—128no—
530C23—128no—
531C23—128no—
532C23—128no—
533C23—122no—
534C23—122no—
535C23—122no—
536C23—122no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23#89312yes—
544C23#91412yes—
545C23#91412yes—
546C23C2×D3×D411yescharacteristic
547D4—38no—
548D4—616no—
549D4—64no—
550D4—64no—
551D4—64no—
552D4—64no—
553D4—128no—
554D4—122no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—32no—
561C2×C4—32no—
562C2×C4—32no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—68no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—1216no—
584C2×C4—1216no—
585C2×C4—124no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—244no—
589C2×C4#89312yes—
590C2×C4C22×C4×D311yescharacteristic
591C2×C4C2×D3×D412yes—
592C2×C4C2×#23911yescharacteristic
593C2×C4D3×#4112yes—
594C8—24no—
595C8—122no—
596C2×C3—24no—
597C2×C3C4×#12011yescharacteristic
598C2×C3C2×#44211yescharacteristic
599C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
600C2×C3#300311yescharacteristic
601C2×C3#300611yescharacteristic
602C2×C3#310911yescharacteristic
603C2×C3#311411yescharacteristic
604D3—48no—
605D3—82no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—68no—
613C22—1232no—
614C22—124no—
615C22—124no—
616C22—124no—
617C22—124no—
618C22—124no—
619C22—124no—
620C22—122no—
621C22—122no—
622C22—122no—
623C22—241no—
624C22—241no—
625C22—241no—
626C22—241no—
627C22#486914yescomplemented
628C22C2×#89311yescharacteristic
629C22D3×#12011yescharacteristic
630C22#565611yescharacteristic
631C22C2×#91411yescharacteristic
632C22C2×#91411yescharacteristic
633C22D3×AGL(1,ℤ/8)11yescharacteristic
634C22#577711yescharacteristic
635C4—22no—
636C4—44no—
637C4—62no—
638C4—62no—
639C4—62no—
640C4—81no—
641C4—81no—
642C4—122no—
643C4—121no—
644C4C4×D3×D412yes—
645C4#562712yes—
646C3#2145211yescharacteristic, complemented, Sylow
647C2—24no—
648C2—128no—
649C2—242no—
650C2#7437911yescharacteristic
651C2C2×#486911yescharacteristic
652C2#7919911yescharacteristic
653C2#8004011yescharacteristic
654C2#8014611yescharacteristic
655C2#8131011yescharacteristic
656C2#8141511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1853C2
socleC23×C3C22×D4
prime radicalC2×#4732C2
cosocleC22×C3×C4C24
hypercenter#1853D3
hypocenterC3#21452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
1. C22×#893C2semidirect
2. C2×#4869C2semidirect
4. #73865C2semidirect
5. C3×#1853C2semidirect
6. #71560C2semidirect
7. #70584C2semidirect
8. #69906C2semidirect
9. #69521C2semidirect
10. #68897C2semidirect
11. #68782C2semidirect
25. C2×#893C22semidirect
27. C2×C3×#106C22semidirect
32. C2×#815C22semidirect
33. #4988C22semidirect
34. Dic3×C4⋊C4C22semidirect
35. #4972C22semidirect
38. C2×#809C22semidirect
41. #4732C4semidirect
44. C3×#340C22semidirect
45. #4550C22semidirect
46. #4529C22semidirect
54. #1853D3semidirect
68. C22×D12C2×C4semidirect
85. #930C2×C4semidirect
104. C2×D24C2×C4semidirect
111. #893D4semidirect
114. C3×C22⋊C8C2×C4semidirect
119. #821C2×C4semidirect
140. C2×#106C2×D3semidirect
156. #340C2×D3semidirect
206. C2×D12C22×C4semidirect
221. C2×C3×C8C22×C4semidirect
235. C4⋊Dic3C22×C4semidirect
307. C22⋊C8C4×D3semidirect
384. D12C4×D4semidirect
457. C2×C8C2×C4×D3semidirect
484. C22×C3#442semidirect
627. C22#4869semidirect
646. C3#21452semidirect

Maximal subgroups

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

subgroupindex
C22×#8932
C2×#48692
C2×#47322
#738652
C3×#18532
#715602
#705842
#699062
#695212
#688972
#687822
#214523