Finite Group Databasebeta

← #10943900

Subgroups — #10943900

Order 768.

3688 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#585C211yesmaximal, characteristic, complemented
2C2×#5245C212yesmaximal, complemented
3C2×#5241C212yesmaximal, complemented
4#84092C211yesmaximal, characteristic, complemented
5#84082C212yesmaximal, complemented
6#72143C212yesmaximal, complemented
7#72140C212yesmaximal, complemented
8#72138C212yesmaximal, complemented
9#72137C211yesmaximal, characteristic, complemented
10#32890—31nomaximal, Sylow
11C3×#585—28no—
12C3×#585—28no—
13C2×C3×2−1+4—22no—
14C2×C3×2−1+4C2212yes—
15C2×C3×H(5,𝔽2)—22no—
16C2×C3×H(5,𝔽2)—22no—
17C2×C3×H(5,𝔽2)C2212yes—
18C22×C3×#41—24no—
19C22×C3×#41—22no—
20C22×C3×#41C2212yes—
21C22×C3×#41C2211yescharacteristic
22C2×#949C2214yescomplemented
23C2×#947C2211yescharacteristic, complemented
24C2×#940C2211yescharacteristic, complemented
25C2×#930C2211yescharacteristic, complemented
26#5245—28no—
27#5241—28no—
28#5238—24no—
29C2×#837C2214yescomplemented
30C2×#837C2214yescomplemented
31#5232C2214yescomplemented
32#5229C2211yescharacteristic, complemented
33#5227—22no—
34C2×#836C2212yescomplemented
35#5223—22no—
36#5221C2211yescharacteristic, complemented
37#5217—22no—
38C2×#833C2212yescomplemented
39#5213—22no—
40C4×#217C2211yescharacteristic, complemented
41#5207C2212yescomplemented
42C2×#585D311yescharacteristic, complemented
43C2×#454—32no—
44C2×#453—32no—
45#3006—32no—
46#2431—31no—
47#1793—32no—
48#1792—32no—
49#1789—32no—
50#1785—31no—
51C3×2−1+4—232no—
52C3×2−1+4—28no—
53C3×2−1+4—24no—
54C3×2−1+4D418yescomplemented
55C3×H(5,𝔽2)—216no—
56C3×H(5,𝔽2)—232no—
57C3×H(5,𝔽2)—216no—
58C3×H(5,𝔽2)—28no—
59C3×H(5,𝔽2)—24no—
60C3×H(5,𝔽2)D418yescomplemented
61C2×C3×#41—24no—
62C2×C3×#41—28no—
63C2×C3×#41—24no—
64C2×C3×#41—232no—
65C2×C3×#41—232no—
66C2×C3×#41—216no—
67C2×C3×#41—216no—
68C2×C3×#41—28no—
69C2×C3×#41—28no—
70C2×C3×#41—24no—
71C2×C3×#41—24no—
72C2×C3×#41—24no—
73C2×C3×#41—24no—
74C2×C3×#41—22no—
75C2×C3×#41—22no—
76C2×C3×#41—24no—
77C2×C3×#41—22no—
78C2×C3×#41—22no—
79C2×C3×#41D414yescomplemented
80C2×C3×#41C2314yes—
81C22×C3×Q8—22no—
82C22×C3×Q8—24no—
83C22×C3×Q8D412yescomplemented
84C22×C3×Q8C2311yescharacteristic
85C22×C3×D4—28no—
86C22×C3×D4—22no—
87C22×C3×D4—24no—
88C22×C3×D4—22no—
89C22×C3×D4—24no—
90C22×C3×D4D412yes—
91C22×C3×D4D412yescomplemented
92C22×C3×D4C2311yescharacteristic
93C23×C3×C4—24no—
94C23×C3×C4—22no—
95C23×C3×C4D412yes—
96C23×C3×C4C2311yescharacteristic
97C22×#67—22no—
98C22×#67—21no—
99C22×D12C2311yescharacteristic
100C2×#217—21no—
101C2×#217—22no—
102#949—28no—
103#949—28no—
104#947—24no—
105C2×#215—24no—
106#940—24no—
107C2×#213—24no—
108#930—22no—
109#930—22no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—22no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C2×#208C2314yes—
116#837—28no—
117#837—28no—
118#836—24no—
119#835—24no—
120#833—24no—
121#832—24no—
122C3⋊#95—21no—
123C3⋊#95—22no—
124#830—22no—
125#830—22no—
126#585—28no—
127#585—28no—
128C2×2−1+4—22no—
129C2×2−1+4C2×D312yes—
130C2×H(5,𝔽2)—22no—
131C2×H(5,𝔽2)—22no—
132C2×H(5,𝔽2)C2×D312yes—
133C22×#41—24no—
134C22×#41—22no—
135C22×#41C2×D312yes—
136C22×#41C2×D311yescharacteristic
137C2×AGL(1,ℤ/8)—34no—
138C2×#127—31no—
139C2×#124—31no—
140C2×#121—31no—
141#454—68no—
142#453—68no—
143#449—64no—
144#448—62no—
145#447—62no—
146C2×#104—34no—
147C2×#104—34no—
148#417—34no—
149C2×#101—32no—
150C2×#100—32no—
151#407—32no—
152#392—62no—
153#391—31no—
154#390—31no—
155#386—62no—
156C4×C22⋊C4—31no—
157C24×C3—24no—
158C24×C3—22no—
159C24×C3C2×D412yes—
160C24×C3C2×D411yescharacteristic
161C23×D3—21no—
162C3×#41—232no—
163C3×#41—28no—
164C3×#41—28no—
165C3×#41—216no—
166C3×#41—264no—
167C3×#41—232no—
168C3×#41—216no—
169C3×#41—216no—
170C3×#41—216no—
171C3×#41—216no—
172C3×#41—216no—
173C3×#41—28no—
174C3×#41—216no—
175C3×#41—28no—
176C3×#41—216no—
177C3×#41—216no—
178C3×#41—24no—
179C3×#41—24no—
180C3×#41C2×D418yes—
181C3×#41C2×D418yes—
182C2×C3×Q8—24no—
183C2×C3×Q8—216no—
184C2×C3×Q8—216no—
185C2×C3×Q8—216no—
186C2×C3×Q8—28no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8C2×D414yes—
191C2×C3×Q8C2×D414yes—
192C2×C3×D4—28no—
193C2×C3×D4—24no—
194C2×C3×D4—264no—
195C2×C3×D4—216no—
196C2×C3×D4—216no—
197C2×C3×D4—28no—
198C2×C3×D4—28no—
199C2×C3×D4—216no—
200C2×C3×D4—24no—
201C2×C3×D4—24no—
202C2×C3×D4—216no—
203C2×C3×D4—24no—
204C2×C3×D4—24no—
205C2×C3×D4—216no—
206C2×C3×D4—28no—
207C2×C3×D4—22no—
208C2×C3×D4—24no—
209C2×C3×D4—22no—
210C2×C3×D4—22no—
211C2×C3×D4C2×D414yes—
212C2×C3×D4C2×D414yes—
213C22×C3×C4—24no—
214C22×C3×C4—28no—
215C22×C3×C4—232no—
216C22×C3×C4—216no—
217C22×C3×C4—22no—
218C22×C3×C4—24no—
219C22×C3×C4—24no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—24no—
223C22×C3×C4—24no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2×D412yes—
230C22×C3×C4C2411yescharacteristic
231C2×#67—44no—
232C2×#67—42no—
233C2×#67—44no—
234C2×#67—82no—
235C2×#67—82no—
236C22×Dic3—21no—
237C22×Dic3—21no—
238C22×Dic3—21no—
239C2×D12—22no—
240C2×D12—22no—
241C2×D12—22no—
242#217—42no—
243#217—82no—
244#215—48no—
245#213—48no—
246(C2×D3)⋊C4—41no—
247(C2×D3)⋊C4—41no—
248(C2×D3)⋊C4—44no—
249C4⋊Dic3—22no—
250C4×Dic3—22no—
251C4×Dic3—22no—
252#208—24no—
253#208—24no—
254C2×C3⋊C8—24no—
2552−1+4—232no—
2562−1+4—28no—
2572−1+4—24no—
2582−1+4#6718yescomplemented
259H(5,𝔽2)—216no—
260H(5,𝔽2)—232no—
261H(5,𝔽2)—216no—
262H(5,𝔽2)—28no—
263H(5,𝔽2)—24no—
264H(5,𝔽2)#6718yescomplemented
265C2×#41—24no—
266C2×#41—28no—
267C2×#41—24no—
268C2×#41—232no—
269C2×#41—232no—
270C2×#41—216no—
271C2×#41—216no—
272C2×#41—28no—
273C2×#41—28no—
274C2×#41—24no—
275C2×#41—24no—
276C2×#41—24no—
277C2×#41—24no—
278C2×#41—22no—
279C2×#41—22no—
280C2×#41—24no—
281C2×#41—22no—
282C2×#41—22no—
283C2×#41#6714yescomplemented
284C2×#41C22×D314yes—
285C22×Q8—22no—
286C22×Q8—24no—
287C22×Q8#6712yescomplemented
288C22×Q8C22×D311yescharacteristic
289C22×D4—28no—
290C22×D4—22no—
291C22×D4—24no—
292C22×D4—22no—
293C22×D4—24no—
294C22×D4—31no—
295C22×D4—62no—
296C22×D4—61no—
297C22×D4#6712yes—
298C22×D4#6712yescomplemented
299C22×D4C22×D311yescharacteristic
300C23×C4—24no—
301C23×C4—22no—
302C23×C4#6712yes—
303C23×C4C22×D311yescharacteristic
304AGL(1,ℤ/8)—68no—
305AGL(1,ℤ/8)—68no—
306C2×SD16—64no—
307C2×D8—64no—
308C2×M16—34no—
309#127—64no—
310#124—64no—
311#121—62no—
312#121—62no—
313C2×C4⋊C4—31no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—62no—
318C2×C42—31no—
319C2×C42—31no—
320#104—68no—
321#104—68no—
322#103—64no—
323#102—64no—
324#101—64no—
325#100—64no—
326C22⋊C8—62no—
327C22⋊C8—62no—
328#95—61no—
329#95—62no—
330C23×C3—28no—
331C23×C3—216no—
332C23×C3—22no—
333C23×C3—24no—
334C23×C3—21no—
335C23×C3—22no—
336C23×C3—21no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—82no—
342C23×C3—82no—
343C23×C3—84no—
344C23×C3—84no—
345C23×C3#12012yes—
346C23×C3#12014yes—
347C23×C3C22×D411yescharacteristic
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C3×Q8—216no—
354C3×Q8—28no—
355C3×Q8—24no—
356C3×Q8—28no—
357C3×Q8#12018yes—
358C3×D4—232no—
359C3×D4—216no—
360C3×D4—216no—
361C3×D4—216no—
362C3×D4—28no—
363C3×D4—28no—
364C3×D4—28no—
365C3×D4—28no—
366C3×D4—24no—
367C3×D4#12018yes—
368C2×C3×C4—216no—
369C2×C3×C4—28no—
370C2×C3×C4—216no—
371C2×C3×C4—28no—
372C2×C3×C4—232no—
373C2×C3×C4—28no—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—24no—
380C2×C3×C4—28no—
381C2×C3×C4—28no—
382C2×C3×C4—24no—
383C2×C3×C4#12011yescharacteristic
384C2×C3×C4#12011yescharacteristic
385C2×C3×C4#12014yes—
386C2×C3×C4C22×D412yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D412yes—
389#67—88no—
390#67—84no—
391C2×Dic3—22no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—42no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399D12—44no—
400C3⋊C8—44no—
401C24—24no—
402C24—22no—
403C24—61no—
404C24C2×#6712yes—
405C24C2×#6711yescharacteristic
406#41—232no—
407#41—28no—
408#41—28no—
409#41—216no—
410#41—264no—
411#41—232no—
412#41—216no—
413#41—216no—
414#41—216no—
415#41—216no—
416#41—216no—
417#41—28no—
418#41—216no—
419#41—28no—
420#41—216no—
421#41—216no—
422#41—24no—
423#41—24no—
424#41C2×#6718yes—
425#41C2×#6718yes—
426C2×Q8—24no—
427C2×Q8—216no—
428C2×Q8—216no—
429C2×Q8—216no—
430C2×Q8—28no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8C2×#6714yes—
435C2×Q8C2×#6714yes—
436C2×D4—28no—
437C2×D4—24no—
438C2×D4—264no—
439C2×D4—216no—
440C2×D4—216no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—216no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—216no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—216no—
450C2×D4—28no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—62no—
456C2×D4—62no—
457C2×D4—62no—
458C2×D4—124no—
459C2×D4—122no—
460C2×D4—124no—
461C2×D4—242no—
462C2×D4—242no—
463C2×D4C2×#6714yes—
464C2×D4C2×#6714yes—
465C22×C4—24no—
466C22×C4—28no—
467C22×C4—232no—
468C22×C4—216no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4C2×#6711yescharacteristic
482C22×C4C2×#6712yes—
483C22×C4C2×#6712yes—
484C22×C4C2×#6712yes—
485C22×C4C23×D311yescharacteristic
486SD16—128no—
487D8—128no—
488M16—64no—
489M16—64no—
490C2×C8—64no—
491C4⋊C4—62no—
492C22⋊C4—122no—
493C22⋊C4—121no—
494C22⋊C4—121no—
495C22⋊C4—124no—
496C22⋊C4—242no—
497C42—62no—
498C42—62no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—22no—
502C22×C3—24no—
503C22×C3—28no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—44no—
509C22×C3—42no—
510C22×C3—44no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—82no—
514C22×C3—88no—
515C22×C3—816no—
516C22×C3C2×#12011yescharacteristic
517C22×C3C2×#12011yescharacteristic
518C22×C3C2×#12011yescharacteristic
519C2×D3—82no—
520C2×D3—82no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C3×C4—21no—
525C3×C4—28no—
526C3×C4—24no—
527C3×C4—21no—
528C3×C4C2×#12014yes—
529Dic3—42no—
530Dic3—81no—
531Dic3—81no—
532C23—28no—
533C23—216no—
534C23—22no—
535C23—24no—
536C23—21no—
537C23—22no—
538C23—21no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—82no—
544C23—82no—
545C23—84no—
546C23—84no—
547C23—122no—
548C23—122no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23#95312yes—
553C23#95314yes—
554C23C22×#6711yescharacteristic
555Q8—216no—
556Q8—28no—
557Q8—24no—
558Q8—28no—
559Q8#95318yes—
560D4—232no—
561D4—216no—
562D4—216no—
563D4—216no—
564D4—28no—
565D4—28no—
566D4—28no—
567D4—28no—
568D4—24no—
569D4—124no—
570D4—248no—
571D4—244no—
572D4#95318yes—
573C2×C4—216no—
574C2×C4—28no—
575C2×C4—216no—
576C2×C4—28no—
577C2×C4—232no—
578C2×C4—28no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—62no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4#95314yes—
597C2×C4#95311yescharacteristic
598C2×C4#95311yescharacteristic
599C2×C4C22×#6712yes—
600C2×C4C22×#6712yes—
601C2×C4C22×#6712yes—
602C8—124no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—42no—
606C2×C3—44no—
607C2×C3—48no—
608C2×C3#284611yescharacteristic
609C2×C3#301811yescharacteristic
610C2×C3#302111yescharacteristic
611D3—161no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—28no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—44no—
624C22—81no—
625C22—81no—
626C22—82no—
627C22—88no—
628C22—816no—
629C22—242no—
630C22—242no—
631C22—241no—
632C22—241no—
633C22—241no—
634C22C2×#95311yescharacteristic
635C22C2×#95311yescharacteristic
636C22C2×#95311yescharacteristic
637C4—21no—
638C4—28no—
639C4—24no—
640C4—21no—
641C4—122no—
642C4—241no—
643C4—241no—
644C4C2×#95314yes—
645C3#3289011yescharacteristic, complemented, Sylow
646C2—21no—
647C2—21no—
648C2—42no—
649C2—44no—
650C2—48no—
651C2—481no—
652C2#8412111yescharacteristic
653C2#8412211yescharacteristic
654C2#8412811yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#585C2
socleC22×C3C2×#120
hypercenterC2×#585D3
hypocenterC3#32890
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#585

Complements

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

NH ≅ G/Ntype
1. C2×C3×#585C2semidirect
2. C2×#5245C2semidirect
3. C2×#5241C2semidirect
4. #84092C2semidirect
5. #84082C2semidirect
6. #72143C2semidirect
7. #72140C2semidirect
8. #72138C2semidirect
9. #72137C2semidirect
22. C2×#949C22semidirect
23. C2×#947C22semidirect
24. C2×#940C22semidirect
25. C2×#930C22semidirect
29. C2×#837C22semidirect
30. C2×#837C22semidirect
31. #5232C22semidirect
32. #5229C22semidirect
34. C2×#836C22semidirect
36. #5221C22semidirect
38. C2×#833C22semidirect
40. C4×#217C22semidirect
41. #5207C22semidirect
42. C2×#585D3semidirect
54. C3×2−1+4D4semidirect
60. C3×H(5,𝔽2)D4semidirect
79. C2×C3×#41D4semidirect
83. C22×C3×Q8D4semidirect
91. C22×C3×D4D4semidirect
258. 2−1+4#67semidirect
264. H(5,𝔽2)#67semidirect
283. C2×#41#67semidirect
287. C22×Q8#67semidirect
298. C22×D4#67semidirect
645. C3#32890semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5852
C2×#52452
C2×#52412
#840922
#840822
#721432
#721402
#721382
#721372
#328903