Finite Group Databasebeta

← #10826219

Subgroups — #10826219

Order 768.

2096 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#213C211yesmaximal, characteristic, complemented
2C2×#5550C211yesmaximal, characteristic, complemented
3C3×#2408C211yesmaximal, characteristic, complemented
4C2×#5218C211yesmaximal, characteristic, complemented
5C2×#5037C212yesmaximal, complemented
6#71684C212yesmaximal, complemented
7#71610C211yesmaximal, characteristic, complemented
8#69659C212yesmaximal, complemented
9#69657C212yesmaximal, complemented
10#69521C211yesmaximal, characteristic, complemented
11#69514C211yesmaximal, characteristic, complemented
12#23851—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic, complemented
15C23×D12C2211yescharacteristic
16C2×C3×#121C2212yescomplemented
17C2×C3×#120—21no—
18C2×C3×C4×D4C2211yescharacteristic
19C2×#937—21no—
20C22×#214C2212yescomplemented
21C22×#213—24no—
22C22×#213—28no—
23C22×#213C2212yes—
24C22×#213C2212yes—
25C2×#930C2211yescharacteristic, complemented
26C2×#924C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28#5550—24no—
29#5550—24no—
30C2×#894—21no—
31C2×#884—21no—
32C2×D3×C22⋊C4—21no—
33C2×#875C2211yescharacteristic, complemented
34C2×C4×D12C2211yescharacteristic, complemented
35C3×#394—21no—
36C3×#392—21no—
37C3×#390C2212yescomplemented
38C3×#380C2211yescharacteristic
39C3×#379—21no—
40#5218—24no—
41C2×#832C2212yescomplemented
42C2×#830C2211yescharacteristic
43C2×#830C2212yescomplemented
44#5037—28no—
45C2×#810C2212yescomplemented
46C2×#809C2212yescomplemented
47C2×#809C2212yescomplemented
48#4529C2212yescomplemented
49#4524C2212yescomplemented
50C23×D8—31no—
51C2×#546—31no—
52C2×#449—32no—
53C2×#450—31no—
54#2408D311yescharacteristic, complemented
55#2012—32no—
56#1999—32no—
57#1892—31no—
58#1879—32no—
59#1869—31no—
60#1786—31no—
61C25×C3—21no—
62C24×D3—21no—
63C22×C3×D4—24no—
64C22×C3×D4—22no—
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—42no—
69C22×C3×D4D411yescharacteristic
70C22×C3×D4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C22×#67—21no—
74C22×#67—21no—
75C2×#235—22no—
76C2×#235—24no—
77C2×#235—22no—
78C2×#235—42no—
79C22×D12—24no—
80C22×D12—21no—
81C22×D12—22no—
82C22×D12—42no—
83C22×D12D414yescomplemented
84C22×D12C2311yescharacteristic
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×C4×D3—21no—
88C22×Dic6C2311yescharacteristic
89C3×#121—24no—
90C3×#121—42no—
91C3×#120—44no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104#937—28no—
105C2×#214—24no—
106C2×#214—24no—
107C2×#214—42no—
108C2×#213—232no—
109C2×#213—24no—
110C2×#213—24no—
111C2×#213—24no—
112C2×#213—24no—
113C2×#213—24no—
114C2×#213D418yescomplemented
115#930—22no—
116#930—22no—
117C2×(C2×D3)⋊C4—21no—
118C2×(C2×D3)⋊C4—21no—
119C2×(C2×D3)⋊C4—21no—
120C2×(C2×D3)⋊C4—21no—
121C2×C4⋊Dic3C2311yescharacteristic
122#924—22no—
123#924D414yescomplemented
124C2×Dic3⋊C4—21no—
125C22×C3⋊C8—22no—
126C22×C3⋊C8—24no—
127C22×C3⋊C8C2312yes—
128C22×C3⋊C8C2312yes—
129#894—28no—
130#884—28no—
131D3×C22⋊C4—28no—
132#875—22no—
133#875—22no—
134C4×D12—22no—
135C4×D12—22no—
136C3×#95—21no—
137#832—24no—
138#830—22no—
139#830—24no—
140#810—24no—
141#809—24no—
142#809—42no—
143C22×#41—31no—
144C23×D4—31no—
145C23×D4C2×D311yescharacteristic
146C22×SD16—32no—
147C22×D8—32no—
148C22×D8—32no—
149C22×D8—68no—
150C22×D8—64no—
151C23×C8—31no—
152#546—64no—
153#546—64no—
154C2×#124—31no—
155C2×#123—61no—
156C2×C22⋊Q8—31no—
157C2×#121—31no—
158C2×#121—61no—
159C2×#121C2×D312yescomplemented
160C2×#120—21no—
161C2×#120—61no—
162C2×C4×D4—31no—
163C2×C4×D4C2×D311yescharacteristic
164C22×C22⋊C4—61no—
165#450—64no—
166#449—68no—
167C2×#103—32no—
168C2×#102—32no—
169C2×#102—32no—
170C2×#102—32no—
171C2×C22⋊C8—31no—
172C2×C22⋊C8—32no—
173#394—21no—
174#392—21no—
175#390C2×D312yescomplemented
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—
189C23×D3—21no—
190C23×D3—24no—
191C23×D3—44no—
192C23×D3—42no—
193C23×D3—42no—
194C23×D3—82no—
195C2×C3×D4—28no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—48no—
205C2×C3×D4—44no—
206C2×C3×D4—44no—
207C2×C3×D4—44no—
208C2×C3×D4—44no—
209C2×C3×D4—42no—
210C2×C3×D4—82no—
211C2×C3×D4C2×D414yes—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4C2×D412yes—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C2×#67—24no—
227C2×#67—24no—
228C2×#67—44no—
229C2×#67—82no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232#235—48no—
233#235—44no—
234#235—44no—
235C2×D12—21no—
236C2×D12—22no—
237C2×D12—22no—
238C2×D12—22no—
239C2×D12—28no—
240C2×D12—24no—
241C2×D12—21no—
242C2×D12—44no—
243C2×D12—48no—
244C2×D12—44no—
245C2×D12C2×D412yes—
246C2×D12C2×D414yes—
247C2×D12C2×D412yes—
248C2×C4×D3—24no—
249C2×C4×D3—24no—
250C2×C4×D3—44no—
251C2×C4×D3—82no—
252C2×Dic6—22no—
253C2×Dic6—22no—
254C2×Dic6—21no—
255C2×Dic6C2×D412yes—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—44no—
262C3×C22⋊C4—44no—
263C3×C22⋊C4—42no—
264C3×C22⋊C4—82no—
265C3×C42—21no—
266C3×C42—21no—
267#217—24no—
268#214—48no—
269#213—216no—
270#213—216no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—24no—
275C4⋊Dic3—21no—
276C4⋊Dic3C2×D412yes—
277Dic3⋊C4—24no—
278C2×C3⋊C8—28no—
279C2×C3⋊C8—22no—
280C2×C3⋊C8—22no—
281C2×C3⋊C8—22no—
282C2×C3⋊C8—22no—
283C2×C3⋊C8—22no—
284C2×C3⋊C8C2×D414yes—
285C25—21no—
286C25—61no—
287C2×#41—62no—
288C2×#41—62no—
289C2×#41—64no—
290C2×#41—122no—
291C22×Q8—31no—
292C22×D4—24no—
293C22×D4—22no—
294C22×D4—22no—
295C22×D4—22no—
296C22×D4—21no—
297C22×D4—34no—
298C22×D4—31no—
299C22×D4—31no—
300C22×D4—42no—
301C22×D4—64no—
302C22×D4—61no—
303C22×D4—61no—
304C22×D4—62no—
305C22×D4—61no—
306C22×D4—122no—
307C22×D4#6711yescharacteristic
308C22×D4C22×D312yes—
309C23×C4—61no—
310C23×C4—61no—
311C23×C4C22×D311yescharacteristic
312C23×C4C22×D311yescharacteristic
313C2×SD16—64no—
314C2×SD16—64no—
315C2×SD16—122no—
316C2×D8—38no—
317C2×D8—64no—
318C2×D8—64no—
319C2×D8—64no—
320C2×D8—64no—
321C2×D8—632no—
322C2×D8—64no—
323C22×C8—32no—
324C22×C8—32no—
325C22×C8—64no—
326C22×C8—62no—
327#124—62no—
328#124—62no—
329#123—68no—
330C22⋊Q8—34no—
331C22⋊Q8—62no—
332#121—24no—
333#121—42no—
334#121—68no—
335#121—62no—
336#121—62no—
337#120—44no—
338#120—68no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—62no—
342C4×D4—62no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4#6711yescharacteristic
346C2×C4⋊C4C22×D312yes—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—68no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C22⋊C4—61no—
355C2×C22⋊C4—61no—
356C2×C22⋊C4—61no—
357C2×C22⋊C4#6711yescharacteristic
358C2×C22⋊C4#6711yescharacteristic
359C2×C42C22×D311yescharacteristic
360#103—64no—
361#102—64no—
362#102—64no—
363#102—122no—
364C22⋊C8—64no—
365C22⋊C8—62no—
366#95—21no—
367C23×C3—22no—
368C23×C3—22no—
369C23×C3—21no—
370C23×C3—24no—
371C23×C3—22no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—22no—
375C23×C3—48no—
376C23×C3—42no—
377C23×C3—42no—
378C23×C3—42no—
379C23×C3—42no—
380C23×C3—42no—
381C23×C3—42no—
382C23×C3—88no—
383C23×C3—88no—
384C23×C3—84no—
385C23×C3—84no—
386C23×C3#12012yes—
387C23×C3#13512yes—
388C23×C3AGL(1,ℤ/8)12yes—
389C23×C3C22×D411yescharacteristic
390C22×D3—22no—
391C22×D3—24no—
392C22×D3—22no—
393C22×D3—48no—
394C22×D3—41no—
395C22×D3—41no—
396C22×D3—42no—
397C22×D3—42no—
398C22×D3—41no—
399C22×D3—41no—
400C22×D3—42no—
401C22×D3—48no—
402C22×D3—84no—
403C22×D3—88no—
404C22×D3—88no—
405C3×D4—24no—
406C3×D4—24no—
407C3×D4—48no—
408C3×D4—44no—
409C3×D4—42no—
410C3×D4—42no—
411C3×D4—88no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—22no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—22no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—42no—
426C2×C3×C4—42no—
427C2×C3×C4—42no—
428C2×C3×C4—42no—
429C2×C3×C4#12012yes—
430C2×C3×C4C22×D411yescharacteristic
431C2×C3×C4C22×D412yes—
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4H(5,𝔽2)12yes—
434#67—48no—
435#67—84no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—42no—
439C2×Dic3—42no—
440D12—22no—
441D12—22no—
442D12—22no—
443D12—24no—
444D12—48no—
445D12—42no—
446D12#12014yescomplemented
447C4×D3—48no—
448C4×D3—84no—
449Dic6—24no—
450Dic6—41no—
451Dic6—41no—
452C3⋊C8—24no—
453C3⋊C8—42no—
454C24—22no—
455C24—44no—
456C24—42no—
457C24—42no—
458C24—42no—
459C24—61no—
460C24—61no—
461C24—64no—
462C24—82no—
463C24—124no—
464C24—122no—
465C24—122no—
466C24—242no—
467C24C2×#6711yescharacteristic
468C24C2×#6711yescharacteristic
469#41—124no—
470#41—128no—
471#41—124no—
472C2×Q8—32no—
473C2×Q8—61no—
474C2×Q8—62no—
475C2×Q8—62no—
476C2×D4—28no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—34no—
486C2×D4—32no—
487C2×D4—32no—
488C2×D4—48no—
489C2×D4—44no—
490C2×D4—44no—
491C2×D4—44no—
492C2×D4—44no—
493C2×D4—42no—
494C2×D4—68no—
495C2×D4—64no—
496C2×D4—64no—
497C2×D4—64no—
498C2×D4—61no—
499C2×D4—62no—
500C2×D4—61no—
501C2×D4—62no—
502C2×D4—62no—
503C2×D4—82no—
504C2×D4—124no—
505C2×D4—128no—
506C2×D4—124no—
507C2×D4—124no—
508C2×D4—242no—
509C2×D4D3×D414yes—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—64no—
518C22×C4—64no—
519C22×C4—61no—
520C22×C4—61no—
521C22×C4—124no—
522C22×C4—242no—
523C22×C4D3×D412yes—
524C22×C4D3×D412yes—
525C22×C4C2×#6711yescharacteristic
526C22×C4C2×#6711yescharacteristic
527C22×C4C2×#6711yescharacteristic
528C22×C4C2×#6711yescharacteristic
529C22×C4C23×D311yescharacteristic
530SD16—128no—
531D8—616no—
532D8—616no—
533C2×C8—34no—
534C2×C8—68no—
535C2×C8—62no—
536C2×C8—62no—
537C2×C8—62no—
538C2×C8—62no—
539C2×C8—62no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—32no—
544C4⋊C4—64no—
545C4⋊C4—61no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—42no—
551C22⋊C4—64no—
552C22⋊C4—64no—
553C22⋊C4—64no—
554C22⋊C4—64no—
555C22⋊C4—64no—
556C22⋊C4—82no—
557C42—21no—
558C42—21no—
559C22×C3—22no—
560C22×C3—22no—
561C22×C3—24no—
562C22×C3—24no—
563C22×C3—41no—
564C22×C3—44no—
565C22×C3—41no—
566C22×C3—41no—
567C22×C3—42no—
568C22×C3—44no—
569C22×C3—42no—
570C22×C3—44no—
571C22×C3—816no—
572C22×C3—82no—
573C22×C3—82no—
574C22×C3—82no—
575C22×C3—82no—
576C22×C3#44914yescomplemented
577C22×C3C2×#12011yescharacteristic
578C22×C3D4212yes—
579C22×C3C2×#13511yescharacteristic
580C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
581C22×C3#57612yes—
582C2×D3—24no—
583C2×D3—42no—
584C2×D3—42no—
585C2×D3—44no—
586C2×D3—44no—
587C2×D3—42no—
588C2×D3—42no—
589C2×D3—42no—
590C2×D3—816no—
591C2×D3—82no—
592C2×D3—82no—
593C3×C4—22no—
594C3×C4—41no—
595C3×C4—41no—
596C3×C4—41no—
597C3×C4—82no—
598C3×C4C2×#12011yescharacteristic
599C3×C4#53411yescharacteristic
600C3×C4D4212yes—
601Dic3—42no—
602Dic3—42no—
603C23—22no—
604C23—22no—
605C23—21no—
606C23—21no—
607C23—24no—
608C23—22no—
609C23—21no—
610C23—22no—
611C23—48no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—62no—
619C23—64no—
620C23—62no—
621C23—88no—
622C23—88no—
623C23—84no—
624C23—84no—
625C23—128no—
626C23—121no—
627C23—121no—
628C23—122no—
629C23—121no—
630C23—122no—
631C23—128no—
632C23—121no—
633C23—122no—
634C23—244no—
635C23—248no—
636C23—248no—
637C23#93212yes—
638C23#93712yes—
639C23#95012yes—
640C23C22×#6711yescharacteristic
641Q8—64no—
642Q8—121no—
643Q8—121no—
644D4—24no—
645D4—24no—
646D4—34no—
647D4—48no—
648D4—44no—
649D4—42no—
650D4—42no—
651D4—62no—
652D4—62no—
653D4—62no—
654D4—64no—
655D4—88no—
656D4—128no—
657D4—128no—
658D4—122no—
659D4—244no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—24no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—22no—
669C2×C4—21no—
670C2×C4—22no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—62no—
678C2×C4—62no—
679C2×C4—128no—
680C2×C4—122no—
681C2×C4—122no—
682C2×C4—244no—
683C2×C4#93712yes—
684C2×C4C2×D3×D411yescharacteristic
685C2×C4C2×D3×D412yes—
686C2×C4C2×D3×D411yescharacteristic
687C2×C4#100412yes—
688C8—64no—
689C8—122no—
690C2×C3—24no—
691C2×C3—81no—
692C2×C3—84no—
693C2×C3#240311yescharacteristic
694C2×C3C2×#44911yescharacteristic
695C2×C3#300811yescharacteristic
696C2×C3#326412yes—
697C2×C3#328012yes—
698D3—44no—
699D3—82no—
700C22—24no—
701C22—24no—
702C22—22no—
703C22—22no—
704C22—42no—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—41no—
709C22—44no—
710C22—41no—
711C22—41no—
712C22—64no—
713C22—816no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—124no—
719C22—124no—
720C22—122no—
721C22—122no—
722C22—122no—
723C22—122no—
724C22—122no—
725C22—2416no—
726C22—242no—
727C22—242no—
728C22#503714yescomplemented
729C22C2×#93211yescharacteristic
730C22C2×#93711yescharacteristic
731C22D4×#6712yes—
732C22C2×#95011yescharacteristic
733C22#583512yes—
734C4—22no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—82no—
739C4—122no—
740C4—122no—
741C4D3×#12011yescharacteristic
742C4#558911yescharacteristic
743C4#559012yes—
744C3#2385111yescharacteristic, complemented, Sylow
745C2—24no—
746C2—81no—
747C2—84no—
748C2—124no—
749C2—242no—
750C2#8260411yescharacteristic
751C2C2×#503711yescharacteristic
752C2#8338311yescharacteristic
753C2#8348812yes—
754C2#8350512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2408C2
socleC23×C3C22×D4
hypercenter#2408D3
hypocenterC3#23851
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×#213C2semidirect
2. C2×#5550C2semidirect
3. C3×#2408C2semidirect
4. C2×#5218C2semidirect
5. C2×#5037C2semidirect
6. #71684C2semidirect
7. #71610C2semidirect
8. #69659C2semidirect
9. #69657C2semidirect
10. #69521C2semidirect
11. #69514C2semidirect
14. C22×#235C22semidirect
16. C2×C3×#121C22semidirect
20. C22×#214C22semidirect
25. C2×#930C22semidirect
33. C2×#875C22semidirect
34. C2×C4×D12C22semidirect
37. C3×#390C22semidirect
41. C2×#832C22semidirect
43. C2×#830C22semidirect
45. C2×#810C22semidirect
46. C2×#809C22semidirect
47. C2×#809C22semidirect
48. #4529C22semidirect
49. #4524C22semidirect
54. #2408D3semidirect
83. C22×D12D4semidirect
114. C2×#213D4semidirect
123. #924D4semidirect
159. C2×#121C2×D3semidirect
175. #390C2×D3semidirect
446. D12#120semidirect
576. C22×C3#449semidirect
728. C22#5037semidirect
744. C3#23851semidirect

Maximal subgroups

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

subgroupindex
C23×#2132
C2×#55502
C3×#24082
C2×#52182
C2×#50372
#716842
#716102
#696592
#696572
#695212
#695142
#238513