Finite Group Databasebeta

← #10826277

Subgroups — #10826277

Order 768.

1930 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#204C211yesmaximal, characteristic, complemented
2C2×#5632C211yesmaximal, characteristic, complemented
3C2×#5046C211yesmaximal, characteristic, complemented
4C2×#4806C212yesmaximal, complemented
5#74016C211yesmaximal, characteristic
6C3×#1892C211yesmaximal, characteristic, complemented
7#71975C211yesmaximal, characteristic, complemented
8#70282C212yesmaximal, complemented
9#69525C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68887C212yesmaximal, complemented
12#23855—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic, complemented
14C23×D12C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×#944C2211yescharacteristic, complemented
19C22×#215C2212yes—
20C2×C4×#67C2211yescharacteristic
21C22×#204—24no—
22C22×#204—28no—
23C22×#204C2212yes—
24C22×#204C2212yes—
25#5632—24no—
26#5632—24no—
27C2×#897C2211yescharacteristic, complemented
28C2×#896—21no—
29C2×#895C2212yes—
30C2×#888—21no—
31C2×#887C2211yescharacteristic, complemented
32C2×#882—21no—
33C2×#67⋊C4—21no—
34C2×#878—21no—
35C2×C3×#103C2212yescomplemented
36C3×#380C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×(C2×D3)⋊C8C2212yescomplemented
39#5046—24no—
40#4988C2212yes—
41C2×#810C2212yescomplemented
42C2×#810C2212yescomplemented
43C2×#809C2212yes—
44#4955—21no—
45#4806—28no—
46#4672—21no—
47#4671—21no—
48C3×#340C2212yescomplemented
49#4550C2212yescomplemented
50C23×SD16—31no—
51C2×D4×Q8—31no—
52C2×#448—32no—
53C2×#450—31no—
54#2408—31no—
55#2013—32no—
56#2000—32no—
57#1892D311yescharacteristic, complemented
58#1880—32no—
59#1869—31no—
60#1787—31no—
61C24×D3—21no—
62C22×C3×Q8C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×#67D411yescharacteristic
65C23×Dic3—21no—
66C2×D3×Q8—22no—
67C2×D3×Q8—22no—
68C2×D3×Q8—24no—
69C2×D3×Q8—42no—
70C22×D12—22no—
71C22×D12—24no—
72C22×D12—22no—
73C22×D12—21no—
74C22×D12—22no—
75C22×D12—42no—
76C22×D12C2312yes—
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×Dic6—24no—
80C22×Dic6—22no—
81C22×Dic6—21no—
82C22×Dic6—21no—
83C22×Dic6—42no—
84C22×Dic6D414yescomplemented
85C22×Dic6C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C22×C3×C8—22no—
88C22×C3×C8—24no—
89C22×C3×C8C2312yes—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8D414yescomplemented
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×#217D411yescharacteristic
98#944—22no—
99#944—22no—
100C2×#215—24no—
101C2×#215—24no—
102C2×#215—42no—
103C4×#67—22no—
104C4×#67—22no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—22no—
107C2×(C2×D3)⋊C4—22no—
108C2×(C2×D3)⋊C4—22no—
109C2×(C2×D3)⋊C4D411yescharacteristic
110C2×C4⋊Dic3—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2312yes—
118C2×#204—232no—
119C2×#204—24no—
120C2×#204—24no—
121C2×#204—24no—
122C2×#204—24no—
123C2×#204—24no—
124C2×#204D418yescomplemented
125#897—22no—
126#897—22no—
127#896—28no—
128#895—24no—
129#895—42no—
130#888—28no—
131#887—22no—
132#887—22no—
133#882—44no—
134#67⋊C4—28no—
135#878—28no—
136C3×#103—24no—
137C3×#95—21no—
138#830—22no—
139(C2×D3)⋊C8—24no—
140#810—24no—
141#810—42no—
142#809—24no—
143C23×Q8—31no—
144C23×Q8—31no—
145C23×D4—31no—
146C22×SD16—32no—
147C22×SD16—32no—
148C22×SD16—32no—
149C22×SD16—68no—
150C22×SD16—64no—
151C23×C8C2×D311yescharacteristic
152D4×Q8—64no—
153D4×Q8—64no—
154C2×C4⋊Q8—31no—
155C2×C4⋊Q8—61no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8—61no—
158C2×C22⋊Q8—61no—
159C2×C22⋊Q8C2×D311yescharacteristic
160C2×#121—32no—
161C2×#120—61no—
162C2×C4×Q8—31no—
163C2×C4×D4—31no—
164C2×C4×D4—61no—
165#450—64no—
166#448—68no—
167C2×#103—32no—
168C2×#103—32no—
169C2×#103C2×D312yescomplemented
170C2×#102—32no—
171C2×C22⋊C8—31no—
172C2×C22⋊C8—32no—
173#394—61no—
174#392—61no—
175#390—32no—
176#380C2×D311yescharacteristic
177#379—61no—
178#340C2×D312yescomplemented
179#336—32no—
180C24×C3C2×D411yescharacteristic
181C23×D3—22no—
182C23×D3—44no—
183C23×D3—42no—
184C23×D3—42no—
185C23×D3—42no—
186C23×D3—82no—
187C23×D3C2×D411yescharacteristic
188C2×C3×Q8—21no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8C2×D412yes—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2411yescharacteristic
201C2×#67—22no—
202C2×#67—22no—
203C2×#67SD1614yescomplemented
204C22×Dic3—21no—
205C22×Dic3—24no—
206C22×Dic3—21no—
207C22×Dic3—44no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211D3×Q8—48no—
212D3×Q8—44no—
213D3×Q8—44no—
214C2×D12—28no—
215C2×D12—22no—
216C2×D12—22no—
217C2×D12—22no—
218C2×D12—22no—
219C2×D12—22no—
220C2×D12—48no—
221C2×D12—44no—
222C2×D12—44no—
223C2×D12—42no—
224C2×D12—44no—
225C2×D12—44no—
226C2×D12—82no—
227C2×D12C2×D414yes—
228C2×C4×D3—24no—
229C2×C4×D3—22no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—22no—
233C2×C4×D3—44no—
234C2×Dic6—21no—
235C2×Dic6—22no—
236C2×Dic6—22no—
237C2×Dic6—22no—
238C2×Dic6—28no—
239C2×Dic6—24no—
240C2×Dic6—24no—
241C2×Dic6—21no—
242C2×Dic6—44no—
243C2×Dic6—48no—
244C2×Dic6—44no—
245C2×Dic6—44no—
246C2×Dic6C2×D412yes—
247C2×Dic6C2×D414yes—
248C2×Dic6C2×D412yes—
249C2×C3×C8—28no—
250C2×C3×C8—22no—
251C2×C3×C8—22no—
252C2×C3×C8C2×D414yes—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4C2×D412yes—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—42no—
260#217—22no—
261#215—48no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—22no—
264(C2×D3)⋊C4—44no—
265(C2×D3)⋊C4—44no—
266(C2×D3)⋊C4—82no—
267C4⋊Dic3—24no—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—24no—
271Dic3⋊C4—22no—
272C4×Dic3—24no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C2×C3⋊C8—22no—
276C2×C3⋊C8—22no—
277C2×C3⋊C8—22no—
278#204—216no—
279#204—216no—
280C25—61no—
281C22×Q8—34no—
282C22×Q8—31no—
283C22×Q8—31no—
284C22×Q8—61no—
285C22×Q8—64no—
286C22×Q8—62no—
287C22×Q8—62no—
288C22×Q8—62no—
289C22×Q8—61no—
290C22×Q8—64no—
291C22×Q8—122no—
292C22×Q8—122no—
293C22×Q8C22×D311yescharacteristic
294C22×D4—31no—
295C22×D4—32no—
296C22×D4—64no—
297C22×D4—62no—
298C22×D4—62no—
299C22×D4—62no—
300C22×D4—61no—
301C22×D4—122no—
302C23×C4—31no—
303C23×C4—61no—
304C23×C4—61no—
305C23×C4C22×D311yescharacteristic
306C2×SD16—38no—
307C2×SD16—64no—
308C2×SD16—64no—
309C2×SD16—64no—
310C2×SD16—64no—
311C2×SD16—632no—
312C2×SD16—64no—
313C2×SD16—64no—
314C2×SD16—64no—
315C2×SD16—122no—
316C22×C8—22no—
317C22×C8—24no—
318C22×C8—32no—
319C22×C8C22×D312yes—
320C4⋊Q8—68no—
321C4⋊Q8—62no—
322C4⋊Q8—62no—
323C22⋊Q8—22no—
324C22⋊Q8—68no—
325C22⋊Q8—62no—
326C22⋊Q8—68no—
327C22⋊Q8—62no—
328C22⋊Q8D1214yescomplemented
329#121—64no—
330#121—122no—
331#120—124no—
332C4×Q8—62no—
333C4×Q8—62no—
334C4×D4—68no—
335C4×D4—62no—
336C4×D4—62no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4C22×D312yes—
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—31no—
348C2×C22⋊C4—31no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—62no—
351C2×C22⋊C4—62no—
352C2×C22⋊C4—62no—
353C2×C42—31no—
354C2×C42—61no—
355#103—24no—
356#103—64no—
357#103—122no—
358#102—64no—
359C22⋊C8—62no—
360C22⋊C8—64no—
361#95—21no—
362C23×C3—22no—
363C23×C3—22no—
364C23×C3#12012yes—
365C23×C3C2×SD1612yes—
366C23×C3#13712yes—
367C23×C3C22×D411yescharacteristic
368C22×D3—21no—
369C22×D3—22no—
370C22×D3—21no—
371C22×D3—22no—
372C22×D3—24no—
373C22×D3—48no—
374C22×D3—42no—
375C22×D3—42no—
376C22×D3—42no—
377C22×D3—42no—
378C22×D3—42no—
379C22×D3—42no—
380C22×D3—88no—
381C22×D3—88no—
382C22×D3—84no—
383C22×D3—84no—
384C22×D3C2×SD1612yescomplemented
385C3×Q8—24no—
386C3×Q8—41no—
387C3×Q8—41no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—24no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—42no—
395C2×C3×C4—42no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4—42no—
399C2×C3×C4#12012yes—
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D412yes—
403C2×C3×C4H(5,𝔽2)12yes—
404#67—28no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—24no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—48no—
415C2×Dic3—42no—
416C2×Dic3C2×SD1612yescomplemented
417D12—24no—
418D12—24no—
419D12—48no—
420D12—42no—
421D12—42no—
422D12—88no—
423C4×D3—48no—
424C4×D3—41no—
425C4×D3—41no—
426C4×D3—42no—
427Dic6—22no—
428Dic6—24no—
429Dic6—22no—
430Dic6—22no—
431Dic6—48no—
432Dic6—48no—
433Dic6—42no—
434Dic6#12014yescomplemented
435C3×C8—24no—
436C3⋊C8—42no—
437C24—31no—
438C24—62no—
439C24—124no—
440C24—122no—
441C24—122no—
442C24—122no—
443C24—242no—
444C24D3×D411yescharacteristic
445C2×Q8—21no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—34no—
449C2×Q8—32no—
450C2×Q8—32no—
451C2×Q8—64no—
452C2×Q8—68no—
453C2×Q8—64no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—62no—
457C2×Q8—62no—
458C2×Q8—62no—
459C2×Q8—124no—
460C2×Q8—124no—
461C2×Q8—128no—
462C2×Q8—124no—
463C2×Q8—128no—
464C2×Q8—124no—
465C2×Q8—124no—
466C2×Q8C2×D1212yes—
467C2×D4—34no—
468C2×D4—34no—
469C2×D4—68no—
470C2×D4—62no—
471C2×D4—62no—
472C2×D4—62no—
473C2×D4—62no—
474C2×D4—62no—
475C2×D4—62no—
476C2×D4—62no—
477C2×D4—128no—
478C2×D4—124no—
479C2×D4—124no—
480C2×D4—124no—
481C2×D4—124no—
482C2×D4—122no—
483C2×D4—242no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—31no—
492C22×C4—64no—
493C22×C4—61no—
494C22×C4—64no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—62no—
498C22×C4—61no—
499C22×C4—62no—
500C22×C4—124no—
501C22×C4—124no—
502C22×C4C2×D1212yes—
503C22×C4D3×D411yescharacteristic
504C22×C4D3×D412yes—
505C22×C4C23×D311yescharacteristic
506SD16—616no—
507SD16—616no—
508SD16—128no—
509C2×C8—28no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—62no—
513C2×C8—62no—
514C2×C8—62no—
515C2×C8D3×D414yes—
516C4⋊C4—24no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—64no—
523C4⋊C4—64no—
524C4⋊C4—62no—
525C4⋊C4C2×D1212yes—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—64no—
529C22⋊C4—62no—
530C22⋊C4—62no—
531C22⋊C4—124no—
532C22⋊C4—124no—
533C22⋊C4—242no—
534C42—64no—
535C42—61no—
536C42—61no—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—24no—
540C22×C3—24no—
541C22×C3#44814yescomplemented
542C22×C3C2×#12011yescharacteristic
543C22×C3D4212yes—
544C22×C3C22×SD1611yescharacteristic
545C22×C3C2×#13711yescharacteristic
546C22×C3#57712yes—
547C2×D3—22no—
548C2×D3—24no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—42no—
552C2×D3—44no—
553C2×D3—44no—
554C2×D3—44no—
555C2×D3—816no—
556C2×D3—82no—
557C2×D3—82no—
558C2×D3—82no—
559C2×D3—82no—
560C3×C4—22no—
561C3×C4—42no—
562C3×C4—42no—
563C3×C4—82no—
564C3×C4C2×#12011yescharacteristic
565C3×C4#53411yescharacteristic
566C3×C4D4212yes—
567Dic3—24no—
568Dic3—22no—
569Dic3—42no—
570Dic3—41no—
571Dic3—41no—
572C23—22no—
573C23—22no—
574C23—32no—
575C23—61no—
576C23—61no—
577C23—62no—
578C23—62no—
579C23—64no—
580C23—128no—
581C23—122no—
582C23—122no—
583C23—122no—
584C23—122no—
585C23—122no—
586C23—122no—
587C23—248no—
588C23—248no—
589C23—244no—
590C23—244no—
591C23#88212yes—
592C23D3×SD1612yes—
593C23#91812yes—
594C23C2×D3×D411yescharacteristic
595Q8—24no—
596Q8—34no—
597Q8—41no—
598Q8—41no—
599Q8—62no—
600Q8—64no—
601Q8—62no—
602Q8—62no—
603Q8—128no—
604Q8—128no—
605Q8—122no—
606D4—68no—
607D4—64no—
608D4—64no—
609D4—128no—
610D4—122no—
611D4—122no—
612D4—248no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—32no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—61no—
626C2×C4—62no—
627C2×C4—61no—
628C2×C4—64no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—128no—
635C2×C4—128no—
636C2×C4—122no—
637C2×C4—121no—
638C2×C4—122no—
639C2×C4—121no—
640C2×C4#88212yes—
641C2×C4C22×D1211yescharacteristic
642C2×C4C2×D3×D411yescharacteristic
643C2×C4C2×D3×D412yes—
644C2×C4#100912yes—
645C8—24no—
646C8—122no—
647C2×C3—24no—
648C2×C3#240311yescharacteristic
649C2×C3C2×#44811yescharacteristic
650C2×C3#301011yescharacteristic
651C2×C3#326612yes—
652C2×C3D4×SD1612yes—
653D3—42no—
654D3—84no—
655C22—24no—
656C22—24no—
657C22—22no—
658C22—22no—
659C22—64no—
660C22—62no—
661C22—122no—
662C22—121no—
663C22—121no—
664C22—124no—
665C22—124no—
666C22—124no—
667C22—2416no—
668C22—242no—
669C22—242no—
670C22—242no—
671C22—242no—
672C22#480614yescomplemented
673C22C2×#88211yescharacteristic
674C22#560912yes—
675C22C2×D3×SD1611yescharacteristic
676C22C2×#91811yescharacteristic
677C22#577012yes—
678C4—22no—
679C4—42no—
680C4—42no—
681C4—64no—
682C4—62no—
683C4—82no—
684C4—121no—
685C4—122no—
686C4—121no—
687C4D4×D1212yes—
688C4D3×#12011yescharacteristic
689C4#565311yescharacteristic
690C3#2385511yescharacteristic, complemented, Sylow
691C2—24no—
692C2—122no—
693C2—244no—
694C2#7488411yescharacteristic
695C2C2×#480611yescharacteristic
696C2#7883611yescharacteristic
697C2#8067612yes—
698C2#8077112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1892C2
socleC23×C3C22×D4
prime radical#74016C2
hypercenter#1892D3
hypocenterC3#23855
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
1. C23×#204C2semidirect
2. C2×#5632C2semidirect
3. C2×#5046C2semidirect
4. C2×#4806C2semidirect
6. C3×#1892C2semidirect
7. #71975C2semidirect
8. #70282C2semidirect
9. #69525C2semidirect
10. #68909C2semidirect
11. #68887C2semidirect
13. C22×D3×Q8C22semidirect
18. C2×#944C22semidirect
27. C2×#897C22semidirect
31. C2×#887C22semidirect
35. C2×C3×#103C22semidirect
38. C2×(C2×D3)⋊C8C22semidirect
41. C2×#810C22semidirect
42. C2×#810C22semidirect
48. C3×#340C22semidirect
49. #4550C22semidirect
57. #1892D3semidirect
84. C22×Dic6D4semidirect
91. C3×C22⋊Q8D4semidirect
124. C2×#204D4semidirect
169. C2×#103C2×D3semidirect
178. #340C2×D3semidirect
203. C2×#67SD16semidirect
328. C22⋊Q8D12semidirect
384. C22×D3C2×SD16semidirect
416. C2×Dic3C2×SD16semidirect
434. Dic6#120semidirect
541. C22×C3#448semidirect
672. C22#4806semidirect
690. C3#23855semidirect

Maximal subgroups

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

subgroupindex
C23×#2042
C2×#56322
C2×#50462
C2×#48062
#740162
C3×#18922
#719752
#702822
#695252
#689092
#688872
#238553