Finite Group Databasebeta

← #10943980

Subgroups — #10943980

Order 768.

2008 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#893C211yesmaximal, characteristic, complemented
2#82700C211yesmaximal, characteristic, complemented
3C2×#5036C211yesmaximal, characteristic, complemented
4C2×#4786C212yesmaximal, complemented
5C2×#4732C211yesmaximal, characteristic, complemented
6C3×#1874C211yesmaximal, characteristic, complemented
7#71560C211yesmaximal, characteristic, complemented
8#70208C212yesmaximal, complemented
9#69908C211yesmaximal, characteristic, complemented
10#68897C212yesmaximal, complemented
11#68892C212yesmaximal, complemented
12#32902—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3—21no—
15C2×C3×#121C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#940C2211yescharacteristic, complemented
18C2×#939—21no—
19C2×#939—21no—
20C22×#213C2212yes—
21C2×#930C2211yescharacteristic
22C22×(C2×D3)⋊C4—21no—
23C22×C4×Dic3C2211yescharacteristic
24C22×D24C2212yes—
25C2×#893—21no—
26C2×#893—21no—
27C2×#893—28no—
28C2×#893C2212yes—
29C2×#893C2212yes—
30C2×C3×#102C2212yescomplemented
31C2×C3×C22⋊C8C2211yescharacteristic
32#5221C2211yescharacteristic, complemented
33C2×#830C2211yescharacteristic
34C2×#821C2212yes—
35C2×Dic3⋊C8C2212yescomplemented
36#5036—24no—
37#4988C2211yescharacteristic, complemented
38#4975C2211yescharacteristic, complemented
39C2×#809C2212yes—
40C2×#809C2212yescomplemented
41#4957—21no—
42#4957—21no—
43#4786—28no—
44#4732—24no—
45C3×#340C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C22×C4×D4—31no—
49C2×#459—32no—
50C2×#447—31no—
51C2×#447—31no—
52#2856—31no—
53#1879—32no—
54#1879—32no—
55#1874—31no—
56#1874D311yescharacteristic, complemented
57#1802—32no—
58#1797—31no—
59C24×D3—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C22×#67—21no—
66C22×#67—21no—
67C23×Dic3D411yescharacteristic
68C23×Dic3D411yescharacteristic
69C22×D12—21no—
70C22×D12—22no—
71C22×D12—24no—
72C22×D12—24no—
73C22×D12D414yescomplemented
74C22×D12C2311yescharacteristic
75C22×D12C2312yes—
76C22×C4×D3—28no—
77C22×C4×D3—22no—
78C22×C4×D3—22no—
79C22×C4×D3—44no—
80C22×C4×D3—44no—
81C22×C3×C8C2312yes—
82C3×#121—22no—
83C3×#121—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—21no—
89C2×#217—21no—
90C2×#217—21no—
91#940—24no—
92#939—44no—
93#939—44no—
94C2×#213—24no—
95C2×#213—24no—
96C2×#213—24no—
97#930—22no—
98#930—22no—
99C2×(C2×D3)⋊C4—28no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—22no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—24no—
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2312yes—
114C2×D24—24no—
115C2×D24—24no—
116C2×D24—24no—
117#893—28no—
118#893—28no—
119#893—24no—
120#893—24no—
121#893—24no—
122#893D418yescomplemented
123C3×#102—24no—
124C3×C22⋊C8—22no—
125#830—22no—
126#821—24no—
127Dic3⋊C8—24no—
128#809—24no—
129#809—24no—
130C23×D4—31no—
131C24×C4—61no—
132C22×D8—32no—
133C22×D8—32no—
134C2×#127—31no—
135C2×#121—31no—
136C2×#121—61no—
137C2×#121—61no—
138C2×#121C2×D311yescharacteristic
139C2×C4×D4—32no—
140C2×C4×D4—32no—
141C2×C4×D4—61no—
142C2×C4×D4—61no—
143C2×C4×D4—68no—
144C22×C4⋊C4C2×D311yescharacteristic
145C22×C22⋊C4—61no—
146C22×C42—31no—
147#459—68no—
148#447—64no—
149#447—64no—
150C2×C4⋊C8—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×#102C2×D312yescomplemented
155C2×C22⋊C8—31no—
156C2×C22⋊C8C2×D311yescharacteristic
157#390—31no—
158#390—31no—
159#390—61no—
160#390—61no—
161#384—31no—
162#340—32no—
163#340C2×D312yescomplemented
164#336—32no—
165C24×C3—21no—
166C24×C3C2×D411yescharacteristic
167C23×D3—24no—
168C23×D3—24no—
169C23×D3—24no—
170C23×D3—22no—
171C23×D3—21no—
172C23×D3—42no—
173C23×D3—42no—
174C23×D3—42no—
175C23×D3—42no—
176C2×C3×D4—22no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D412yes—
192C22×C3×C4#4112yes—
193C22×C3×C4C2411yescharacteristic
194C2×#67—42no—
195C2×#67—42no—
196C2×#67—42no—
197C2×#67—42no—
198C2×#67—42no—
199C2×#67—42no—
200C2×#67—42no—
201C2×#67—42no—
202C2×#67—42no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—21no—
208C22×Dic3—41no—
209C22×Dic3D814yescomplemented
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—24no—
215C2×D12—24no—
216C2×D12—216no—
217C2×D12—24no—
218C2×D12—22no—
219C2×D12—22no—
220C2×D12—22no—
221C2×D12—216no—
222C2×D12—22no—
223C2×D12—21no—
224C2×D12—22no—
225C2×D12—21no—
226C2×D12C2×D414yes—
227C2×D12C2×D414yes—
228C2×D12#4114yes—
229C2×C4×D3—28no—
230C2×C4×D3—28no—
231C2×C4×D3—44no—
232C2×C4×D3—432no—
233C2×C4×D3—44no—
234C2×C4×D3—44no—
235C2×C4×D3—44no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C2×C3×C8—22no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4C2×D414yes—
244C3×C22⋊C4—42no—
245#217—42no—
246#217—42no—
247#213—48no—
248(C2×D3)⋊C4—28no—
249(C2×D3)⋊C4—28no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257C4×Dic3—22no—
258C4×Dic3—22no—
259C4×Dic3—22no—
260C4×Dic3C2×D414yes—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—22no—
264D24—48no—
265C25—61no—
266C22×D4—21no—
267C22×D4—34no—
268C22×D4—32no—
269C22×D4—31no—
270C22×D4—64no—
271C22×D4—64no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4—61no—
275C22×D4—62no—
276C22×D4—61no—
277C22×D4C22×D311yescharacteristic
278C23×C4—21no—
279C23×C4—31no—
280C23×C4—31no—
281C23×C4—62no—
282C23×C4—62no—
283C23×C4—68no—
284C23×C4—124no—
285C23×C4—124no—
286C23×C4C22×D311yescharacteristic
287C2×D8—64no—
288C2×D8—64no—
289C2×D8—64no—
290C2×D8—64no—
291C2×D8—64no—
292C2×D8—64no—
293C22×C8—32no—
294C22×C8C22×D312yes—
295#127—64no—
296#121—22no—
297#121—22no—
298#121—62no—
299#121—62no—
300#121—124no—
301#121—124no—
302C4×D4—38no—
303C4×D4—64no—
304C4×D4—64no—
305C4×D4—68no—
306C4×D4—68no—
307C4×D4—64no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D312yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—62no—
318C2×C22⋊C4—68no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—64no—
329C4⋊C8—64no—
330#102—24no—
331#102—64no—
332#102—64no—
333#102—64no—
334C22⋊C8—22no—
335C22⋊C8—62no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3#12112yes—
348C23×C3C2×D812yes—
349C23×C3AGL(1,ℤ/8)12yes—
350C23×C3C22×D411yescharacteristic
351C22×D3—216no—
352C22×D3—24no—
353C22×D3—24no—
354C22×D3—24no—
355C22×D3—48no—
356C22×D3—48no—
357C22×D3—48no—
358C22×D3—48no—
359C22×D3—48no—
360C22×D3—42no—
361C22×D3—42no—
362C22×D3—42no—
363C22×D3—42no—
364C22×D3—48no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C3×D4—42no—
373C3×D4—42no—
374C3×D4—84no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—24no—
380C2×C3×C4—48no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4#12112yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×D412yes—
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4112yes—
391#67—84no—
392#67—84no—
393#67—84no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—28no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—44no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3#12012yes—
407C2×Dic3#12012yes—
408C2×Dic3C2×D812yes—
409C2×Dic3C2×D812yes—
410C2×Dic3AGL(1,ℤ/8)12yes—
411D12—216no—
412D12—24no—
413D12—24no—
414D12—24no—
415D12—42no—
416D12—42no—
417D12#12118yescomplemented
418C4×D3—416no—
419C4×D3—416no—
420C3×C8—42no—
421C3⋊C8—42no—
422C24—21no—
423C24—64no—
424C24—64no—
425C24—64no—
426C24—62no—
427C24—61no—
428C24—122no—
429C24—122no—
430C24—122no—
431C24—122no—
432C24D3×D411yescharacteristic
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—34no—
438C2×D4—34no—
439C2×D4—34no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—616no—
444C2×D4—64no—
445C2×D4—64no—
446C2×D4—616no—
447C2×D4—64no—
448C2×D4—62no—
449C2×D4—61no—
450C2×D4—62no—
451C2×D4—62no—
452C2×D4—61no—
453C2×D4—62no—
454C2×D4—62no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4—122no—
460C2×D4—122no—
461C2×D4—122no—
462C2×D4—122no—
463C2×D4—122no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—31no—
470C22×C4—34no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—68no—
477C22×C4—68no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—61no—
483C22×C4—124no—
484C22×C4—124no—
485C22×C4—124no—
486C22×C4—1232no—
487C22×C4—124no—
488C22×C4—121no—
489C22×C4#23512yes—
490C22×C4D3×D411yescharacteristic
491C22×C4D3×D412yes—
492C22×C4C23×D311yescharacteristic
493D8—128no—
494D8—128no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—62no—
499C2×C8—62no—
500C2×C8—62no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4D3×D414yes—
510C22⋊C4—42no—
511C22⋊C4—68no—
512C22⋊C4—68no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C42—34no—
519C42—62no—
520C42—62no—
521C42—62no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3#45914yescomplemented
536C22×C3C2×#12011yescharacteristic
537C22×C3C2×#12111yescharacteristic
538C22×C3#52511yescharacteristic
539C22×C3C22×D811yescharacteristic
540C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
541C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
542C22×C3#57511yescharacteristic
543C2×D3—28no—
544C2×D3—432no—
545C2×D3—44no—
546C2×D3—44no—
547C2×D3—44no—
548C2×D3—44no—
549C2×D3—44no—
550C2×D3—44no—
551C2×D3—81no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C3×C4—22no—
559C3×C4—44no—
560C3×C4—81no—
561C3×C4C2×#12112yes—
562C3×C4#52512yes—
563Dic3—22no—
564Dic3—22no—
565Dic3—41no—
566Dic3—81no—
567Dic3#44714yescomplemented
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—616no—
580C23—64no—
581C23—64no—
582C23—64no—
583C23—128no—
584C23—128no—
585C23—128no—
586C23—128no—
587C23—128no—
588C23—122no—
589C23—122no—
590C23—122no—
591C23—122no—
592C23—128no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23#88412yes—
601C23D3×D812yes—
602C23#91412yes—
603C23C2×D3×D411yescharacteristic
604D4—38no—
605D4—42no—
606D4—42no—
607D4—616no—
608D4—64no—
609D4—64no—
610D4—64no—
611D4—84no—
612D4—122no—
613D4—122no—
614D4—244no—
615D4—244no—
616D4—244no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—32no—
623C2×C4—32no—
624C2×C4—32no—
625C2×C4—32no—
626C2×C4—32no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—68no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—1216no—
642C2×C4—1216no—
643C2×C4—124no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4#88412yes—
648C2×C4C2×#23511yescharacteristic
649C2×C4C2×D3×D412yes—
650C2×C4C2×D3×D411yescharacteristic
651C2×C4D3×#4112yes—
652C8—42no—
653C8—122no—
654C2×C3—24no—
655C2×C3—161no—
656C2×C3#284711yescharacteristic
657C2×C3C2×#44711yescharacteristic
658C2×C3#300611yescharacteristic
659C2×C3C2×#45911yescharacteristic
660C2×C3#303611yescharacteristic
661C2×C3#310011yescharacteristic
662C2×C3#310911yescharacteristic
663D3—48no—
664D3—161no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—68no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—1232no—
680C22—124no—
681C22—124no—
682C22—124no—
683C22—124no—
684C22—124no—
685C22—124no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22—241no—
691C22—241no—
692C22—241no—
693C22#478614yescomplemented
694C22C2×#88411yescharacteristic
695C22D3×#12011yescharacteristic
696C22#565611yescharacteristic
697C22C2×D3×D811yescharacteristic
698C22C2×#91411yescharacteristic
699C22D3×AGL(1,ℤ/8)11yescharacteristic
700C22#577711yescharacteristic
701C4—22no—
702C4—34no—
703C4—44no—
704C4—62no—
705C4—62no—
706C4—81no—
707C4—121no—
708C4—241no—
709C4#554712yes—
710C4D3×#12112yes—
711C3#3290211yescharacteristic, complemented, Sylow
712C2—24no—
713C2—128no—
714C2—161no—
715C2—481no—
716C2#7493511yescharacteristic
717C2C2×#478611yescharacteristic
718C2#7891011yescharacteristic
719C2D3×#44711yescharacteristic
720C2#8004011yescharacteristic
721C2#8126811yescharacteristic
722C2#8131011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1874C2
socleC23×C3C22×D4
hypercenter#1874D3
hypocenterC3#32902
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#893C2semidirect
2. #82700C2semidirect
3. C2×#5036C2semidirect
4. C2×#4786C2semidirect
5. C2×#4732C2semidirect
6. C3×#1874C2semidirect
7. #71560C2semidirect
8. #70208C2semidirect
9. #69908C2semidirect
10. #68897C2semidirect
11. #68892C2semidirect
17. C2×#940C22semidirect
30. C2×C3×#102C22semidirect
32. #5221C22semidirect
35. C2×Dic3⋊C8C22semidirect
37. #4988C22semidirect
38. #4975C22semidirect
40. C2×#809C22semidirect
45. C3×#340C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
56. #1874D3semidirect
73. C22×D12D4semidirect
122. #893D4semidirect
154. C2×#102C2×D3semidirect
163. #340C2×D3semidirect
209. C22×Dic3D8semidirect
417. D12#121semidirect
535. C22×C3#459semidirect
567. Dic3#447semidirect
693. C22#4786semidirect
711. C3#32902semidirect

Maximal subgroups

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

subgroupindex
C22×#8932
#827002
C2×#50362
C2×#47862
C2×#47322
C3×#18742
#715602
#702082
#699082
#688972
#688922
#329023