Finite Group Databasebeta

← #10800834

Subgroups — #10800834

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
2C4×#930C211yesmaximal, characteristic, complemented
3C2×#4732C211yesmaximal, characteristic
4C2×#4711C212yesmaximal, complemented
5C3×#1749C211yesmaximal, characteristic, complemented
6#69959C211yesmaximal, characteristic, complemented
7#69932C211yesmaximal, characteristic, complemented
8#69928C211yesmaximal, characteristic, complemented
9#69812C212yesmaximal, complemented
10#68673C212yesmaximal, complemented
11#68671C212yesmaximal, complemented
12#21452—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3—21no—
15C22×C3×C42C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×C4×#67—21no—
18C22×(C2×D3)⋊C4—21no—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×D24C2212yes—
21C2×C4×D12—21no—
22C2×C4×D12—21no—
23C2×C4×D12—28no—
24C2×C4×D12C2212yes—
25C2×C4×D12C2212yes—
26C2×C4×D12C2211yescharacteristic, complemented
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×#97C2212yescomplemented
30#5213—21no—
31#5212C2211yescharacteristic
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#821C2212yescomplemented
36C2×#817C2212yescomplemented
37#4940—21no—
38#4939C2211yescharacteristic, complemented
39C4×(C2×D3)⋊C4—21no—
40C4×C4⋊Dic3C2211yescharacteristic, complemented
41#4928C2211yescharacteristic, complemented
42#4732C418yescomplemented
43#4711—28no—
44C3×#336C2212yescomplemented
45#4551C2212yescomplemented
46#4551C2212yescomplemented
47C22×C4×D4—31no—
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#1749D311yescharacteristic, complemented
58C24×D3—21no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C23×Dic3—21no—
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—
78C22×C3×C8C2312yes—
79C2×C3×C42—24no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85#930—22no—
86#930C2×C414yescomplemented
87C4×#67—44no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—24no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2312yes—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×D24—24no—
104C2×D24—24no—
105C2×D24C2×C418yescomplemented
106C4×D12—28no—
107C4×D12—28no—
108C4×D12—24no—
109C4×D12—24no—
110C4×D12—24no—
111C4×D12—24no—
112C4×D12D418yescomplemented
113C3×C22⋊C8—22no—
114C3×C22⋊C8C2×C414yescomplemented
115C3×#97—24no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119#821—24no—
120#821—24no—
121#821—24no—
122#821C2×C418yescomplemented
123#817—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—68no—
132C2×C4×D4—61no—
133C2×C4×D4—61no—
134C2×C4×D4—61no—
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⋊C8C2×D311yescharacteristic
146C2×C22⋊C8C2×D311yescharacteristic
147C2×#97C2×D312yescomplemented
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#336C2×D312yescomplemented
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—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—41no—
176C22×C3×C4C22×C412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4#4112yes—
183C22×C3×C4C2411yescharacteristic
184C2×#67—24no—
185C2×#67—42no—
186C2×#67—42no—
187C22×Dic3—24no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—42no—
193C22×Dic3—42no—
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×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8C22×C414yescomplemented
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42C2×D414yes—
229#217—42no—
230(C2×D3)⋊C4—28no—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—28no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236C4⋊Dic3—24no—
237C4⋊Dic3—22no—
238C4⋊Dic3—21no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3C22×C412yescomplemented
242C4⋊Dic3C2×D414yes—
243Dic3⋊C4—42no—
244C4×Dic3—42no—
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—68no—
256C23×C4—61no—
257C23×C4—62no—
258C23×C4—62no—
259C23×C4—61no—
260C23×C4—124no—
261C23×C4—124no—
262C23×C4D1211yescharacteristic
263C23×C4D1211yescharacteristic
264C23×C4C22×D311yescharacteristic
265C2×D8—38no—
266C2×D8—64no—
267C2×D8—64no—
268C22×C8C22×D312yes—
269C22×C8C22×D312yes—
270#121—34no—
271#121—62no—
272C4×D4—38no—
273C4×D4—64no—
274C4×D4—64no—
275C4×D4—68no—
276C4×D4—68no—
277C4×D4—64no—
278C4×D4—64no—
279C4×D4—124no—
280C2×C4⋊C4—32no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—64no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C22⋊C4—68no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4—62no—
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×C42D1211yescharacteristic
298C2×C42D1211yescharacteristic
299C2×C42C22×D311yescharacteristic
300C2×C42C22×D311yescharacteristic
301#106—64no—
302#102—38no—
303#102—64no—
304#102—64no—
305#102—64no—
306C22⋊C8—22no—
307C22⋊C8C4×D314yescomplemented
308#97—24no—
309#95—61no—
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—28no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—44no—
356C2×C3×C4C4×D412yes—
357C2×C3×C4#12012yes—
358C2×C3×C4#12012yes—
359C2×C3×C4AGL(1,ℤ/8)12yes—
360C2×C3×C4C23×C411yescharacteristic
361C2×C3×C4C22×D412yes—
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4112yes—
364#67—48no—
365C2×Dic3—22no—
366C2×Dic3—24no—
367C2×Dic3—48no—
368C2×Dic3—41no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
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—
399C24C2×D1211yescharacteristic
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—61no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—122no—
417C2×D4—122no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—68no—
427C22×C4—64no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—68no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—1232no—
434C22×C4—124no—
435C22×C4—124no—
436C22×C4—124no—
437C22×C4—122no—
438C22×C4—122no—
439C22×C4—122no—
440C22×C4—122no—
441C22×C4—124no—
442C22×C4—122no—
443C22×C4C2×C4×D312yes—
444C22×C4C2×D1211yescharacteristic
445C22×C4C2×D1211yescharacteristic
446C22×C4C2×D1211yescharacteristic
447C22×C4C2×D1211yescharacteristic
448C22×C4C2×D1211yescharacteristic
449C22×C4#23512yes—
450C22×C4C23×D311yescharacteristic
451D8—616no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8C2×C4×D314yescomplemented
458C4⋊C4—32no—
459C4⋊C4—34no—
460C4⋊C4—64no—
461C4⋊C4—61no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—122no—
466C22⋊C4—68no—
467C22⋊C4—64no—
468C22⋊C4—68no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—122no—
477C42D3×D414yes—
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—22no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—41no—
512C3×C4C2×C4×D412yes—
513C3×C4#52512yes—
514Dic3—42no—
515Dic3—44no—
516Dic3—81no—
517Dic3—81no—
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—
543C23C4×D1212yes—
544C23#90812yes—
545C23#90812yes—
546C23C22×D1211yescharacteristic
547D4—38no—
548D4—616no—
549D4—64no—
550D4—64no—
551D4—64no—
552D4—64no—
553D4—128no—
554D4—122no—
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—44no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—1216no—
573C2×C4—1216no—
574C2×C4—128no—
575C2×C4—121no—
576C2×C4—122no—
577C2×C4—122no—
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×D1212yes—
587C2×C4#88212yes—
588C2×C4#88212yes—
589C2×C4#90812yes—
590C2×C4C22×C4×D311yescharacteristic
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×D3×D412yes—
593C2×C4D3×#4112yes—
594C8—24no—
595C8—42no—
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#471114yescomplemented
628C22C2×C4×D1211yescharacteristic
629C22C2×#88211yescharacteristic
630C22#552411yescharacteristic
631C22C2×#90811yescharacteristic
632C22C2×#90811yescharacteristic
633C22C2×#90811yescharacteristic
634C22#574811yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—124no—
641C4—122no—
642C4—241no—
643C4—241no—
644C4C4×D3×D412yes—
645C4#554712yes—
646C3#2145211yescharacteristic, complemented, Sylow
647C2—24no—
648C2—128no—
649C2—242no—
650C2C4×#88211yescharacteristic
651C2C2×#471111yescharacteristic
652C2C4×#90811yescharacteristic
653C2#7863711yescharacteristic
654C2#7863911yescharacteristic
655C2#7934311yescharacteristic
656C2#7934611yescharacteristic
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1749C2
socleC23×C3C22×D4
prime radicalC2×#4732C2
cosocleC22×C3×C4C24
hypercenter#1749D3
hypocenterC3#21452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
1. C22×C4×D12C2semidirect
2. C4×#930C2semidirect
4. C2×#4711C2semidirect
5. C3×#1749C2semidirect
6. #69959C2semidirect
7. #69932C2semidirect
8. #69928C2semidirect
9. #69812C2semidirect
10. #68673C2semidirect
11. #68671C2semidirect
26. C2×C4×D12C22semidirect
29. C2×C3×#97C22semidirect
35. C2×#821C22semidirect
36. C2×#817C22semidirect
38. #4939C22semidirect
40. C4×C4⋊Dic3C22semidirect
41. #4928C22semidirect
42. #4732C4semidirect
44. C3×#336C22semidirect
45. #4551C22semidirect
46. #4551C22semidirect
57. #1749D3semidirect
68. C22×D12C2×C4semidirect
86. #930C2×C4semidirect
105. C2×D24C2×C4semidirect
112. C4×D12D4semidirect
114. C3×C22⋊C8C2×C4semidirect
122. #821C2×C4semidirect
147. C2×#97C2×D3semidirect
157. #336C2×D3semidirect
206. C2×D12C22×C4semidirect
224. C2×C3×C8C22×C4semidirect
241. C4⋊Dic3C22×C4semidirect
307. C22⋊C8C4×D3semidirect
384. D12C4×D4semidirect
457. C2×C8C2×C4×D3semidirect
484. C22×C3#442semidirect
627. C22#4711semidirect
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
C4×#9302
C2×#47322
C2×#47112
C3×#17492
#699592
#699322
#699282
#698122
#686732
#686712
#214523