Finite Group Databasebeta

← #10828131

Subgroups — #10828131

Order 768.

1822 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×D12C211yesmaximal, characteristic
2C22×#821C211yesmaximal, characteristic, complemented
3#82373C211yesmaximal, characteristic, complemented
4C2×#4736C211yesmaximal, characteristic, complemented
5C3×#1869C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#69959C212yesmaximal, complemented
8#69959C211yesmaximal, characteristic, complemented
9#69934C212yesmaximal, complemented
10#68670C212yesmaximal, complemented
11#68629C212yesmaximal, complemented
12#24032—31nomaximal, Sylow
13C22×D3×D4—21no—
14C23×D12C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic, complemented
17C2×C3×C4×D4C2211yescharacteristic
18C2×#935—21no—
19C2×#930C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic, complemented
22D4×D12—24no—
23D4×D12—24no—
24C2×#895—21no—
25C2×#882—21no—
26C2×#882—21no—
27C2×#874C2211yescharacteristic
28C2×C4×D12C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic, complemented
30C2×C3×C22⋊C8C2212yescomplemented
31#5212C2211yescharacteristic, complemented
32#5212C2212yescomplemented
33C2×#821—24no—
34C2×#821C2212yes—
35C2×#821C2212yes—
36C2×#821C2212yescomplemented
37C2×#821C2212yescomplemented
38C2×#821C2212yescomplemented
39C2×#821C2212yescomplemented
40C2×#818C2212yescomplemented
41#4981—21no—
42#4953—21no—
43#4943—21no—
44#4928C2212yescomplemented
45#4736—24no—
46C3×#336C2212yescomplemented
47#4551C2212yescomplemented
48C2×D42—31no—
49C2×#480—31no—
50C22×#102—31no—
51#2486—31no—
52#2041—32no—
53#2033—32no—
54#1893—31no—
55#1869D311yescharacteristic, complemented
56#1862—32no—
57#1786—31no—
58#1786—32no—
59C24×D3—21no—
60C24×D3—21no—
61C22×C3×D4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C22×#67—21no—
66C23×Dic3—21no—
67C2×D3×D4—28no—
68C2×D3×D4—44no—
69C2×D3×D4—44no—
70C22×D12—24no—
71C22×D12—21no—
72C22×D12—21no—
73C22×D12—22no—
74C22×D12—21no—
75C22×D12—21no—
76C22×D12—22no—
77C22×D12—22no—
78C22×D12—24no—
79C22×D12—42no—
80C22×D12—42no—
81C22×D12D414yes—
82C22×D12C2311yescharacteristic
83C22×D12C2311yescharacteristic
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×C3×C8—22no—
87C22×C3×C8—42no—
88C22×C3×C8C2312yes—
89C22×C3×C8C2312yes—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C22⋊C4D411yescharacteristic
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—22no—
98#935—44no—
99#930—22no—
100#930—22no—
101#930—22no—
102#930D414yes—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—22no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2312yes—
110C2×C4⋊Dic3C2312yes—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—22no—
113#895—28no—
114#882—28no—
115#882—28no—
116#874—22no—
117#874—22no—
118C4×D12—22no—
119C4×D12—22no—
120C3×C22⋊C8—22no—
121C3×C22⋊C8—42no—
122C3⋊#95—22no—
123C3⋊#95—22no—
124C3⋊#95—21no—
125#821—24no—
126#821—24no—
127#821—24no—
128#821—24no—
129#821—42no—
130#821—42no—
131#818—24no—
132C23×D4—31no—
133C23×D4—31no—
134C23×D4—61no—
135C23×C8C2×D311yescharacteristic, complemented
136D42—64no—
137D42—64no—
138C2×#127—31no—
139C2×#123—61no—
140C2×#121—31no—
141C2×#121—31no—
142C2×#121—61no—
143C2×#120—61no—
144C2×#120—61no—
145C2×C4×D4—31no—
146C2×C4×D4C2×D311yescharacteristic
147C22×C4⋊C4—31no—
148#480—64no—
149C2×#107—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×#102—32no—
155C2×#102—32no—
156C2×#102—64no—
157C2×C22⋊C8C2×D311yescharacteristic, complemented
158C2×C22⋊C8C2×D312yescomplemented
159#398—61no—
160#397—61no—
161#389—32no—
162#381—61no—
163#380—31no—
164#380—32no—
165#340—32no—
166#336C2×D312yescomplemented
167C24×C3C2×D411yescharacteristic
168C24×C3C2×D411yescharacteristic
169C23×D3—21no—
170C23×D3—21no—
171C23×D3—21no—
172C23×D3—24no—
173C23×D3—24no—
174C23×D3—44no—
175C23×D3—44no—
176C23×D3—42no—
177C23×D3—42no—
178C23×D3—42no—
179C23×D3—42no—
180C23×D3—82no—
181C23×D3—82no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4D814yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2411yescharacteristic
198C2×#67—24no—
199C2×#67—24no—
200C2×#67—42no—
201C2×#67—42no—
202C2×#67—42no—
203C2×#67—42no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—42no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210D3×D4—432no—
211C2×D12—21no—
212C2×D12—22no—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—22no—
217C2×D12—28no—
218C2×D12—24no—
219C2×D12—24no—
220C2×D12—24no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—24no—
224C2×D12—44no—
225C2×D12—48no—
226C2×D12—48no—
227C2×D12—44no—
228C2×D12—44no—
229C2×D12—44no—
230C2×D12—42no—
231C2×D12—42no—
232C2×D12—44no—
233C2×D12—82no—
234C2×D12C2×D412yes—
235C2×D12C2×D412yes—
236C2×D12C2×D412yes—
237C2×D12#4114yes—
238C2×C4×D3—24no—
239C2×C4×D3—42no—
240C2×C4×D3—42no—
241C2×C3×C8—22no—
242C2×C3×C8—22no—
243C2×C3×C8—22no—
244C2×C3×C8—22no—
245C2×C3×C8—22no—
246C2×C3×C8—22no—
247C2×C3×C8—44no—
248C3×C4⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—22no—
251C3×C42—21no—
252C3×C42—21no—
253#217—42no—
254#217—44no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—22no—
260C4⋊Dic3—22no—
261C4⋊Dic3—22no—
262C4⋊Dic3—22no—
263C4⋊Dic3—42no—
264C4⋊Dic3C2×D412yes—
265Dic3⋊C4—44no—
266C25—61no—
267C25—61no—
268C22×D4—34no—
269C22×D4—31no—
270C22×D4—31no—
271C22×D4—31no—
272C22×D4—64no—
273C22×D4—62no—
274C22×D4—62no—
275C22×D4—64no—
276C22×D4—68no—
277C22×D4—61no—
278C22×D4—61no—
279C22×D4—61no—
280C22×D4—61no—
281C22×D4—61no—
282C22×D4—61no—
283C22×D4—62no—
284C22×D4—122no—
285C22×D4—122no—
286C22×D4—124no—
287C22×D4—124no—
288C22×D4D1211yescharacteristic
289C23×C4—61no—
290C23×C4—61no—
291C23×C4C22×D311yescharacteristic
292C23×C4C22×D311yescharacteristic
293C22×C8—22no—
294C22×C8—42no—
295C22×C8C22×D312yes—
296C22×C8C22×D312yes—
297#127—62no—
298#127—62no—
299#123—124no—
300#121—34no—
301#121—68no—
302#121—62no—
303#121—62no—
304#121—62no—
305#120—68no—
306#120—68no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—62no—
310C4×D4—62no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—62no—
315C2×C4⋊C4—62no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—62no—
318C2×C4⋊C4D1211yescharacteristic
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—62no—
324C2×C22⋊C4D1211yescharacteristic
325C2×C22⋊C4D1211yescharacteristic
326C2×C42C22×D311yescharacteristic
327#107—64no—
328#102—64no—
329#102—64no—
330#102—64no—
331#102—64no—
332#102—122no—
333#102—122no—
334C22⋊C8—22no—
335C22⋊C8—42no—
336#95—62no—
337#95—61no—
338#95—62no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—22no—
343C23×C3—21no—
344C23×C3—22no—
345C23×C3#12012yes—
346C23×C3C2×D812yes—
347C23×C3C2×D812yes—
348C23×C3AGL(1,ℤ/8)12yes—
349C23×C3C22×D411yescharacteristic
350C22×D3—22no—
351C22×D3—24no—
352C22×D3—22no—
353C22×D3—22no—
354C22×D3—48no—
355C22×D3—48no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—42no—
359C22×D3—48no—
360C22×D3—42no—
361C22×D3—42no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—42no—
367C22×D3—42no—
368C22×D3—48no—
369C22×D3—84no—
370C22×D3—84no—
371C22×D3—88no—
372C22×D3—88no—
373C22×D3—88no—
374C22×D3—88no—
375C3×D4—28no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—24no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—42no—
389C2×C3×C4#12012yes—
390C2×C3×C4C2×D812yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4112yes—
394C2×C3×C4H(5,𝔽2)12yes—
395#67—48no—
396#67—48no—
397C2×Dic3—22no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—84no—
407D12—22no—
408D12—22no—
409D12—22no—
410D12—24no—
411D12—24no—
412D12—48no—
413D12—48no—
414D12—48no—
415D12—41no—
416D12—41no—
417D12—42no—
418D12—84no—
419D12#12314yescomplemented
420C4×D3—48no—
421C3×C8—42no—
422C3×C8—42no—
423C24—64no—
424C24—61no—
425C24—61no—
426C24—61no—
427C24—64no—
428C24—124no—
429C24—124no—
430C24—122no—
431C24—122no—
432C24—122no—
433C24—122no—
434C24—242no—
435C24—242no—
436C24C2×D1211yescharacteristic
437C24C2×D1211yescharacteristic
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—32no—
441C2×D4—32no—
442C2×D4—32no—
443C2×D4—34no—
444C2×D4—68no—
445C2×D4—64no—
446C2×D4—64no—
447C2×D4—64no—
448C2×D4—64no—
449C2×D4—64no—
450C2×D4—64no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—62no—
455C2×D4—62no—
456C2×D4—62no—
457C2×D4—62no—
458C2×D4—62no—
459C2×D4—124no—
460C2×D4—128no—
461C2×D4—124no—
462C2×D4—124no—
463C2×D4—122no—
464C2×D4—124no—
465C2×D4—128no—
466C2×D4—124no—
467C2×D4—1232no—
468C2×D4—122no—
469C2×D4—122no—
470C2×D4—122no—
471C2×D4—122no—
472C2×D4—122no—
473C2×D4—242no—
474C2×D4D2414yes—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—64no—
482C22×C4—61no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4—122no—
489C22×C4—122no—
490C22×C4C2×D1211yescharacteristic
491C22×C4C2×D1211yescharacteristic
492C22×C4C2×D1211yescharacteristic
493C22×C4C2×D1211yescharacteristic
494C22×C4D3×D412yes—
495C22×C4D3×D412yes—
496C22×C4C23×D311yescharacteristic
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—44no—
504C4⋊C4—22no—
505C4⋊C4—32no—
506C4⋊C4—61no—
507C4⋊C4—62no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—62no—
511C4⋊C4—122no—
512C4⋊C4—124no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—64no—
516C22⋊C4—64no—
517C22⋊C4—64no—
518C22⋊C4—122no—
519C22⋊C4—124no—
520C42—21no—
521C42—21no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—24no—
525C22×C3—24no—
526C22×C3—22no—
527C22×C3—24no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—42no—
531C22×C3#44714yes—
532C22×C3C2×#12011yescharacteristic
533C22×C3#54612yes—
534C22×C3C22×D811yescharacteristic
535C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
536C22×C3#57712yes—
537C2×D3—24no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—44no—
541C2×D3—44no—
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—816no—
549C2×D3—816no—
550C2×D3—82no—
551C2×D3—82no—
552C2×D3—82no—
553C2×D3—82no—
554C3×C4—22no—
555C3×C4—22no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4C2×#12311yescharacteristic
559C3×C4#53411yescharacteristic
560C3×C4#54612yes—
561Dic3—42no—
562Dic3—82no—
563Dic3—82no—
564C23—22no—
565C23—22no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—62no—
571C23—64no—
572C23—62no—
573C23—62no—
574C23—128no—
575C23—128no—
576C23—128no—
577C23—121no—
578C23—121no—
579C23—122no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23—122no—
585C23—122no—
586C23—128no—
587C23—122no—
588C23—122no—
589C23—244no—
590C23—248no—
591C23—248no—
592C23—248no—
593C23—244no—
594C23—248no—
595C23#88212yes—
596C23C2×D2412yes—
597C23C2×D2412yes—
598C23#90812yes—
599C23C22×D1211yescharacteristic
600D4—28no—
601D4—34no—
602D4—62no—
603D4—62no—
604D4—62no—
605D4—64no—
606D4—64no—
607D4—128no—
608D4—128no—
609D4—128no—
610D4—128no—
611D4—128no—
612D4—121no—
613D4—121no—
614D4—122no—
615D4—244no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—62no—
630C2×C4—128no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4—122no—
638C2×C4—122no—
639C2×C4—244no—
640C2×C4#88212yes—
641C2×C4C2×D2412yes—
642C2×C4C2×D3×D411yescharacteristic
643C2×C4C2×D3×D411yescharacteristic
644C2×C4#100412yes—
645C2×C4D3×#4112yes—
646C8—42no—
647C8—42no—
648C2×C3—24no—
649C2×C3—42no—
650C2×C3#240511yescharacteristic
651C2×C3C2×#44711yescharacteristic
652C2×C3#300911yescharacteristic
653C2×C3#329412yes—
654C2×C3#331012yes—
655D3—44no—
656D3—82no—
657D3—82no—
658C22—24no—
659C22—22no—
660C22—24no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—41no—
665C22—42no—
666C22—41no—
667C22—64no—
668C22—122no—
669C22—124no—
670C22—122no—
671C22—124no—
672C22—122no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—122no—
677C22—122no—
678C22—2416no—
679C22—2416no—
680C22—242no—
681C22—242no—
682C22—242no—
683C22—242no—
684C22#473214yes—
685C22C2×#88211yescharacteristic
686C22#555412yes—
687C22C22×D2411yescharacteristic
688C22C2×#90811yescharacteristic
689C22#575112yes—
690C4—22no—
691C4—22no—
692C4—41no—
693C4—41no—
694C4—122no—
695C4—242no—
696C4—242no—
697C4#558911yescharacteristic
698C4#561012yes—
699C4D3×#12311yescharacteristic
700C3#2403211yescharacteristic, complemented, Sylow
701C2—24no—
702C2—42no—
703C2—124no—
704C2—242no—
705C2—242no—
706C2#7468611yescharacteristic
707C2C2×#473211yescharacteristic
708C2#7867111yescharacteristic
709C2#7943812yes—
710C2#7950112yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1869C2
socleC23×C3C22×D4
prime radicalC2×D4×D12C2
hypercenter#1869D3
hypocenterC3#24032
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C22×#821C2semidirect
3. #82373C2semidirect
4. C2×#4736C2semidirect
5. C3×#1869C2semidirect
6. #71973C2semidirect
7. #69959C2semidirect
8. #69959C2semidirect
9. #69934C2semidirect
10. #68670C2semidirect
11. #68629C2semidirect
16. C23×C3×C8C22semidirect
21. C22×C4⋊Dic3C22semidirect
29. C2×C3×C22⋊C8C22semidirect
30. C2×C3×C22⋊C8C22semidirect
31. #5212C22semidirect
32. #5212C22semidirect
36. C2×#821C22semidirect
37. C2×#821C22semidirect
38. C2×#821C22semidirect
39. C2×#821C22semidirect
40. C2×#818C22semidirect
44. #4928C22semidirect
46. C3×#336C22semidirect
47. #4551C22semidirect
55. #1869D3semidirect
135. C23×C8C2×D3semidirect
157. C2×C22⋊C8C2×D3semidirect
158. C2×C22⋊C8C2×D3semidirect
166. #336C2×D3semidirect
419. D12#123semidirect
700. C3#24032semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D122
C22×#8212
#823732
C2×#47362
C3×#18692
#719732
#699592
#699342
#686702
#686292
#240323