Finite Group Databasebeta

← #10826312

Subgroups — #10826312

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
1C2×#5817C211yesmaximal, characteristic, complemented
2C23×#215C211yesmaximal, characteristic, complemented
3C2×#4810C212yesmaximal, complemented
4C2×#4732C211yesmaximal, characteristic, complemented
5#74016C211yesmaximal, characteristic, complemented
6C3×#1787C211yesmaximal, characteristic, complemented
7#70277C212yesmaximal, complemented
8#70272C212yesmaximal, complemented
9#70240C212yesmaximal, complemented
10#69952C211yesmaximal, characteristic, complemented
11#69515C211yesmaximal, characteristic, complemented
12#23858—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#239C2211yescharacteristic, complemented
15C23×D12C2211yescharacteristic
16#5817—24no—
17#5817—24no—
18C2×#947—21no—
19C2×#947C2211yescharacteristic, complemented
20C2×#946—21no—
21C2×#946C2211yescharacteristic, complemented
22C2×Q8×Dic3C2211yescharacteristic, complemented
23C22×#215—24no—
24C22×#215—28no—
25C22×#215C2212yes—
26C22×#215C2212yes—
27C2×#930—21no—
28C2×#930C2211yescharacteristic
29C2×C4×#67—21no—
30C2×C4×#67C2211yescharacteristic
31C23×C3⋊C8C2211yescharacteristic
32C22×D24C2212yes—
33C2×#895C2212yes—
34C2×#882—21no—
35C2×C3×#103C2212yescomplemented
36C2×C3×#103C2212yescomplemented
37C2×C3×C22⋊C8C2211yescharacteristic
38C3×#380C2211yescharacteristic
39C2×#821C2212yes—
40C2×(C2×D3)⋊C8C2212yescomplemented
41#4988C2212yes—
42C2×#809C2212yescomplemented
43#4955—21no—
44#4810—28no—
45#4732—24no—
46#4672—21no—
47#4671—21no—
48#4550C2212yescomplemented
49#4529C2212yescomplemented
50C23×SD16—31no—
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#1787D311yescharacteristic, complemented
61C24×D3—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
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C22×#67—21no—
72C22×#67D411yescharacteristic
73C2×#239—22no—
74C2×#239—22no—
75C2×#239—24no—
76C2×#239—42no—
77C22×D12—22no—
78C22×D12—24no—
79C22×D12—21no—
80C22×D12—22no—
81C22×D12—21no—
82C22×D12—22no—
83C22×D12—42no—
84C22×D12C2312yes—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×C4×D3C2311yescharacteristic
88C22×C3×C8C2312yes—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C22⋊C4—21no—
91C2×#217D411yescharacteristic
92#947—28no—
93#947—22no—
94#947—22no—
95#946—28no—
96#946—22no—
97#946—22no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100C2×#215—232no—
101C2×#215—24no—
102C2×#215—24no—
103C2×#215—24no—
104C2×#215—24no—
105C2×#215—24no—
106C2×#215D418yescomplemented
107#930—28no—
108#930—22no—
109#930D414yescomplemented
110C4×#67—28no—
111C4×#67—22no—
112C4×#67—22no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×(C2×D3)⋊C4—22no—
118C2×(C2×D3)⋊C4—22no—
119C2×(C2×D3)⋊C4—22no—
120C2×(C2×D3)⋊C4D411yescharacteristic
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3C2311yescharacteristic
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4D411yescharacteristic
125C2×C4×Dic3—21no—
126C2×C4×Dic3C2311yescharacteristic
127C22×C3⋊C8—22no—
128C22×C3⋊C8—24no—
129C22×C3⋊C8C2312yes—
130C2×D24—24no—
131C2×D24—24no—
132C2×D24—42no—
133#895—24no—
134#895—42no—
135#882—44no—
136C3×#103—24no—
137C3×#103—42no—
138C3×C22⋊C8—22no—
139C3×#95—21no—
140#821—24no—
141(C2×D3)⋊C8—24no—
142#809—24no—
143C22×#41—31no—
144C23×Q8C2×D311yescharacteristic
145C23×D4—31no—
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—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×#103C2×D312yescomplemented
168C2×#103C2×D312yescomplemented
169C2×#102—32no—
170C2×#102—32no—
171C2×C22⋊C8—32no—
172C2×C22⋊C8C2×D311yescharacteristic
173#394—61no—
174#392—61no—
175#390—32no—
176#380C2×D311yescharacteristic
177#379—61no—
178#340—32no—
179#336—32no—
180C24×C3C2×D411yescharacteristic
181C23×D3—21no—
182C23×D3—22no—
183C23×D3—44no—
184C23×D3—42no—
185C23×D3—42no—
186C23×D3—42no—
187C23×D3—82no—
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×D412yes—
201C2×C3×Q8C2×D414yes—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—24no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—44no—
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2411yescharacteristic
213C2×#67—22no—
214C2×#67—24no—
215C2×#67—22no—
216C2×#67—22no—
217C2×#67—42no—
218C2×#67—42no—
219C22×Dic3—21no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223#239—48no—
224#239—44no—
225#239—44no—
226C2×D12—28no—
227C2×D12—22no—
228C2×D12—22no—
229C2×D12—22no—
230C2×D12—22no—
231C2×D12—22no—
232C2×D12—22no—
233C2×D12—24no—
234C2×D12—21no—
235C2×D12—22no—
236C2×D12—48no—
237C2×D12—44no—
238C2×D12—44no—
239C2×D12—44no—
240C2×D12—42no—
241C2×D12—44no—
242C2×D12—44no—
243C2×D12—82no—
244C2×D12C2×D414yes—
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—
252C2×C3×C8—22no—
253C2×C3×C8—22no—
254C2×C3×C8—22no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—22no—
257C3×C22⋊C4—42no—
258#217—22no—
259#215—216no—
260#215—216no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—22no—
266(C2×D3)⋊C4—44no—
267(C2×D3)⋊C4—44no—
268(C2×D3)⋊C4—82no—
269C4⋊Dic3—24no—
270C4⋊Dic3—21no—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—22no—
274C4×Dic3—24no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C2×C3⋊C8—28no—
278C2×C3⋊C8—22no—
279C2×C3⋊C8—22no—
280C2×C3⋊C8C2×D414yes—
281D24—48no—
282C25—61no—
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×Q8D1214yescomplemented
292C22×Q8C22×D311yescharacteristic
293C22×Q8C22×D311yescharacteristic
294C22×D4—31no—
295C22×D4—31no—
296C22×D4—32no—
297C22×D4—64no—
298C22×D4—61no—
299C22×D4—62no—
300C22×D4—61no—
301C22×D4—62no—
302C22×D4—62no—
303C22×D4—61no—
304C22×D4—122no—
305C23×C4—21no—
306C23×C4—31no—
307C23×C4—61no—
308C23×C4C22×D311yescharacteristic
309C2×SD16—38no—
310C2×SD16—64no—
311C2×SD16—64no—
312C2×SD16—632no—
313C2×SD16—64no—
314C2×SD16—64no—
315C2×SD16—64no—
316C2×D8—64no—
317C2×D8—64no—
318C2×D8—122no—
319C22×C8—32no—
320C22×C8—64no—
321C22×C8—62no—
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—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4C22×D312yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—31no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—62no—
353C2×C22⋊C4—62no—
354C2×C22⋊C4—62no—
355C2×C42—31no—
356C2×C42—61no—
357#103—24no—
358#103—42no—
359#102—64no—
360#102—64no—
361C22⋊C8—22no—
362C22⋊C8—64no—
363#95—21no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3#12012yes—
367C23×C3#13512yes—
368C23×C3AGL(1,ℤ/8)12yes—
369C23×C3C22×D411yescharacteristic
370C22×D3—21no—
371C22×D3—22no—
372C22×D3—24no—
373C22×D3—22no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—22no—
377C22×D3—48no—
378C22×D3—42no—
379C22×D3—42no—
380C22×D3—42no—
381C22×D3—41no—
382C22×D3—42no—
383C22×D3—42no—
384C22×D3—41no—
385C22×D3—42no—
386C22×D3—42no—
387C22×D3—42no—
388C22×D3—88no—
389C22×D3—88no—
390C22×D3—84no—
391C22×D3—84no—
392C3×Q8—22no—
393C3×Q8—24no—
394C3×Q8—22no—
395C3×Q8—22no—
396C3×Q8—48no—
397C3×Q8—42no—
398C3×Q8#12014yescomplemented
399C2×C3×C4—22no—
400C2×C3×C4—24no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—48no—
407C2×C3×C4—42no—
408C2×C3×C4—42no—
409C2×C3×C4—42no—
410C2×C3×C4—42no—
411C2×C3×C4#12012yes—
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C22×D412yes—
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4H(5,𝔽2)12yes—
416#67—44no—
417#67—48no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—22no—
421C2×Dic3—21no—
422C2×Dic3—22no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—22no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428D12—24no—
429D12—24no—
430D12—24no—
431D12—48no—
432D12—48no—
433D12—41no—
434D12—42no—
435D12—41no—
436D12—42no—
437D12—88no—
438C4×D3—48no—
439C4×D3—41no—
440C4×D3—41no—
441C4×D3—42no—
442C3×C8—42no—
443C3⋊C8—24no—
444C24—31no—
445C24—61no—
446C24—62no—
447C24—124no—
448C24—122no—
449C24—122no—
450C24—122no—
451C24—242no—
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×D1212yes—
467C2×Q8C2×D1212yes—
468C2×Q8D3×D414yes—
469C2×D4—32no—
470C2×D4—34no—
471C2×D4—68no—
472C2×D4—64no—
473C2×D4—64no—
474C2×D4—62no—
475C2×D4—62no—
476C2×D4—62no—
477C2×D4—62no—
478C2×D4—61no—
479C2×D4—62no—
480C2×D4—62no—
481C2×D4—62no—
482C2×D4—62no—
483C2×D4—62no—
484C2×D4—62no—
485C2×D4—128no—
486C2×D4—124no—
487C2×D4—124no—
488C2×D4—124no—
489C2×D4—124no—
490C2×D4—124no—
491C2×D4—122no—
492C2×D4—122no—
493C2×D4—122no—
494C2×D4—242no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—31no—
504C22×C4—44no—
505C22×C4—64no—
506C22×C4—61no—
507C22×C4—61no—
508C22×C4—62no—
509C22×C4—61no—
510C22×C4—62no—
511C22×C4—124no—
512C22×C4C2×D1212yes—
513C22×C4D3×D411yescharacteristic
514C22×C4D3×D412yes—
515C22×C4C23×D311yescharacteristic
516SD16—616no—
517SD16—616no—
518D8—128no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—34no—
523C2×C8—68no—
524C2×C8—62no—
525C2×C8—62no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—32no—
529C4⋊C4—64no—
530C4⋊C4—64no—
531C4⋊C4—62no—
532C4⋊C4—61no—
533C22⋊C4—42no—
534C22⋊C4—64no—
535C22⋊C4—64no—
536C22⋊C4—64no—
537C22⋊C4—62no—
538C22⋊C4—62no—
539C22⋊C4—64no—
540C22⋊C4—124no—
541C22⋊C4—124no—
542C22⋊C4—242no—
543C42—64no—
544C42—61no—
545C42—61no—
546C22×C3—22no—
547C22×C3—24no—
548C22×C3—22no—
549C22×C3—24no—
550C22×C3#44914yescomplemented
551C22×C3C2×#12011yescharacteristic
552C22×C3D4212yes—
553C22×C3C2×#13511yescharacteristic
554C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
555C22×C3#57712yes—
556C2×D3—42no—
557C2×D3—41no—
558C2×D3—44no—
559C2×D3—41no—
560C2×D3—42no—
561C2×D3—42no—
562C2×D3—44no—
563C2×D3—41no—
564C2×D3—42no—
565C2×D3—42no—
566C2×D3—44no—
567C2×D3—816no—
568C2×D3—82no—
569C2×D3—82no—
570C2×D3—82no—
571C2×D3—82no—
572C2×D3—82no—
573C2×D3—82no—
574C3×C4—24no—
575C3×C4—22no—
576C3×C4—42no—
577C3×C4—82no—
578C3×C4C2×#12011yescharacteristic
579C3×C4#53411yescharacteristic
580C3×C4D4212yes—
581Dic3—41no—
582Dic3—41no—
583Dic3—42no—
584Dic3—41no—
585C23—22no—
586C23—22no—
587C23—62no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—62no—
592C23—64no—
593C23—62no—
594C23—128no—
595C23—122no—
596C23—122no—
597C23—121no—
598C23—121no—
599C23—122no—
600C23—122no—
601C23—122no—
602C23—122no—
603C23—122no—
604C23—122no—
605C23—248no—
606C23—248no—
607C23—244no—
608C23—244no—
609C23#88212yes—
610C23#91412yes—
611C23#91912yes—
612C23C2×D3×D411yescharacteristic
613Q8—22no—
614Q8—24no—
615Q8—22no—
616Q8—22no—
617Q8—48no—
618Q8—42no—
619Q8#88214yescomplemented
620D4—64no—
621D4—64no—
622D4—64no—
623D4—128no—
624D4—128no—
625D4—124no—
626D4—128no—
627D4—121no—
628D4—122no—
629D4—121no—
630D4—122no—
631D4—248no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—48no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—61no—
645C2×C4—62no—
646C2×C4—61no—
647C2×C4—61no—
648C2×C4—61no—
649C2×C4—61no—
650C2×C4—62no—
651C2×C4—62no—
652C2×C4—128no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—122no—
658C2×C4#88212yes—
659C2×C4C22×D1211yescharacteristic
660C2×C4C2×D3×D412yes—
661C2×C4C2×D3×D411yescharacteristic
662C2×C4#100412yes—
663C8—42no—
664C8—64no—
665C2×C3—24no—
666C2×C3#240311yescharacteristic
667C2×C3C2×#44911yescharacteristic
668C2×C3#301011yescharacteristic
669C2×C3#326512yes—
670C2×C3#328212yes—
671D3—81no—
672D3—82no—
673D3—84no—
674C22—24no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—122no—
679C22—124no—
680C22—121no—
681C22—122no—
682C22—122no—
683C22—121no—
684C22—124no—
685C22—122no—
686C22—124no—
687C22—121no—
688C22—122no—
689C22—2416no—
690C22—242no—
691C22—242no—
692C22—242no—
693C22—242no—
694C22—242no—
695C22—242no—
696C22#481014yescomplemented
697C22C2×#88211yescharacteristic
698C22#559012yes—
699C22C2×#91411yescharacteristic
700C22C2×#91911yescharacteristic
701C22#577812yes—
702C4—22no—
703C4—24no—
704C4—42no—
705C4—82no—
706C4—121no—
707C4—121no—
708C4—121no—
709C4—122no—
710C4C2×#88211yescharacteristic
711C4#549311yescharacteristic
712C4#559012yes—
713C3#2385811yescharacteristic, complemented, Sylow
714C2—24no—
715C2—241no—
716C2—244no—
717C2—242no—
718C2#7385211yescharacteristic
719C2C2×#481011yescharacteristic
720C2#7896011yescharacteristic
721C2#8006012yes—
722C2#8020712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1787C2
socleC23×C3C22×D4
hypercenter#1787D3
hypocenterC3#23858
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C2×#5817C2semidirect
2. C23×#215C2semidirect
3. C2×#4810C2semidirect
4. C2×#4732C2semidirect
5. #74016C2semidirect
6. C3×#1787C2semidirect
7. #70277C2semidirect
8. #70272C2semidirect
9. #70240C2semidirect
10. #69952C2semidirect
11. #69515C2semidirect
14. C22×#239C22semidirect
19. C2×#947C22semidirect
21. C2×#946C22semidirect
22. C2×Q8×Dic3C22semidirect
35. C2×C3×#103C22semidirect
36. C2×C3×#103C22semidirect
40. C2×(C2×D3)⋊C8C22semidirect
42. C2×#809C22semidirect
48. #4550C22semidirect
49. #4529C22semidirect
60. #1787D3semidirect
66. C22×C3×Q8D4semidirect
106. C2×#215D4semidirect
109. #930D4semidirect
167. C2×#103C2×D3semidirect
168. C2×#103C2×D3semidirect
291. C22×Q8D12semidirect
398. C3×Q8#120semidirect
550. C22×C3#449semidirect
619. Q8#882semidirect
696. C22#4810semidirect
713. C3#23858semidirect

Maximal subgroups

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

subgroupindex
C2×#58172
C23×#2152
C2×#48102
C2×#47322
#740162
C3×#17872
#702772
#702722
#702402
#699522
#695152
#238583