Finite Group Databasebeta

← #10800830

Subgroups — #10800830

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×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#5217C211yesmaximal, characteristic
3C2×#5014C212yesmaximal, complemented
4C4×#930C211yesmaximal, characteristic, complemented
5#72128C211yesmaximal, characteristic, complemented
6#71613C212yesmaximal, complemented
7#71609C212yesmaximal, complemented
8#71607C212yesmaximal, complemented
9#71560C211yesmaximal, characteristic, complemented
10#71556C211yesmaximal, characteristic, complemented
11#71529C211yesmaximal, characteristic, complemented
12#21452—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×C3×C4—21no—
15C2×C3×C4×D4—28no—
16C2×C3×C4×D4—21no—
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4C2212yes—
19C2×C3×C4×D4C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4—21no—
22C22×C3×C42C2211yescharacteristic
23C22×#213C2212yes—
24C2×#930C2211yescharacteristic
25C2×C4×#67—21no—
26C2×C4×D12C2211yescharacteristic, complemented
27#5217C418yescomplemented
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31#5213—21no—
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5014—28no—
36C2×#809C2212yescomplemented
37C2×#808C2212yescomplemented
38#4940—21no—
39#4939C2211yescharacteristic, complemented
40C4×(C2×D3)⋊C4—21no—
41C4×C4⋊Dic3C2211yescharacteristic, complemented
42#4928C2211yescharacteristic, complemented
43C2×#803C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46#4524C2212yescomplemented
47C22×C4×D4D311yescharacteristic, complemented
48C2×#447—31no—
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—22no—
63C22×C3×D4C2×C414yescomplemented
64C22×C3×D4C2312yes—
65C23×C3×C4—28no—
66C23×C3×C4—21no—
67C23×C3×C4—22no—
68C23×C3×C4—22no—
69C23×C3×C4—44no—
70C23×C3×C4—44no—
71C23×C3×C4D411yescharacteristic
72C23×C3×C4D411yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C22×#67—21no—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C3×C4×D4—28no—
78C3×C4×D4—28no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×C4×D4—24no—
82C3×C4×D4D418yescomplemented
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—28no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C42—24no—
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#213—24no—
96C2×#213—24no—
97C2×#213C2×C418yescomplemented
98#930—22no—
99#930C2×C414yescomplemented
100C4×#67—44no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C22×C3⋊C8C2312yes—
111C22×C3⋊C8C2312yes—
112C4×D12—24no—
113#832—24no—
114#832—24no—
115#832C2×C418yescomplemented
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119#830—22no—
120#830C2×C414yescomplemented
121#809—24no—
122#808—24no—
123#803—24no—
124C23×D4C2×D311yescharacteristic
125C24×C4—21no—
126C22×D8—32no—
127C2×#121—31no—
128C2×C4×D4—28no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—31no—
132C2×C4×D4—61no—
133C2×C4×D4C2×D312yes—
134C2×C4×D4C2×D312yes—
135C22×C4⋊C4C2×D311yescharacteristic
136C22×C22⋊C4—21no—
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#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—42no—
163C24×C3—42no—
164C24×C3—42no—
165C24×C3—42no—
166C24×C3C2×D411yescharacteristic
167C23×D3—21no—
168C2×C3×D4—24no—
169C2×C3×D4—24no—
170C2×C3×D4—216no—
171C2×C3×D4—216no—
172C2×C3×D4—24no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4C22×C414yes—
177C2×C3×D4C2×D414yes—
178C2×C3×D4#4114yes—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—24no—
185C22×C3×C4—28no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—28no—
190C22×C3×C4—41no—
191C22×C3×C4—432no—
192C22×C3×C4—44no—
193C22×C3×C4—44no—
194C22×C3×C4—44no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—44no—
198C22×C3×C4C22×C412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4#4112yes—
205C22×C3×C4C2411yescharacteristic
206C2×#67—24no—
207C2×#67—42no—
208C2×#67—42no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C2×D12—22no—
213C2×D12—22no—
214C2×D12—21no—
215C2×D12C22×C412yescomplemented
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×C4×D3—42no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4C2×D414yes—
222C3×C22⋊C4—28no—
223C3×C22⋊C4—28no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42C2×D414yes—
228#217—42no—
229#213—216no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—42no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234C4⋊Dic3—21no—
235C4⋊Dic3—22no—
236C4⋊Dic3—22no—
237C4⋊Dic3C22×C412yescomplemented
238Dic3⋊C4—42no—
239C4×Dic3—42no—
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—21no—
247C22×D4—24no—
248C22×D4—24no—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—31no—
252C22×D4—61no—
253C22×D4C4×D314yescomplemented
254C22×D4C22×D312yes—
255C23×C4—28no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4—61no—
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—34no—
271#121—62no—
272C4×D4—28no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—64no—
278C4×D4—124no—
279C4×D4#6718yescomplemented
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D312yes—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C42—24no—
296C2×C42—61no—
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—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—48no—
321C23×C3—48no—
322C23×C3—48no—
323C23×C3—48no—
324C23×C3—48no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—48no—
330C23×C3C4×D412yes—
331C23×C3AGL(1,ℤ/8)12yes—
332C23×C3AGL(1,ℤ/8)12yes—
333C23×C3C22×D411yescharacteristic
334C22×D3—22no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C3×D4—216no—
342C3×D4—24no—
343C3×D4—24no—
344C3×D4—24no—
345C3×D4C4×D418yescomplemented
346C2×C3×C4—28no—
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—21no—
354C2×C3×C4—21no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—21no—
358C2×C3×C4—24no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—44no—
362C2×C3×C4—416no—
363C2×C3×C4—416no—
364C2×C3×C4—48no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4C4×D412yes—
368C2×C3×C4#12012yes—
369C2×C3×C4#12012yes—
370C2×C3×C4AGL(1,ℤ/8)12yes—
371C2×C3×C4C23×C411yescharacteristic
372C2×C3×C4C22×D412yes—
373C2×C3×C4C2×#4112yes—
374C2×C3×C4C2×#4111yescharacteristic
375#67—48no—
376C2×Dic3—22no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385D12—24no—
386D12—42no—
387C4×D3—84no—
388C3⋊C8—24no—
389C3⋊C8—42no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—22no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—61no—
399C24C2×#6711yescharacteristic
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—32no—
409C2×D4—64no—
410C2×D4—62no—
411C2×D4—61no—
412C2×D4—62no—
413C2×D4—122no—
414C2×D4—122no—
415C2×D4C2×C4×D314yes—
416C2×D4#23514yes—
417C2×D4C2×#6714yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—41no—
430C22×C4—432no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—122no—
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—24no—
459C4⋊C4—22no—
460C4⋊C4—32no—
461C4⋊C4—61no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—122no—
465C4⋊C4C2×#6714yes—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—64no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—122no—
477C42C2×#6714yes—
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#44214yescomplemented
493C22×C3C2×C4×D411yescharacteristic
494C22×C3C2×#12011yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
497C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
498C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
499C22×C3#57511yescharacteristic
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—22no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—44no—
512C3×C4—41no—
513C3×C4C2×C4×D412yes—
514C3×C4#52512yes—
515Dic3—42no—
516Dic3—81no—
517Dic3—81no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—216no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—48no—
527C23—48no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—62no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23C4×#6712yes—
544C23#93212yes—
545C23#94912yes—
546C23C22×#6711yescharacteristic
547D4—216no—
548D4—24no—
549D4—24no—
550D4—24no—
551D4—64no—
552D4—128no—
553D4—122no—
554D4C4×#6718yescomplemented
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—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—44no—
571C2×C4—416no—
572C2×C4—416no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—62no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—244no—
586C2×C4C4×#6712yes—
587C2×C4#93212yes—
588C2×C4#95312yes—
589C2×C4#95312yes—
590C2×C4C22×C4×D311yescharacteristic
591C2×C4C2×#23512yes—
592C2×C4C2×#23511yescharacteristic
593C2×C4C22×#6712yes—
594C8—64no—
595C8—122no—
596C2×C3—24no—
597C2×C3—48no—
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—82no—
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—122no—
621C22—122no—
622C22—122no—
623C22—241no—
624C22—241no—
625C22—241no—
626C22—241no—
627C22#501414yescomplemented
628C22C2×C4×#6711yescharacteristic
629C22#579111yescharacteristic
630C22C2×#93211yescharacteristic
631C22C2×#93211yescharacteristic
632C22C2×#94911yescharacteristic
633C22#582211yescharacteristic
634C22C2×#95311yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—44no—
640C4—41no—
641C4—122no—
642C4—241no—
643C4—241no—
644C4C2×C4×#6712yes—
645C4#579112yes—
646C3#2145211yescharacteristic, complemented, Sylow
647C2—24no—
648C2—48no—
649C2—242no—
650C2C2×#501411yescharacteristic
651C2C4×#93211yescharacteristic
652C2#8231011yescharacteristic
653C2#8231111yescharacteristic
654C2C4×#95311yescharacteristic
655C2#8403811yescharacteristic
656C2#8408211yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC22×C3×C4×D4C2
socleC23×C3C22×D4
prime radicalC2×#5217C2
cosocleC22×C3×C4C24
hypercenterC22×C4×D4D3
hypocenterC3#21452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×D4C2semidirect
3. C2×#5014C2semidirect
4. C4×#930C2semidirect
5. #72128C2semidirect
6. #71613C2semidirect
7. #71609C2semidirect
8. #71607C2semidirect
9. #71560C2semidirect
10. #71556C2semidirect
11. #71529C2semidirect
26. C2×C4×D12C22semidirect
27. #5217C4semidirect
36. C2×#809C22semidirect
37. C2×#808C22semidirect
39. #4939C22semidirect
41. C4×C4⋊Dic3C22semidirect
42. #4928C22semidirect
43. C2×#803C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
46. #4524C22semidirect
47. C22×C4×D4D3semidirect
63. C22×C3×D4C2×C4semidirect
82. C3×C4×D4D4semidirect
97. C2×#213C2×C4semidirect
99. #930C2×C4semidirect
115. #832C2×C4semidirect
120. #830C2×C4semidirect
215. C2×D12C22×C4semidirect
237. C4⋊Dic3C22×C4semidirect
245. C2×C3⋊C8C22×C4semidirect
253. C22×D4C4×D3semidirect
279. C4×D4#67semidirect
345. C3×D4C4×D4semidirect
492. C22×C3#442semidirect
554. D4C4×#67semidirect
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×C3×C4×D42
C2×#52172
C2×#50142
C4×#9302
#721282
#716132
#716092
#716072
#715602
#715562
#715292
#214523