Finite Group Databasebeta

← #10800828

Subgroups — #10800828

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×C4×D12C211yesmaximal, characteristic, complemented
2C3×C4×#121C211yesmaximal, characteristic, complemented
3C2×#5036C211yesmaximal, characteristic
4C2×#5014C212yesmaximal, complemented
5#71677C211yesmaximal, characteristic, complemented
6#71529C211yesmaximal, characteristic, complemented
7#69657C212yesmaximal, complemented
8#69626C212yesmaximal, complemented
9#69521C211yesmaximal, characteristic, complemented
10#69505C211yesmaximal, characteristic, complemented
11#69474C212yesmaximal, complemented
12#21452—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3—21no—
15C2×C3×#121C2211yescharacteristic
16C2×C3×C4×D4—21no—
17C2×C3×C4×D4C2211yescharacteristic, complemented
18C22×C3×C42C2211yescharacteristic
19C22×#213C2212yes—
20C22×(C2×D3)⋊C4—21no—
21C22×C4⋊Dic3C2211yescharacteristic
22C2×C4×D12—21no—
23C2×C4×D12—21no—
24C2×C4×D12—28no—
25C2×C4×D12C2212yes—
26C2×C4×D12C2212yes—
27C3×#390C2211yescharacteristic, complemented
28C3×#389C2211yescharacteristic, complemented
29C3×#388—21no—
30C3×#386—21no—
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic, complemented
33C3×C4×C22⋊C4—21no—
34C2×#832C2212yescomplemented
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5036C418yescomplemented
38#5014—28no—
39C2×#809C2212yes—
40C2×#809C2212yes—
41C2×#809C2212yes—
42C2×#808C2212yescomplemented
43C2×#803C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46#4524C2212yescomplemented
47C22×C4×D4—31no—
48C2×#447—31no—
49C2×#442—32no—
50C4×#121D311yescharacteristic, complemented
51#1879—32no—
52#1874—31no—
53#1864—32no—
54#1853—31no—
55#1786—31no—
56#1762—32no—
57#1749—31no—
58C24×D3—21no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×Dic3—21no—
66C22×D12—21no—
67C22×D12—22no—
68C22×D12—24no—
69C22×D12—24no—
70C22×D12C2×C414yescomplemented
71C22×D12C2312yes—
72C22×C4×D3—28no—
73C22×C4×D3—22no—
74C22×C4×D3—22no—
75C22×C4×D3—44no—
76C22×C4×D3—44no—
77C3×#121—22no—
78C3×#121C2×C414yescomplemented
79C3×C4×D4—24no—
80C3×C4×D4—44no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—24no—
91C2×C3×C42—21no—
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#213—24no—
97C2×#213—24no—
98C2×#213C2×C418yescomplemented
99C2×(C2×D3)⋊C4—28no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—24no—
104C2×C4⋊Dic3C2312yes—
105C22×C3⋊C8C2312yes—
106C22×C3⋊C8C2312yes—
107C4×D12—28no—
108C4×D12—28no—
109C4×D12—24no—
110C4×D12—24no—
111C4×D12—24no—
112C4×D12D418yescomplemented
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116#832—24no—
117#830—22no—
118#830C2×C414yescomplemented
119#809—24no—
120#809—24no—
121#809C2×C418yescomplemented
122#808—24no—
123#803—24no—
124C23×D4—31no—
125C24×C4—61no—
126C22×D8—32no—
127C2×#121C2×D311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4—32no—
130C2×C4×D4—32no—
131C2×C4×D4—68no—
132C2×C4×D4—61no—
133C2×C4×D4—61no—
134C2×C4×D4C2×D311yescharacteristic, complemented
135C22×C4⋊C4—31no—
136C22×C22⋊C4—61no—
137C22×C42C2×D311yescharacteristic
138#447—38no—
139#442—68no—
140C2×#106—32no—
141C2×#102—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—31no—
147C2×#97—32no—
148#390C2×D311yescharacteristic, complemented
149#389C2×D311yescharacteristic, complemented
150#388—21no—
151#386—21no—
152#380C2×D311yescharacteristic
153C4×C4⋊C4C2×D311yescharacteristic, complemented
154C4×C22⋊C4—21no—
155#340—32no—
156#340—32no—
157#336—32no—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C23×D3—24no—
161C23×D3—24no—
162C23×D3—24no—
163C23×D3—22no—
164C23×D3—42no—
165C23×D3—42no—
166C23×D3—42no—
167C23×D3—42no—
168C2×C3×D4—24no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4C22×C412yescomplemented
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—41no—
186C22×C3×C4—42no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4C22×C412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—24no—
198C22×Dic3—22no—
199C22×Dic3—42no—
200C22×Dic3—42no—
201C2×D12—24no—
202C2×D12—24no—
203C2×D12—216no—
204C2×D12—24no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—216no—
209C2×D12C22×C414yes—
210C2×D12C2×D414yes—
211C2×D12#4114yes—
212C2×C4×D3—28no—
213C2×C4×D3—28no—
214C2×C4×D3—432no—
215C2×C4×D3—44no—
216C2×C4×D3—44no—
217C2×C4×D3—44no—
218C2×C4×D3—44no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—42no—
223C3×C4⋊C4C22×C412yescomplemented
224C3×C22⋊C4—24no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—22no—
232C3×C42—42no—
233C3×C42C2×D414yes—
234#213—216no—
235(C2×D3)⋊C4—28no—
236(C2×D3)⋊C4—28no—
237C4⋊Dic3—24no—
238C4⋊Dic3—22no—
239C4⋊Dic3C2×D414yes—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8C22×C414yescomplemented
246C25—61no—
247C22×D4—21no—
248C22×D4—34no—
249C22×D4—32no—
250C22×D4—64no—
251C22×D4—64no—
252C22×D4—62no—
253C22×D4—61no—
254C22×D4C22×D311yescharacteristic
255C23×C4—21no—
256C23×C4—68no—
257C23×C4—61no—
258C23×C4—62no—
259C23×C4—62no—
260C23×C4—124no—
261C23×C4—124no—
262C23×C4#6711yescharacteristic
263C23×C4#6711yescharacteristic
264C23×C4C22×D311yescharacteristic
265C2×D8—38no—
266C2×D8—64no—
267C2×D8—64no—
268C22×C8—32no—
269C22×C8—32no—
270#121—22no—
271#121C4×D314yescomplemented
272C4×D4—24no—
273C4×D4—38no—
274C4×D4—44no—
275C4×D4—64no—
276C4×D4—64no—
277C4×D4—68no—
278C4×D4—68no—
279C4×D4—64no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—68no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4—62no—
295C2×C42—24no—
296C2×C42—21no—
297C2×C42#6711yescharacteristic
298C2×C42#6711yescharacteristic
299C2×C42C22×D311yescharacteristic
300C2×C42C22×D311yescharacteristic
301#106—64no—
302#102—38no—
303#102—64no—
304#102—64no—
305#102—64no—
306C22⋊C8—34no—
307C22⋊C8—62no—
308#97—64no—
309#95—21no—
310#95—21no—
311#95—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3C4×D412yes—
324C23×C3AGL(1,ℤ/8)12yes—
325C23×C3AGL(1,ℤ/8)12yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—216no—
328C22×D3—24no—
329C22×D3—24no—
330C22×D3—24no—
331C22×D3—48no—
332C22×D3—48no—
333C22×D3—48no—
334C22×D3—48no—
335C22×D3—48no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—48no—
341C3×D4—24no—
342C3×D4—48no—
343C3×D4—42no—
344C2×C3×C4—28no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—44no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—84no—
369C2×C3×C4C4×D412yes—
370C2×C3×C4#12012yes—
371C2×C3×C4#12012yes—
372C2×C3×C4AGL(1,ℤ/8)12yes—
373C2×C3×C4C23×C411yescharacteristic
374C2×C3×C4C22×D412yes—
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4112yes—
377C2×Dic3—24no—
378C2×Dic3—48no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381D12—216no—
382D12—24no—
383D12—24no—
384D12—24no—
385D12C4×D418yescomplemented
386C4×D3—416no—
387C4×D3—416no—
388C3⋊C8—24no—
389C3⋊C8—42no—
390C24—21no—
391C24—64no—
392C24—64no—
393C24—64no—
394C24—62no—
395C24—122no—
396C24—122no—
397C24—122no—
398C24—122no—
399C24C2×#6711yescharacteristic
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—34no—
405C2×D4—34no—
406C2×D4—34no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—616no—
410C2×D4—64no—
411C2×D4—64no—
412C2×D4—616no—
413C2×D4—64no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4C2×C4×D312yescomplemented
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—68no—
433C22×C4—64no—
434C22×C4—62no—
435C22×C4—68no—
436C22×C4—1232no—
437C22×C4—124no—
438C22×C4—124no—
439C22×C4—124no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—124no—
443C22×C4C2×C4×D312yes—
444C22×C4#23512yes—
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C23×D311yescharacteristic
451D8—616no—
452C2×C8—34no—
453C2×C8—62no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8—62no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—34no—
462C4⋊C4—42no—
463C4⋊C4—64no—
464C4⋊C4—62no—
465C4⋊C4C2×C4×D312yescomplemented
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—68no—
472C22⋊C4—68no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—42no—
477C42D3×D414yes—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3#44214yescomplemented
492C22×C3C2×C4×D411yescharacteristic
493C22×C3C2×#12011yescharacteristic
494C22×C3#52511yescharacteristic
495C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
496C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
497C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
498C22×C3#57511yescharacteristic
499C2×D3—28no—
500C2×D3—432no—
501C2×D3—44no—
502C2×D3—44no—
503C2×D3—44no—
504C2×D3—44no—
505C2×D3—44no—
506C2×D3—44no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—42no—
512C3×C4—41no—
513C3×C4—81no—
514C3×C4—81no—
515C3×C4C2×C4×D412yes—
516C3×C4#52512yes—
517Dic3—44no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—616no—
530C23—64no—
531C23—64no—
532C23—64no—
533C23—128no—
534C23—128no—
535C23—128no—
536C23—128no—
537C23—128no—
538C23—128no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23C4×#6712yes—
544C23#93212yes—
545C23#94912yes—
546C23C22×#6711yescharacteristic
547D4—24no—
548D4—38no—
549D4—48no—
550D4—42no—
551D4—616no—
552D4—64no—
553D4—64no—
554D4—64no—
555C2×C4—28no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—64no—
580C2×C4—84no—
581C2×C4—1216no—
582C2×C4—1216no—
583C2×C4—128no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4C4×#6712yes—
587C2×C4#93712yes—
588C2×C4#93712yes—
589C2×C4#94912yes—
590C2×C4C22×C4×D311yescharacteristic
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×D3×D412yes—
593C2×C4D3×#4112yes—
594C8—64no—
595C8—122no—
596C2×C3—24no—
597C2×C3—82no—
598C2×C3C4×#12011yescharacteristic
599C2×C3C2×#44211yescharacteristic
600C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
601C2×C3#300311yescharacteristic
602C2×C3#300611yescharacteristic
603C2×C3#310911yescharacteristic
604C2×C3#311411yescharacteristic
605D3—48no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—68no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—1232no—
621C22—124no—
622C22—124no—
623C22—124no—
624C22—124no—
625C22—124no—
626C22—124no—
627C22#501414yescomplemented
628C22C2×C4×#6711yescharacteristic
629C22C2×#93211yescharacteristic
630C22C2×#93711yescharacteristic
631C22C2×#94911yescharacteristic
632C22C2×#94911yescharacteristic
633C22#582211yescharacteristic
634C22#582811yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—42no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—124no—
644C4C4×D3×D412yes—
645C4#554712yes—
646C3#2145211yescharacteristic, complemented, Sylow
647C2—24no—
648C2—82no—
649C2—128no—
650C2C2×#501411yescharacteristic
651C2C4×#93711yescharacteristic
652C2C4×#94911yescharacteristic
653C2#8262411yescharacteristic
654C2#8292711yescharacteristic
655C2#8314311yescharacteristic
656C2#8330011yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#121C2
socleC23×C3C22×D4
prime radicalC2×#5036C2
cosocleC22×C3×C4C24
hypercenterC4×#121D3
hypocenterC3#21452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C22×C4×D12C2semidirect
2. C3×C4×#121C2semidirect
4. C2×#5014C2semidirect
5. #71677C2semidirect
6. #71529C2semidirect
7. #69657C2semidirect
8. #69626C2semidirect
9. #69521C2semidirect
10. #69505C2semidirect
11. #69474C2semidirect
17. C2×C3×C4×D4C22semidirect
27. C3×#390C22semidirect
28. C3×#389C22semidirect
32. C3×C4×C4⋊C4C22semidirect
34. C2×#832C22semidirect
37. #5036C4semidirect
42. C2×#808C22semidirect
43. C2×#803C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
46. #4524C22semidirect
50. C4×#121D3semidirect
70. C22×D12C2×C4semidirect
78. C3×#121C2×C4semidirect
98. C2×#213C2×C4semidirect
112. C4×D12D4semidirect
118. #830C2×C4semidirect
121. #809C2×C4semidirect
134. C2×C4×D4C2×D3semidirect
148. #390C2×D3semidirect
149. #389C2×D3semidirect
153. C4×C4⋊C4C2×D3semidirect
174. C2×C3×D4C22×C4semidirect
223. C3×C4⋊C4C22×C4semidirect
245. C2×C3⋊C8C22×C4semidirect
271. #121C4×D3semidirect
385. D12C4×D4semidirect
417. C2×D4C2×C4×D3semidirect
465. C4⋊C4C2×C4×D3semidirect
491. C22×C3#442semidirect
627. C22#5014semidirect
646. C3#21452semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D122
C3×C4×#1212
C2×#50362
C2×#50142
#716772
#715292
#696572
#696262
#695212
#695052
#694742
#214523