Finite Group Databasebeta

← #10826305

Subgroups — #10826305

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×C3×SD16C211yesmaximal, characteristic, complemented
2C2×#5817C211yesmaximal, characteristic, complemented
3#84053C211yesmaximal, characteristic, complemented
4C2×#5217C211yesmaximal, characteristic, complemented
5C2×#5172C212yesmaximal, complemented
6#72132C211yesmaximal, characteristic, complemented
7#72055C212yesmaximal, complemented
8#72054C212yesmaximal, complemented
9#72048C212yesmaximal, complemented
10#71975C211yesmaximal, characteristic, complemented
11#71973C211yesmaximal, characteristic, complemented
12#23858—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×#239C2211yescharacteristic, complemented
16C22×C3×SD16—24no—
17C22×C3×SD16—28no—
18C22×C3×SD16C2212yes—
19C22×C3×SD16C2212yes—
20C23×C3×C8C2211yescharacteristic
21C2×#953—21no—
22#5817—24no—
23#5817—24no—
24C2×#947—21no—
25C2×#947C2211yescharacteristic, complemented
26C2×#946—21no—
27C2×#946C2211yescharacteristic, complemented
28C2×Q8×Dic3C2211yescharacteristic, complemented
29C2×#938C2212yes—
30C22×#213C2212yes—
31C2×#930—21no—
32C2×#930C2211yescharacteristic
33C2×C4×#67—21no—
34C2×C4×#67C2211yescharacteristic
35#5247—21no—
36C2×#835C2212yescomplemented
37C2×#835C2212yescomplemented
38#5223—21no—
39#5222—21no—
40#5221C2212yes—
41#5217—24no—
42C2×#832C2212yes—
43#5212C2211yescharacteristic
44C2×#830C2211yescharacteristic
45#5172—28no—
46C2×#821C2212yescomplemented
47C2×(C2×D3)⋊C8C2212yescomplemented
48#4551C2212yescomplemented
49#4550C2212yescomplemented
50C23×SD16D311yescharacteristic, complemented
51C2×#548—31no—
52C2×#449—32no—
53C2×#447—31no—
54#2408—31no—
55#2012—32no—
56#2000—32no—
57#1893—31no—
58#1880—32no—
59#1869—31no—
60#1787—31no—
61C25×C3—21no—
62C22×C3×Q8—21no—
63C22×C3×Q8—22no—
64C22×C3×Q8—24no—
65C22×C3×Q8—42no—
66C22×C3×Q8D414yescomplemented
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—24no—
70C22×C3×D4—22no—
71C22×C3×D4—22no—
72C22×C3×D4—21no—
73C22×C3×D4—42no—
74C22×C3×D4C2312yes—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C22×#67—22no—
79C22×#67D411yescharacteristic
80C2×#239—22no—
81C2×#239—22no—
82C2×#239—24no—
83C2×#239—42no—
84C22×D12—21no—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×C4×D3C2311yescharacteristic
88C2×C3×SD16—232no—
89C2×C3×SD16—24no—
90C2×C3×SD16—24no—
91C2×C3×SD16—24no—
92C2×C3×SD16—24no—
93C2×C3×SD16—24no—
94C2×C3×SD16D418yescomplemented
95C22×C3×C8—22no—
96C22×C3×C8—24no—
97C22×C3×C8C2312yes—
98#953—44no—
99C2×#217—21no—
100C2×#217—22no—
101C2×#217—22no—
102C2×#217—22no—
103C2×#217D411yescharacteristic
104#947—28no—
105#947—22no—
106#947—22no—
107#946—28no—
108#946—22no—
109#946—22no—
110Q8×Dic3—22no—
111Q8×Dic3—22no—
112#938—24no—
113#938—42no—
114C2×#213—24no—
115C2×#213—24no—
116C2×#213—42no—
117#930—28no—
118#930—22no—
119#930D414yescomplemented
120C4×#67—28no—
121C4×#67—22no—
122C4×#67—22no—
123C2×(C2×D3)⋊C4—21no—
124C2×(C2×D3)⋊C4—21no—
125C2×(C2×D3)⋊C4—21no—
126C2×(C2×D3)⋊C4—21no—
127C2×(C2×D3)⋊C4D411yescharacteristic
128C2×C4⋊Dic3—21no—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×Dic3⋊C4—21no—
132C2×Dic3⋊C4D411yescharacteristic
133C2×C4×Dic3—21no—
134C2×C4×Dic3C2311yescharacteristic
135C22×C3⋊C8C2312yes—
136#835—24no—
137#835—42no—
138#832—24no—
139C3⋊#95—21no—
140#830—22no—
141#821—24no—
142(C2×D3)⋊C8—24no—
143C22×#41—31no—
144C23×Q8C2×D311yescharacteristic
145C23×D4C2×D311yescharacteristic
146C22×SD16—24no—
147C22×SD16—28no—
148C22×SD16C2×D312yes—
149C22×SD16C2×D312yes—
150C22×D8—32no—
151C23×C8C2×D311yescharacteristic
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—32no—
160C2×#121—61no—
161C2×#120—61no—
162C2×C4×Q8—31no—
163C2×C4×D4—31no—
164C2×C4×D4—61no—
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—61no—
174#392—61no—
175#390—32no—
176#380—31no—
177#379—61no—
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
187C23×D3—21no—
188C23×D3C2×D411yescharacteristic
189C2×C3×Q8—24no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—22no—
193C2×C3×Q8—28no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—44no—
197C2×C3×Q8—44no—
198C2×C3×Q8—48no—
199C2×C3×Q8C2×D412yes—
200C2×C3×Q8C2×D414yes—
201C2×C3×Q8C2×D412yes—
202C2×C3×D4—28no—
203C2×C3×D4—22no—
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—22no—
207C2×C3×D4—22no—
208C2×C3×D4—48no—
209C2×C3×D4—44no—
210C2×C3×D4—44no—
211C2×C3×D4—42no—
212C2×C3×D4C2×D414yes—
213C22×C3×C4—21no—
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—24no—
217C22×C3×C4—21no—
218C22×C3×C4—44no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D412yes—
222C22×C3×C4C2411yescharacteristic
223C2×#67—22no—
224C2×#67—24no—
225C2×#67—22no—
226C2×#67—22no—
227C2×#67—42no—
228C2×#67—42no—
229C2×#67—44no—
230C2×#67—44no—
231C2×#67—82no—
232C22×Dic3—21no—
233C22×Dic3—22no—
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237#239—48no—
238#239—44no—
239#239—44no—
240C2×D12—22no—
241C2×D12—22no—
242C2×D12—24no—
243C2×D12—21no—
244C2×D12—44no—
245C2×D12C2×D412yes—
246C2×C4×D3—24no—
247C2×C4×D3—22no—
248C2×C4×D3—21no—
249C2×C4×D3—21no—
250C2×C4×D3—22no—
251C2×C4×D3—44no—
252C3×SD16—216no—
253C3×SD16—216no—
254C2×C3×C8—28no—
255C2×C3×C8—22no—
256C2×C3×C8—22no—
257C2×C3×C8C2×D414yes—
258#217—22no—
259#217—44no—
260#217—44no—
261#217—42no—
262#217—82no—
263#213—48no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—22no—
269C4⋊Dic3—24no—
270C4⋊Dic3—21no—
271C4⋊Dic3—22no—
272C4⋊Dic3—22no—
273C4⋊Dic3C2×D412yes—
274Dic3⋊C4—24no—
275Dic3⋊C4—22no—
276C4×Dic3—24no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C2×C3⋊C8—22no—
280C2×C3⋊C8—22no—
281C2×C3⋊C8—22no—
282C25—21no—
283C2×#41—62no—
284C2×#41—62no—
285C2×#41—64no—
286C2×#41—122no—
287C22×Q8—21no—
288C22×Q8—22no—
289C22×Q8—24no—
290C22×Q8—42no—
291C22×Q8#6714yescomplemented
292C22×Q8C22×D311yescharacteristic
293C22×Q8C22×D311yescharacteristic
294C22×D4—24no—
295C22×D4—22no—
296C22×D4—21no—
297C22×D4—22no—
298C22×D4—31no—
299C22×D4—31no—
300C22×D4—42no—
301C22×D4—61no—
302C22×D4—61no—
303C22×D4—62no—
304C22×D4C22×D312yes—
305C23×C4—21no—
306C23×C4—31no—
307C23×C4—61no—
308C23×C4C22×D311yescharacteristic
309C2×SD16—232no—
310C2×SD16—24no—
311C2×SD16—24no—
312C2×SD16—24no—
313C2×SD16—24no—
314C2×SD16—24no—
315C2×SD16#6718yescomplemented
316C2×D8—64no—
317C2×D8—64no—
318C2×D8—122no—
319C22×C8—22no—
320C22×C8—24no—
321C22×C8—32no—
322C22×C8C22×D312yes—
323#124—62no—
324#124—68no—
325#124—62no—
326C22⋊Q8—62no—
327C22⋊Q8—68no—
328C22⋊Q8—62no—
329#121—34no—
330#121—68no—
331#121—62no—
332#121—64no—
333#121—122no—
334#120—124no—
335C4×Q8—62no—
336C4×Q8—62no—
337C4×D4—68no—
338C4×D4—62no—
339C4×D4—62no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C22⋊C4—31no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—62no—
352C2×C22⋊C4—62no—
353C2×C22⋊C4—61no—
354C2×C22⋊C4—62no—
355C2×C42—31no—
356C2×C42—61no—
357#103—64no—
358#103—122no—
359#102—64no—
360#102—64no—
361C22⋊C8—62no—
362C22⋊C8—64no—
363#95—61no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—24no—
367C23×C3—48no—
368C23×C3—42no—
369C23×C3—42no—
370C23×C3—42no—
371C23×C3—42no—
372C23×C3—42no—
373C23×C3—42no—
374C23×C3—88no—
375C23×C3—88no—
376C23×C3—84no—
377C23×C3—84no—
378C23×C3#12012yes—
379C23×C3#13512yes—
380C23×C3AGL(1,ℤ/8)12yes—
381C23×C3C22×D411yescharacteristic
382C22×D3—21no—
383C22×D3—22no—
384C22×D3—22no—
385C22×D3—21no—
386C22×D3—21no—
387C22×D3—22no—
388C22×D3—42no—
389C22×D3—41no—
390C22×D3—41no—
391C22×D3—42no—
392C3×Q8—22no—
393C3×Q8—24no—
394C3×Q8—22no—
395C3×Q8—22no—
396C3×Q8—48no—
397C3×Q8—42no—
398C3×Q8#12014yescomplemented
399C3×D4—24no—
400C3×D4—24no—
401C3×D4—48no—
402C3×D4—42no—
403C3×D4—42no—
404C2×C3×C4—22no—
405C2×C3×C4—24no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—24no—
410C2×C3×C4—22no—
411C2×C3×C4—48no—
412C2×C3×C4—42no—
413C2×C3×C4#12012yes—
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C22×D412yes—
417C2×C3×C4H(5,𝔽2)12yes—
418#67—44no—
419#67—48no—
420#67—88no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—22no—
424C2×Dic3—21no—
425C2×Dic3—22no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—22no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—42no—
432C2×Dic3—42no—
433C2×Dic3—42no—
434D12—24no—
435D12—48no—
436D12—41no—
437D12—41no—
438C4×D3—48no—
439C4×D3—41no—
440C4×D3—41no—
441C4×D3—42no—
442C3×C8—24no—
443C3⋊C8—42no—
444C24—22no—
445C24—31no—
446C24—44no—
447C24—42no—
448C24—42no—
449C24—42no—
450C24—61no—
451C24—82no—
452C24D3×D411yescharacteristic
453#41—128no—
454#41—124no—
455#41—124no—
456C2×Q8—24no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—28no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—44no—
464C2×Q8—44no—
465C2×Q8—48no—
466C2×Q8C2×#6712yes—
467C2×Q8C2×#6714yes—
468C2×Q8C2×#6712yes—
469C2×D4—28no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—32no—
476C2×D4—48no—
477C2×D4—44no—
478C2×D4—44no—
479C2×D4—42no—
480C2×D4—64no—
481C2×D4—62no—
482C2×D4—62no—
483C2×D4—64no—
484C2×D4—62no—
485C2×D4—62no—
486C2×D4—62no—
487C2×D4—61no—
488C2×D4—124no—
489C2×D4—122no—
490C2×D4—122no—
491C2×D4—124no—
492C2×D4—124no—
493C2×D4—242no—
494C2×D4C2×#6714yes—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—31no—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—44no—
504C22×C4—64no—
505C22×C4—61no—
506C22×C4—61no—
507C22×C4—62no—
508C22×C4—62no—
509C22×C4—61no—
510C22×C4—62no—
511C22×C4—124no—
512C22×C4D3×D411yescharacteristic
513C22×C4D3×D412yes—
514C22×C4C2×#6712yes—
515C22×C4C23×D311yescharacteristic
516SD16—216no—
517SD16—216no—
518D8—128no—
519C2×C8—28no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—62no—
523C2×C8—62no—
524C2×C8—62no—
525C2×C8C2×#6714yes—
526C4⋊C4—32no—
527C4⋊C4—64no—
528C4⋊C4—64no—
529C4⋊C4—62no—
530C4⋊C4—61no—
531C4⋊C4—62no—
532C4⋊C4—62no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C22⋊C4—64no—
536C22⋊C4—62no—
537C22⋊C4—62no—
538C22⋊C4—64no—
539C22⋊C4—124no—
540C22⋊C4—124no—
541C22⋊C4—122no—
542C22⋊C4—242no—
543C42—64no—
544C42—61no—
545C42—61no—
546C22×C3—22no—
547C22×C3—24no—
548C22×C3—22no—
549C22×C3—24no—
550C22×C3—44no—
551C22×C3—44no—
552C22×C3—44no—
553C22×C3—816no—
554C22×C3—82no—
555C22×C3—82no—
556C22×C3—82no—
557C22×C3—82no—
558C22×C3#44914yescomplemented
559C22×C3C2×#12011yescharacteristic
560C22×C3D4212yes—
561C22×C3C2×#13511yescharacteristic
562C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
563C22×C3#57712yes—
564C2×D3—42no—
565C2×D3—41no—
566C2×D3—41no—
567C2×D3—42no—
568C2×D3—42no—
569C2×D3—41no—
570C2×D3—42no—
571C2×D3—42no—
572C2×D3—82no—
573C2×D3—82no—
574C3×C4—24no—
575C3×C4—22no—
576C3×C4—42no—
577C3×C4C2×#12011yescharacteristic
578C3×C4#53411yescharacteristic
579C3×C4D4212yes—
580Dic3—41no—
581Dic3—41no—
582Dic3—42no—
583Dic3—41no—
584Dic3—82no—
585C23—22no—
586C23—22no—
587C23—24no—
588C23—48no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—61no—
596C23—61no—
597C23—62no—
598C23—62no—
599C23—61no—
600C23—62no—
601C23—88no—
602C23—88no—
603C23—84no—
604C23—84no—
605C23—121no—
606C23—121no—
607C23—122no—
608C23—122no—
609C23#91412yes—
610C23#91612yes—
611C23#93712yes—
612C23C2×D3×D411yescharacteristic
613Q8—24no—
614Q8—22no—
615Q8—22no—
616Q8—22no—
617Q8—48no—
618Q8—42no—
619Q8#95314yescomplemented
620D4—24no—
621D4—24no—
622D4—48no—
623D4—42no—
624D4—42no—
625D4—64no—
626D4—128no—
627D4—124no—
628D4—128no—
629D4—121no—
630D4—121no—
631D4—248no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—48no—
640C2×C4—42no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—62no—
644C2×C4—61no—
645C2×C4—62no—
646C2×C4—61no—
647C2×C4—61no—
648C2×C4—62no—
649C2×C4—128no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—122no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4—122no—
658C2×C4#93712yes—
659C2×C4C2×D3×D411yescharacteristic
660C2×C4#100412yes—
661C2×C4C22×#6712yes—
662C2×C4C22×#6711yescharacteristic
663C8—24no—
664C8—122no—
665C2×C3—24no—
666C2×C3—84no—
667C2×C3#240311yescharacteristic
668C2×C3C2×#44911yescharacteristic
669C2×C3#301011yescharacteristic
670C2×C3#326512yes—
671C2×C3#328212yes—
672D3—81no—
673D3—82no—
674C22—22no—
675C22—22no—
676C22—24no—
677C22—24no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—816no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—122no—
687C22—121no—
688C22—121no—
689C22—122no—
690C22—121no—
691C22—122no—
692C22—122no—
693C22—122no—
694C22—242no—
695C22—242no—
696C22#517214yescomplemented
697C22C2×#91411yescharacteristic
698C22C2×#91611yescharacteristic
699C22#576212yes—
700C22C2×#93711yescharacteristic
701C22D4×#6712yes—
702C4—22no—
703C4—24no—
704C4—42no—
705C4—121no—
706C4—121no—
707C4—122no—
708C4—121no—
709C4—242no—
710C4D4×#6712yes—
711C4#580411yescharacteristic
712C4C2×#95311yescharacteristic
713C3#2385811yescharacteristic, complemented, Sylow
714C2—24no—
715C2—84no—
716C2—241no—
717C2—242no—
718C2C2×#517211yescharacteristic
719C2#8386011yescharacteristic
720C2#8386412yes—
721C2#8386712yes—
722C2#8405211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×SD16C2
socleC23×C3C22×D4
hypercenterC23×SD16D3
hypocenterC3#23858
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×SD16

Complements

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

NH ≅ G/Ntype
1. C23×C3×SD16C2semidirect
2. C2×#5817C2semidirect
3. #84053C2semidirect
4. C2×#5217C2semidirect
5. C2×#5172C2semidirect
6. #72132C2semidirect
7. #72055C2semidirect
8. #72054C2semidirect
9. #72048C2semidirect
10. #71975C2semidirect
11. #71973C2semidirect
15. C22×#239C22semidirect
25. C2×#947C22semidirect
27. C2×#946C22semidirect
28. C2×Q8×Dic3C22semidirect
36. C2×#835C22semidirect
37. C2×#835C22semidirect
46. C2×#821C22semidirect
47. C2×(C2×D3)⋊C8C22semidirect
48. #4551C22semidirect
49. #4550C22semidirect
50. C23×SD16D3semidirect
66. C22×C3×Q8D4semidirect
94. C2×C3×SD16D4semidirect
119. #930D4semidirect
291. C22×Q8#67semidirect
315. C2×SD16#67semidirect
398. C3×Q8#120semidirect
558. C22×C3#449semidirect
619. Q8#953semidirect
696. C22#5172semidirect
713. C3#23858semidirect

Maximal subgroups

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

subgroupindex
C23×C3×SD162
C2×#58172
#840532
C2×#52172
C2×#51722
#721322
#720552
#720542
#720482
#719752
#719732
#238583