Finite Group Databasebeta

← #10850477

Subgroups — #10850477

Order 768.

1770 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3496C211yesmaximal, characteristic, complemented
2C2×#5244C212yesmaximal, complemented
3C2×#5243C212yesmaximal, complemented
4#82708C212yesmaximal, complemented
5#82414C211yesmaximal, characteristic, complemented
6#71851C212yesmaximal, complemented
7#71849C212yesmaximal, complemented
8#71686C212yesmaximal, complemented
9C4×#834C211yesmaximal, characteristic, complemented
10#25773—31nomaximal, Sylow
11C2×C3×H(5,𝔽2)C2212yes—
12C3×#547—22no—
13C3×#546—24no—
14C3×D42—22no—
15C3×#537—21no—
16C3×#534—21no—
17C3×#534C2212yes—
18C3×#532—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C2×#937C2211yescharacteristic
23C2×#937C2214yescomplemented
24C2×#935C2211yescharacteristic
25C2×#935C2214yescomplemented
26C22×(C2×D3)⋊C4C2211yescharacteristic, complemented
27#5244—28no—
28#5243—28no—
29#5223—22no—
30C2×#834C2212yes—
31C2×#834C2214yescomplemented
32C2×#834C2214yescomplemented
33#5213—21no—
34#5212C2211yescharacteristic, complemented
35C4×#217C2212yescomplemented
36#4994—21no—
37#4993—22no—
38#4961—21no—
39#4957—24no—
40#4955C2211yescharacteristic, complemented
41#4943C2211yescharacteristic, complemented
42#4940—21no—
43#4526C2212yescomplemented
44#3496D311yescharacteristic, complemented
45C2×#458—32no—
46C2×B(4,𝔽2)—32no—
47#2539—32no—
48#2534—31no—
49#2022—32no—
50#2021—32no—
51#1964—32no—
52C4×#99—31no—
53C24×D3C2311yescharacteristic
54C3×H(5,𝔽2)—24no—
55C3×H(5,𝔽2)—24no—
56C3×H(5,𝔽2)—44no—
57C2×C3×#41—22no—
58C2×C3×#41—24no—
59C22×C3×D4—22no—
60C22×C3×D4—24no—
61C22×C3×D4—21no—
62C22×C3×D4D412yescomplemented
63C22×C3×D4C2314yes—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4D411yescharacteristic, complemented
66C23×C3×C4C2311yescharacteristic
67C22×#67—24no—
68C22×#67—21no—
69C23×Dic3C2311yescharacteristic
70C3×C4⋊Q8—21no—
71C3×#127—21no—
72C3×#124—22no—
73C3×#123—24no—
74C3×#123—24no—
75C3×C22⋊Q8—24no—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—24no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121—22no—
84C3×#121—22no—
85C3×#120—24no—
86C3×#120—24no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D412yescomplemented
93C2×C3×C4⋊C4D411yescharacteristic, complemented
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4D414yescomplemented
96C2×C3×C22⋊C4D411yescharacteristic, complemented
97C2×C3×C22⋊C4C2311yescharacteristic
98C2×C3×C42D411yescharacteristic, complemented
99C2×#217—24no—
100C2×#217—21no—
101C2×#217C2311yescharacteristic
102C2×#217C2311yescharacteristic
103C2×#217C2314yes—
104#937—24no—
105#937—28no—
106#937—44no—
107#935—24no—
108#935—28no—
109#935—44no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—22no—
114C2×(C2×D3)⋊C4—24no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—22no—
117C2×(C2×D3)⋊C4—21no—
118C2×C4⋊Dic3—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—24no—
121C2×C4×Dic3—22no—
122#834—28no—
123#834—28no—
124#834D418yescomplemented
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—22no—
129C2×H(5,𝔽2)C2×D312yes—
130#547—22no—
131#546—24no—
132D42—22no—
133#537—21no—
134#534—21no—
135#534C2×D312yes—
136#532—21no—
137C2×#123—31no—
138C2×#123—34no—
139C2×#121—21no—
140C2×#121C2×D312yes—
141C2×#120—31no—
142C2×#120—34no—
143C2×C4×D4C2×D311yescharacteristic
144C22×C22⋊C4—31no—
145#458—68no—
146B(4,𝔽2)—68no—
147C2×#99—32no—
148C2×#99—34no—
149C2×#99—34no—
150#399—62no—
151#396—61no—
152#394—61no—
153#392—62no—
154#390—64no—
155#388—61no—
156#386—61no—
157#381—31no—
158#380—31no—
159#379—31no—
160C4×C22⋊C4—32no—
161#342—32no—
162C24×C3—22no—
163C24×C3C2×D414yes—
164C24×C3C2×D411yescharacteristic
165C24×C3C2411yescharacteristic
166C23×D3—21no—
167C23×D3—22no—
168C23×D3—22no—
169C23×D3—22no—
170C23×D3—42no—
171C23×D3—81no—
172C3×#41—48no—
173C3×#41—48no—
174C3×#41—44no—
175C3×#41—44no—
176C2×C3×Q8—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—24no—
180C2×C3×D4—22no—
181C2×C3×D4—24no—
182C2×C3×D4—24no—
183C2×C3×D4—24no—
184C2×C3×D4—24no—
185C2×C3×D4—24no—
186C2×C3×D4—24no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—48no—
192C2×C3×D4—48no—
193C2×C3×D4—44no—
194C2×C3×D4—44no—
195C2×C3×D4—82no—
196C2×C3×D4—82no—
197C2×C3×D4C2×D414yes—
198C22×C3×C4—22no—
199C22×C3×C4—24no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—81no—
203C22×C3×C4—81no—
204C22×C3×C4C2×D414yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D411yescharacteristic
212C2×#67—48no—
213C2×#67—48no—
214C2×#67—42no—
215C2×#67—42no—
216C2×#67—84no—
217C2×#67—81no—
218C22×Dic3—24no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—24no—
236C3×C42—21no—
237C3×C42—21no—
238#217—24no—
239#217—22no—
240#217—22no—
241#217—48no—
242#217—42no—
243#217C2×D418yescomplemented
244(C2×D3)⋊C4—48no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—41no—
247(C2×D3)⋊C4—41no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—81no—
251C4⋊Dic3—81no—
252Dic3⋊C4—48no—
253Dic3⋊C4—42no—
254C4×Dic3—82no—
255C25—31no—
256H(5,𝔽2)—24no—
257H(5,𝔽2)—24no—
258H(5,𝔽2)—44no—
259C2×#41—22no—
260C2×#41—24no—
261C22×D4—22no—
262C22×D4—24no—
263C22×D4—21no—
264C22×D4—61no—
265C22×D4—64no—
266C22×D4#6712yescomplemented
267C22×D4C22×D314yes—
268C22×D4C22×D311yescharacteristic
269C23×C4—31no—
270C23×C4#6711yescharacteristic, complemented
271C23×C4C22×D311yescharacteristic
272C4⋊Q8—21no—
273#127—21no—
274#124—22no—
275#123—24no—
276#123—24no—
277#123—68no—
278#123—64no—
279#123—124no—
280C22⋊Q8—24no—
281#121—22no—
282#121—22no—
283#121—24no—
284#121—22no—
285#121—22no—
286#121—22no—
287#121—22no—
288#121—22no—
289#121—22no—
290#120—24no—
291#120—24no—
292#120—68no—
293#120—64no—
294#120—124no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—64no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4#6711yescharacteristic, complemented
304C2×C4⋊C4#6712yescomplemented
305C2×C22⋊C4—22no—
306C2×C22⋊C4—34no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—64no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—62no—
313C2×C22⋊C4—62no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—64no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4#6711yescharacteristic, complemented
320C2×C22⋊C4#6714yescomplemented
321C2×C22⋊C4C22×D311yescharacteristic
322C2×C42—62no—
323C2×C42#6711yescharacteristic, complemented
324#99—38no—
325#99—68no—
326#99—68no—
327#95—62no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—24no—
334C23×C3—24no—
335C23×C3—24no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—44no—
341C23×C3—44no—
342C23×C3—44no—
343C23×C3—82no—
344C23×C3—82no—
345C23×C3#12712yes—
346C23×C3C22×D412yes—
347C23×C3C22×D411yescharacteristic
348C23×C3C22×D412yes—
349C23×C3H(5,𝔽2)12yes—
350C22×D3—22no—
351C22×D3—24no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—44no—
355C22×D3—41no—
356C22×D3—42no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—44no—
361C22×D3—84no—
362C22×D3—84no—
363C3×Q8—44no—
364C3×D4—28no—
365C3×D4—44no—
366C3×D4—44no—
367C3×D4—48no—
368C3×D4—44no—
369C3×D4—44no—
370C3×D4—44no—
371C3×D4—44no—
372C3×D4—82no—
373C3×D4—82no—
374C3×D4—82no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—42no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—48no—
388C2×C3×C4—44no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—81no—
392C2×C3×C4#12011yescharacteristic
393C2×C3×C4#12011yescharacteristic
394C2×C3×C4#12012yes—
395C2×C3×C4#12712yes—
396#67—816no—
397#67—84no—
398C2×Dic3—28no—
399C2×Dic3—44no—
400C2×Dic3—44no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—44no—
411C24—22no—
412C24—61no—
413C24—62no—
414C24—62no—
415C24—62no—
416C24—122no—
417C24—241no—
418C24C2×#6714yes—
419C24C2×#6711yescharacteristic
420C24C23×D311yescharacteristic
421#41—48no—
422#41—48no—
423#41—44no—
424#41—44no—
425C2×Q8—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—48no—
441C2×D4—48no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—82no—
445C2×D4—82no—
446C2×D4—128no—
447C2×D4—128no—
448C2×D4—122no—
449C2×D4—122no—
450C2×D4—244no—
451C2×D4—241no—
452C2×D4D3×D414yes—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—64no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—62no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4C2×#6714yes—
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6712yes—
472C22×C4C2×#6712yes—
473C22×C4C2×#6711yescharacteristic
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—128no—
480C4⋊C4—122no—
481C4⋊C4—241no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—22no—
485C22⋊C4—24no—
486C22⋊C4—22no—
487C22⋊C4—24no—
488C22⋊C4—38no—
489C22⋊C4—64no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—128no—
493C22⋊C4—122no—
494C22⋊C4—128no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—121no—
498C22⋊C4—121no—
499C22⋊C4—122no—
500C22⋊C4—241no—
501C42—21no—
502C42—21no—
503C42—242no—
504C22×C3—21no—
505C22×C3—22no—
506C22×C3—21no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3—44no—
517C22×C3—81no—
518C22×C3—84no—
519C22×C3—84no—
520C22×C3—164no—
521C22×C3C2×#12011yescharacteristic
522C22×C3C2×#12711yescharacteristic
523C22×C3#53511yescharacteristic
524C22×C3D4214yes—
525C2×D3—48no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—84no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C3×C4—22no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—22no—
538C3×C4—44no—
539C3×C4—41no—
540C3×C4—41no—
541Dic3—48no—
542Dic3—81no—
543Dic3—81no—
544Dic3—81no—
545C23—21no—
546C23—21no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—62no—
558C23—64no—
559C23—82no—
560C23—82no—
561C23—121no—
562C23—121no—
563C23—124no—
564C23—121no—
565C23—122no—
566C23—124no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—244no—
571C23—244no—
572C23#94012yes—
573C23C2×D3×D412yes—
574C23#100412yes—
575C23C22×#6712yes—
576C23C22×#6711yescharacteristic
577Q8—44no—
578D4—28no—
579D4—44no—
580D4—44no—
581D4—48no—
582D4—44no—
583D4—44no—
584D4—44no—
585D4—44no—
586D4—82no—
587D4—82no—
588D4—82no—
589D4—2416no—
590D4—244no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—68no—
608C2×C4—81no—
609C2×C4—124no—
610C2×C4—124no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—124no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4#94012yes—
622C2×C4#95311yescharacteristic
623C2×C4#95311yescharacteristic
624C2×C4#95312yes—
625C2×C3—24no—
626C2×C3—21no—
627C2×C3—21no—
628C2×C3—42no—
629C2×C3—84no—
630C2×C3#243311yescharacteristic
631C2×C3#302511yescharacteristic
632C2×C3#302611yescharacteristic
633D3—82no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—41no—
643C22—41no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—81no—
648C22—84no—
649C22—84no—
650C22—128no—
651C22—122no—
652C22—122no—
653C22—122no—
654C22—164no—
655C22—244no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22D4×#6714yes—
661C22C2×#94011yescharacteristic
662C22#580511yescharacteristic
663C22C2×#95311yescharacteristic
664C4—22no—
665C4—21no—
666C4—21no—
667C4—22no—
668C4—44no—
669C4—41no—
670C4—41no—
671C4—128no—
672C4—241no—
673C4—241no—
674C4—241no—
675C3#2577311yescharacteristic, complemented, Sylow
676C2—24no—
677C2—21no—
678C2—21no—
679C2—42no—
680C2—84no—
681C2—242no—
682C2#8404911yescharacteristic
683C2#8411011yescharacteristic
684C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#3496C2
socleC22×C3C2×#120
hypercenter#3496D3
hypocenterC3#25773
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C3×#3496C2semidirect
2. C2×#5244C2semidirect
3. C2×#5243C2semidirect
4. #82708C2semidirect
5. #82414C2semidirect
6. #71851C2semidirect
7. #71849C2semidirect
8. #71686C2semidirect
9. C4×#834C2semidirect
23. C2×#937C22semidirect
25. C2×#935C22semidirect
26. C22×(C2×D3)⋊C4C22semidirect
31. C2×#834C22semidirect
32. C2×#834C22semidirect
34. #5212C22semidirect
35. C4×#217C22semidirect
40. #4955C22semidirect
41. #4943C22semidirect
43. #4526C22semidirect
44. #3496D3semidirect
62. C22×C3×D4D4semidirect
65. C23×C3×C4D4semidirect
92. C2×C3×C4⋊C4D4semidirect
93. C2×C3×C4⋊C4D4semidirect
95. C2×C3×C22⋊C4D4semidirect
96. C2×C3×C22⋊C4D4semidirect
98. C2×C3×C42D4semidirect
124. #834D4semidirect
243. #217C2×D4semidirect
266. C22×D4#67semidirect
270. C23×C4#67semidirect
303. C2×C4⋊C4#67semidirect
304. C2×C4⋊C4#67semidirect
319. C2×C22⋊C4#67semidirect
320. C2×C22⋊C4#67semidirect
323. C2×C42#67semidirect
675. C3#25773semidirect

Maximal subgroups

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

subgroupindex
C3×#34962
C2×#52442
C2×#52432
#827082
#824142
#718512
#718492
#716862
C4×#8342
#257733