Finite Group Databasebeta

← #10826308

Subgroups — #10826308

Order 768.

1956 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
1C23×#214C211yesmaximal, characteristic, complemented
2C2×#5552C211yesmaximal, characteristic, complemented
3C3×#2408C211yesmaximal, characteristic, complemented
4C2×#5217C211yesmaximal, characteristic, complemented
5C2×#5037C212yesmaximal, complemented
6#71685C212yesmaximal, complemented
7#71610C211yesmaximal, characteristic, complemented
8#69660C212yesmaximal, complemented
9#69659C212yesmaximal, complemented
10#69525C211yesmaximal, characteristic, complemented
11#69515C211yesmaximal, characteristic, complemented
12#23858—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic, complemented
15C23×Dic6C2211yescharacteristic
16C2×C3×#121C2212yes—
17C2×C3×#120—21no—
18C2×C3×C4×D4C2211yescharacteristic
19C2×#939—21no—
20C22×#214—24no—
21C22×#214—28no—
22C22×#214C2212yes—
23C22×#214C2212yes—
24C22×#213C2212yes—
25C2×#930C2211yescharacteristic
26C2×#924C2211yescharacteristic, complemented
27C23×C3⋊C8C2211yescharacteristic
28#5552—24no—
29#5552—24no—
30C2×#896—21no—
31C2×#885—21no—
32C2×#67⋊C4—21no—
33C2×#875C2211yescharacteristic, complemented
34C2×C4×Dic6C2211yescharacteristic, complemented
35C3×#394—21no—
36C3×#392—21no—
37C3×#390C2212yes—
38C3×#380C2211yescharacteristic
39C3×#379—21no—
40#5217—24no—
41C2×#832C2212yes—
42C2×#830C2211yescharacteristic
43C2×#830C2212yescomplemented
44#5037—28no—
45C2×#810C2212yescomplemented
46C2×#810C2212yescomplemented
47C2×#809C2212yescomplemented
48#4529C2212yescomplemented
49#4524C2212yescomplemented
50C23×SD16—31no—
51C2×#548—31no—
52C2×#449—32no—
53C2×#447—31no—
54#2408D311yescharacteristic, complemented
55#2012—32no—
56#2000—32no—
57#1893—31no—
58#1880—32no—
59#1869—31no—
60#1787—31no—
61C25×C3—21no—
62C22×C3×D4—24no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4—22no—
66C22×C3×D4—21no—
67C22×C3×D4—42no—
68C22×C3×D4D411yescharacteristic
69C22×C3×D4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C22×#67—21no—
73C22×#67—21no—
74C23×Dic3—21no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235—24no—
78C2×#235—42no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6—24no—
82C22×Dic6—21no—
83C22×Dic6—22no—
84C22×Dic6—42no—
85C22×Dic6D414yescomplemented
86C22×Dic6C2311yescharacteristic
87C22×Dic6C2311yescharacteristic
88C3×#121—24no—
89C3×#121—42no—
90C3×#120—44no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×#217—21no—
103#939—28no—
104C2×#214—232no—
105C2×#214—24no—
106C2×#214—24no—
107C2×#214—24no—
108C2×#214—24no—
109C2×#214—24no—
110C2×#214D418yescomplemented
111C2×#213—24no—
112C2×#213—24no—
113C2×#213—42no—
114#930—22no—
115#930D414yescomplemented
116C2×(C2×D3)⋊C4—21no—
117C2×(C2×D3)⋊C4—21no—
118C2×(C2×D3)⋊C4—21no—
119C2×C4⋊Dic3C2311yescharacteristic
120#924—22no—
121#924—22no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×C4×Dic3—21no—
125C22×C3⋊C8—22no—
126C22×C3⋊C8—24no—
127C22×C3⋊C8C2312yes—
128C22×C3⋊C8C2312yes—
129#896—28no—
130#885—28no—
131#67⋊C4—28no—
132#875—22no—
133#875—22no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C3×#95—21no—
137#832—24no—
138#830—22no—
139#830—24no—
140#810—24no—
141#810—42no—
142#809—24no—
143C22×#41—31no—
144C23×Q8—31no—
145C23×D4C2×D311yescharacteristic
146C22×SD16—32no—
147C22×SD16—32no—
148C22×SD16—68no—
149C22×SD16—64no—
150C22×D8—32no—
151C23×C8—31no—
152#548—64no—
153#548—64no—
154C2×#124—31no—
155C2×#124—61no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8—61no—
158C2×#121—31no—
159C2×#121—61no—
160C2×#121C2×D312yes—
161C2×#120—21no—
162C2×C4×Q8—31no—
163C2×C4×D4—61no—
164C2×C4×D4C2×D311yescharacteristic
165#449—68no—
166#447—64no—
167C2×#103—32no—
168C2×#103—32no—
169C2×#102—32no—
170C2×#102—32no—
171C2×C22⋊C8—31no—
172C2×C22⋊C8—32no—
173#394—21no—
174#392—21no—
175#390C2×D312yes—
176#380C2×D311yescharacteristic
177#379—21no—
178#340—32no—
179#336—32no—
180C24×C3—22no—
181C24×C3—44no—
182C24×C3—42no—
183C24×C3—42no—
184C24×C3—42no—
185C24×C3—82no—
186C24×C3C2×D411yescharacteristic
187C24×C3C2×D411yescharacteristic
188C23×D3—21no—
189C2×C3×D4—28no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—48no—
199C2×C3×D4—44no—
200C2×C3×D4—44no—
201C2×C3×D4—44no—
202C2×C3×D4—44no—
203C2×C3×D4—42no—
204C2×C3×D4—82no—
205C2×C3×D4C2×D414yes—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C2×#67—24no—
221C2×#67—24no—
222C2×#67—44no—
223C2×#67—42no—
224C2×#67—42no—
225C22×Dic3—21no—
226C22×Dic3—24no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—44no—
230#235—48no—
231#235—44no—
232#235—44no—
233C2×D12—22no—
234C2×D12—22no—
235C2×D12—21no—
236C2×D12C2×D412yes—
237C2×C4×D3—24no—
238C2×C4×D3—44no—
239C2×Dic6—21no—
240C2×Dic6—22no—
241C2×Dic6—22no—
242C2×Dic6—22no—
243C2×Dic6—28no—
244C2×Dic6—24no—
245C2×Dic6—21no—
246C2×Dic6—44no—
247C2×Dic6—48no—
248C2×Dic6—44no—
249C2×Dic6C2×D412yes—
250C2×Dic6C2×D414yes—
251C2×Dic6C2×D412yes—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—44no—
258C3×C22⋊C4—44no—
259C3×C22⋊C4—42no—
260C3×C22⋊C4—82no—
261C3×C42—21no—
262C3×C42—21no—
263#217—24no—
264#214—216no—
265#214—216no—
266#213—48no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—24no—
270C4⋊Dic3—21no—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274C4×Dic3—24no—
275C2×C3⋊C8—28no—
276C2×C3⋊C8—22no—
277C2×C3⋊C8—22no—
278C2×C3⋊C8—22no—
279C2×C3⋊C8—22no—
280C2×C3⋊C8—22no—
281C2×C3⋊C8C2×D414yes—
282C25—21no—
283C2×#41—62no—
284C2×#41—62no—
285C2×#41—64no—
286C2×#41—122no—
287C22×Q8—34no—
288C22×Q8—31no—
289C22×Q8—31no—
290C22×Q8—64no—
291C22×Q8—62no—
292C22×Q8—61no—
293C22×Q8—122no—
294C22×D4—24no—
295C22×D4—22no—
296C22×D4—22no—
297C22×D4—22no—
298C22×D4—21no—
299C22×D4—31no—
300C22×D4—42no—
301C22×D4—61no—
302C22×D4—61no—
303C22×D4#6711yescharacteristic
304C22×D4C22×D312yes—
305C23×C4—61no—
306C23×C4—61no—
307C23×C4C22×D311yescharacteristic
308C23×C4C22×D311yescharacteristic
309C2×SD16—38no—
310C2×SD16—64no—
311C2×SD16—64no—
312C2×SD16—64no—
313C2×SD16—64no—
314C2×SD16—632no—
315C2×SD16—64no—
316C2×D8—64no—
317C2×D8—64no—
318C2×D8—122no—
319C22×C8—32no—
320C22×C8—32no—
321C22×C8—64no—
322C22×C8—62no—
323#124—62no—
324#124—62no—
325#124—68no—
326C22⋊Q8—68no—
327C22⋊Q8—62no—
328C22⋊Q8—62no—
329#121—24no—
330#121—34no—
331#121—42no—
332#121—68no—
333#121—62no—
334#120—44no—
335C4×Q8—62no—
336C4×Q8—62no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—68no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4#6711yescharacteristic
344C2×C4⋊C4C22×D312yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4#6711yescharacteristic
354C2×C22⋊C4#6711yescharacteristic
355C2×C42—61no—
356C2×C42C22×D311yescharacteristic
357#103—64no—
358#103—122no—
359#102—64no—
360#102—64no—
361C22⋊C8—64no—
362C22⋊C8—62no—
363#95—21no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—21no—
367C23×C3—24no—
368C23×C3—22no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—22no—
372C23×C3—48no—
373C23×C3—42no—
374C23×C3—42no—
375C23×C3—42no—
376C23×C3—42no—
377C23×C3—42no—
378C23×C3—42no—
379C23×C3—88no—
380C23×C3—88no—
381C23×C3—84no—
382C23×C3—84no—
383C23×C3#12012yes—
384C23×C3#13512yes—
385C23×C3AGL(1,ℤ/8)12yes—
386C23×C3C22×D411yescharacteristic
387C22×D3—22no—
388C22×D3—42no—
389C22×D3—41no—
390C22×D3—41no—
391C22×D3—42no—
392C3×D4—24no—
393C3×D4—24no—
394C3×D4—48no—
395C3×D4—44no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—88no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—24no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—22no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—42no—
413C2×C3×C4—42no—
414C2×C3×C4—42no—
415C2×C3×C4—42no—
416C2×C3×C4#12012yes—
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4H(5,𝔽2)12yes—
421#67—48no—
422#67—48no—
423C2×Dic3—22no—
424C2×Dic3—24no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—48no—
428C2×Dic3—42no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431D12—24no—
432D12—41no—
433D12—41no—
434C4×D3—48no—
435Dic6—22no—
436Dic6—24no—
437Dic6—22no—
438Dic6—22no—
439Dic6—48no—
440Dic6—42no—
441Dic6#12014yescomplemented
442C3⋊C8—24no—
443C3⋊C8—42no—
444C24—22no—
445C24—44no—
446C24—42no—
447C24—42no—
448C24—42no—
449C24—61no—
450C24—82no—
451C24C2×#6711yescharacteristic
452C24C2×#6711yescharacteristic
453#41—128no—
454#41—124no—
455#41—124no—
456C2×Q8—34no—
457C2×Q8—32no—
458C2×Q8—32no—
459C2×Q8—68no—
460C2×Q8—64no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—62no—
464C2×Q8—62no—
465C2×Q8—62no—
466C2×Q8—124no—
467C2×Q8—124no—
468C2×Q8—128no—
469C2×D4—28no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—32no—
479C2×D4—48no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—44no—
484C2×D4—42no—
485C2×D4—64no—
486C2×D4—64no—
487C2×D4—62no—
488C2×D4—61no—
489C2×D4—62no—
490C2×D4—82no—
491C2×D4—124no—
492C2×D4—122no—
493C2×D4—122no—
494C2×D4D3×D414yes—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—64no—
503C22×C4—61no—
504C22×C4—64no—
505C22×C4—61no—
506C22×C4—61no—
507C22×C4—124no—
508C22×C4—124no—
509C22×C4D3×D412yes—
510C22×C4D3×D412yes—
511C22×C4C2×#6711yescharacteristic
512C22×C4C2×#6711yescharacteristic
513C22×C4C2×#6711yescharacteristic
514C22×C4C2×#6711yescharacteristic
515C22×C4C23×D311yescharacteristic
516SD16—616no—
517SD16—616no—
518D8—128no—
519C2×C8—34no—
520C2×C8—68no—
521C2×C8—62no—
522C2×C8—62no—
523C2×C8—62no—
524C2×C8—62no—
525C2×C8—62no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—32no—
530C4⋊C4—64no—
531C4⋊C4—64no—
532C4⋊C4—61no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—64no—
539C22⋊C4—64no—
540C22⋊C4—64no—
541C22⋊C4—64no—
542C22⋊C4—82no—
543C42—21no—
544C42—21no—
545C42—64no—
546C22×C3—22no—
547C22×C3—24no—
548C22×C3—22no—
549C22×C3—24no—
550C22×C3—41no—
551C22×C3—44no—
552C22×C3—41no—
553C22×C3—42no—
554C22×C3—44no—
555C22×C3—41no—
556C22×C3—42no—
557C22×C3—44no—
558C22×C3—816no—
559C22×C3—82no—
560C22×C3—82no—
561C22×C3—82no—
562C22×C3—82no—
563C22×C3#44914yescomplemented
564C22×C3C2×#12011yescharacteristic
565C22×C3D4212yes—
566C22×C3C2×#13511yescharacteristic
567C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
568C22×C3#57712yes—
569C2×D3—42no—
570C2×D3—42no—
571C2×D3—42no—
572C2×D3—82no—
573C2×D3—82no—
574C3×C4—22no—
575C3×C4—41no—
576C3×C4—41no—
577C3×C4—41no—
578C3×C4—82no—
579C3×C4C2×#12011yescharacteristic
580C3×C4#53411yescharacteristic
581C3×C4D4212yes—
582Dic3—24no—
583Dic3—42no—
584Dic3—42no—
585C23—22no—
586C23—22no—
587C23—21no—
588C23—21no—
589C23—24no—
590C23—22no—
591C23—21no—
592C23—22no—
593C23—48no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—62no—
601C23—88no—
602C23—88no—
603C23—84no—
604C23—84no—
605C23—121no—
606C23—122no—
607C23—121no—
608C23—122no—
609C23#93212yes—
610C23#93712yes—
611C23#95012yes—
612C23C22×#6711yescharacteristic
613Q8—34no—
614Q8—62no—
615Q8—64no—
616Q8—62no—
617Q8—62no—
618Q8—128no—
619Q8—122no—
620D4—24no—
621D4—24no—
622D4—48no—
623D4—44no—
624D4—42no—
625D4—42no—
626D4—64no—
627D4—88no—
628D4—128no—
629D4—128no—
630D4—121no—
631D4—121no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—62no—
650C2×C4—64no—
651C2×C4—62no—
652C2×C4—62no—
653C2×C4—128no—
654C2×C4—128no—
655C2×C4—121no—
656C2×C4—122no—
657C2×C4—121no—
658C2×C4#93712yes—
659C2×C4C2×D3×D411yescharacteristic
660C2×C4C2×D3×D412yes—
661C2×C4C2×D3×D411yescharacteristic
662C2×C4#100412yes—
663C8—64no—
664C8—122no—
665C2×C3—24no—
666C2×C3—81no—
667C2×C3—84no—
668C2×C3#240311yescharacteristic
669C2×C3C2×#44911yescharacteristic
670C2×C3#301011yescharacteristic
671C2×C3#326512yes—
672C2×C3#328212yes—
673D3—82no—
674C22—24no—
675C22—24no—
676C22—22no—
677C22—22no—
678C22—42no—
679C22—44no—
680C22—42no—
681C22—44no—
682C22—41no—
683C22—44no—
684C22—41no—
685C22—41no—
686C22—816no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—122no—
692C22—122no—
693C22—122no—
694C22—242no—
695C22—242no—
696C22#503714yescomplemented
697C22C2×#93211yescharacteristic
698C22C2×#93711yescharacteristic
699C22D4×#6712yes—
700C22C2×#95011yescharacteristic
701C22#583612yes—
702C4—22no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—64no—
707C4—82no—
708C4—122no—
709C4—122no—
710C4D3×#12011yescharacteristic
711C4#558911yescharacteristic
712C4#559012yes—
713C3#2385811yescharacteristic, complemented, Sylow
714C2—24no—
715C2—81no—
716C2—84no—
717C2—242no—
718C2#8260411yescharacteristic
719C2C2×#503711yescharacteristic
720C2#8339011yescharacteristic
721C2#8349112yes—
722C2#8350912yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2408C2
socleC23×C3C22×D4
hypercenter#2408D3
hypocenterC3#23858
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C23×#214C2semidirect
2. C2×#5552C2semidirect
3. C3×#2408C2semidirect
4. C2×#5217C2semidirect
5. C2×#5037C2semidirect
6. #71685C2semidirect
7. #71610C2semidirect
8. #69660C2semidirect
9. #69659C2semidirect
10. #69525C2semidirect
11. #69515C2semidirect
14. C22×#235C22semidirect
26. C2×#924C22semidirect
33. C2×#875C22semidirect
34. C2×C4×Dic6C22semidirect
43. C2×#830C22semidirect
45. C2×#810C22semidirect
46. C2×#810C22semidirect
47. C2×#809C22semidirect
48. #4529C22semidirect
49. #4524C22semidirect
54. #2408D3semidirect
85. C22×Dic6D4semidirect
110. C2×#214D4semidirect
115. #930D4semidirect
441. Dic6#120semidirect
563. C22×C3#449semidirect
696. C22#5037semidirect
713. C3#23858semidirect

Maximal subgroups

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

subgroupindex
C23×#2142
C2×#55522
C3×#24082
C2×#52172
C2×#50372
#716852
#716102
#696602
#696592
#695252
#695152
#238583