Finite Group Databasebeta

← #10900846

Subgroups — #10900846

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic, complemented
2C2×#5238C211yesmaximal, characteristic, complemented
3C2×#5217C211yesmaximal, characteristic, complemented
4#82288C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic, complemented
6#72128C211yesmaximal, characteristic, complemented
7#71767C212yesmaximal, complemented
8#71762C212yesmaximal, complemented
9#71728C212yesmaximal, complemented
10#71722C212yesmaximal, complemented
11#71584C211yesmaximal, characteristic, complemented
12#29630—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C3×#535—24no—
16C3×#535—24no—
17C2×C3×#127C2212yes—
18C2×C3×#124C2212yes—
19C2×C3×#121—21no—
20C2×C3×#121—21no—
21C2×C3×#120—22no—
22C2×C3×#117C2211yescharacteristic
23C22×#215C2211yescharacteristic, complemented
24C22×#213C2212yes—
25C22×#213C2211yescharacteristic, complemented
26C2×#930C2211yescharacteristic
27#5238—24no—
28C2×#835C2211yescharacteristic, complemented
29C2×#835C2212yescomplemented
30#5217—24no—
31C2×#832C2212yes—
32C2×#832C2212yes—
33C2×#832C2212yes—
34C2×#832C2211yescharacteristic, complemented
35C2×#832C2212yescomplemented
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39#4993—21no—
40#4992—21no—
41#4959—22no—
42#4939C2212yescomplemented
43#4928C2212yescomplemented
44#4524C2212yescomplemented
45#4524C2212yescomplemented
46C2×#535D311yescharacteristic, complemented
47C2×#449—31no—
48C2×#447—31no—
49#2624—31no—
50#2042—32no—
51#2041—32no—
52#2012—32no—
53#2011—32no—
54#1818—31no—
55#1788—31no—
56#1786—31no—
57C25×C3—21no—
58C2×C3×#41—22no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C2×C3×#41D414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4—42no—
69C22×C3×D4—42no—
70C22×C3×D4D414yes—
71C22×C3×D4C2312yes—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C22×#67—21no—
76C22×D12C2311yescharacteristic
77C3×#127—24no—
78C3×#127—24no—
79C3×#124—24no—
80C3×#124—24no—
81C3×#121—28no—
82C3×#121—28no—
83C3×#120—216no—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D412yes—
91C2×C3×C42C2312yes—
92C2×#217—21no—
93C2×#215—22no—
94C2×#215—42no—
95C2×#213—24no—
96C2×#213—22no—
97C2×#213—42no—
98C2×#213—42no—
99C2×#213—42no—
100#930—22no—
101#930—22no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—22no—
104C2×(C2×D3)⋊C4—22no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2312yes—
107C2×Dic3⋊C4—22no—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2312yes—
111#835—22no—
112#835—42no—
113#832—24no—
114#832—22no—
115#832—42no—
116#832—42no—
117#832—42no—
118C3⋊#95—22no—
119C3⋊#95—21no—
120#830—22no—
121#830—22no—
122C22×#41C2×D311yescharacteristic
123C23×D4C2×D311yescharacteristic
124C22×SD16—31no—
125C22×D8—32no—
126C22×D8—31no—
127#535—24no—
128#535—24no—
129C2×#127C2×D312yes—
130C2×#124C2×D312yes—
131C2×#121—21no—
132C2×#121—21no—
133C2×#121—31no—
134C2×#120—22no—
135C2×#117C2×D311yescharacteristic
136#449—64no—
137#447—64no—
138C2×#103—31no—
139C2×#103—32no—
140C2×#102—32no—
141C2×#102—31no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—31no—
147#399—61no—
148#397—61no—
149#394—62no—
150#390—32no—
151#389—32no—
152#380—31no—
153#336—32no—
154#336—32no—
155C24×C3—22no—
156C24×C3—21no—
157C24×C3—24no—
158C24×C3—44no—
159C24×C3—81no—
160C24×C3—82no—
161C24×C3—81no—
162C24×C3C2×D411yescharacteristic
163C23×D3—21no—
164C3×#41—44no—
165C3×#41—44no—
166C3×#41—48no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8C2×D412yes—
171C2×C3×D4—24no—
172C2×C3×D4—28no—
173C2×C3×D4—28no—
174C2×C3×D4—24no—
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—48no—
184C2×C3×D4—44no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—82no—
189C2×C3×D4—82no—
190C2×C3×D4—82no—
191C2×C3×D4—82no—
192C2×C3×D4C2×D414yes—
193C2×C3×D4C2×D412yes—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—24no—
198C22×C3×C4—44no—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2411yescharacteristic
205C2×#67—42no—
206C2×#67—82no—
207C22×Dic3—21no—
208C22×Dic3—22no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—22no—
212C2×D12—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C22⋊C4—28no—
216C3×C22⋊C4—28no—
217C3×C22⋊C4—24no—
218C3×C42—22no—
219C3×C42—22no—
220#217—81no—
221#215—44no—
222#213—48no—
223#213—44no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—82no—
226(C2×D3)⋊C4—82no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—22no—
230C4⋊Dic3—22no—
231Dic3⋊C4—82no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C25—21no—
240C2×#41—22no—
241C2×#41—42no—
242C2×#41—42no—
243C2×#41#6714yes—
244C22×Q8C22×D311yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—31no—
251C22×D4—42no—
252C22×D4—42no—
253C22×D4—61no—
254C22×D4#6714yes—
255C22×D4C22×D312yes—
256C22×D4C22×D311yescharacteristic
257C23×C4—21no—
258C23×C4C22×D311yescharacteristic
259C2×SD16—62no—
260C2×SD16—122no—
261C2×D8—64no—
262C2×D8—62no—
263C2×D8—122no—
264C2×D8—122no—
265C2×D8—122no—
266C22×C8—31no—
267C22×C8—32no—
268C22×C8—31no—
269#127—24no—
270#127—24no—
271#124—24no—
272#124—24no—
273#121—28no—
274#121—28no—
275#121—62no—
276#121—62no—
277#120—216no—
278#117—22no—
279#117—22no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4#6711yescharacteristic
284C2×C4⋊C4#6711yescharacteristic
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4#6712yes—
292C2×C42C22×D312yes—
293#103—62no—
294#103—122no—
295#102—64no—
296#102—62no—
297#102—122no—
298#102—122no—
299#102—122no—
300C22⋊C8—62no—
301C22⋊C8—62no—
302#95—61no—
303#95—62no—
304C23×C3—22no—
305C23×C3—21no—
306C23×C3—22no—
307C23×C3—21no—
308C23×C3—24no—
309C23×C3—22no—
310C23×C3—48no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—84no—
322C23×C3—84no—
323C23×C3—88no—
324C23×C3—88no—
325C23×C3#12012yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—41no—
328C22×D3—42no—
329C22×D3—41no—
330C22×D3—42no—
331C22×D3—41no—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—42no—
335C3×D4—24no—
336C3×D4—22no—
337C3×D4—24no—
338C3×D4—22no—
339C3×D4—48no—
340C3×D4—48no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—88no—
345C3×D4—88no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—48no—
358C2×C3×C4—42no—
359C2×C3×C4#12012yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4H(5,𝔽2)12yes—
363C2×C3×C4H(5,𝔽2)12yes—
364#67—84no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370D12—42no—
371D12—41no—
372D12—41no—
373C3⋊C8—42no—
374C3⋊C8—41no—
375C3⋊C8—41no—
376C24—22no—
377C24—21no—
378C24—24no—
379C24—44no—
380C24—61no—
381C24—81no—
382C24—82no—
383C24—81no—
384C24C2×#6711yescharacteristic
385#41—44no—
386#41—44no—
387#41—48no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8C2×#6712yes—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—122no—
418C2×D4—242no—
419C2×D4C2×#6714yes—
420C2×D4C2×#6712yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4C2×#6712yes—
429C22×C4C2×#6712yes—
430C22×C4C2×#6711yescharacteristic
431C22×C4C2×#6712yes—
432C22×C4C2×#6712yes—
433C22×C4C23×D311yescharacteristic
434SD16—124no—
435D8—128no—
436D8—124no—
437C2×C8—62no—
438C2×C8—62no—
439C2×C8—62no—
440C2×C8—62no—
441C2×C8—62no—
442C2×C8—61no—
443C2×C8—61no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—62no—
447C4⋊C4—62no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—242no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—122no—
455C22⋊C4—242no—
456C22⋊C4—242no—
457C22⋊C4—241no—
458C42—22no—
459C42—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—44no—
467C22×C3—42no—
468C22×C3—44no—
469C22×C3—42no—
470C22×C3—44no—
471C22×C3—816no—
472C22×C3—82no—
473C22×C3—82no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3C2×#12011yescharacteristic
479C22×C3#53512yes—
480C22×C3#57611yescharacteristic
481C22×C3#57612yes—
482C22×C3#57711yescharacteristic
483C2×D3—81no—
484C2×D3—82no—
485C2×D3—82no—
486C2×D3—81no—
487C2×D3—81no—
488C3×C4—22no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4#53412yes—
493C3×C4#53411yescharacteristic
494C3×C4#53411yescharacteristic
495Dic3—82no—
496Dic3—81no—
497C23—22no—
498C23—21no—
499C23—22no—
500C23—21no—
501C23—24no—
502C23—22no—
503C23—48no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—84no—
515C23—84no—
516C23—88no—
517C23—88no—
518C23—122no—
519C23—121no—
520C23—122no—
521C23—121no—
522C23—121no—
523C23#95312yes—
524C23C22×#6711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—42no—
528D4—24no—
529D4—22no—
530D4—24no—
531D4—22no—
532D4—48no—
533D4—48no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—88no—
538D4—88no—
539D4—122no—
540D4—121no—
541D4—121no—
542D4—244no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—121no—
561C2×C4#95312yes—
562C2×C4#100412yes—
563C2×C4#100412yes—
564C2×C4C22×#6711yescharacteristic
565C2×C4C22×#6711yescharacteristic
566C8—122no—
567C8—121no—
568C8—121no—
569C2×C3—42no—
570C2×C3—82no—
571C2×C3—84no—
572C2×C3#261311yescharacteristic
573C2×C3#300811yescharacteristic
574C2×C3#301211yescharacteristic
575C2×C3#322012yes—
576C2×C3#322412yes—
577D3—161no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22—816no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—241no—
597C22—241no—
598C22—241no—
599C22—242no—
600C22—242no—
601C22#580512yes—
602C22#583512yes—
603C22#583511yescharacteristic
604C22#583611yescharacteristic
605C22C2×#95311yescharacteristic
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—241no—
611C4—242no—
612C4#580412yes—
613C4#580411yescharacteristic
614C4#580411yescharacteristic
615C3#2963011yescharacteristic, complemented, Sylow
616C2—42no—
617C2—84no—
618C2—82no—
619C2—481no—
620C2#8338812yes—
621C2#8338912yes—
622C2#8405411yescharacteristic
623C2#8410111yescharacteristic
624C2#8410211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#535C2
socleC23×C3C22×D4
hypercenterC2×#535D3
hypocenterC3#29630
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. C2×#5238C2semidirect
3. C2×#5217C2semidirect
4. #82288C2semidirect
5. #72135C2semidirect
6. #72128C2semidirect
7. #71767C2semidirect
8. #71762C2semidirect
9. #71728C2semidirect
10. #71722C2semidirect
11. #71584C2semidirect
23. C22×#215C22semidirect
25. C22×#213C22semidirect
28. C2×#835C22semidirect
29. C2×#835C22semidirect
34. C2×#832C22semidirect
35. C2×#832C22semidirect
42. #4939C22semidirect
43. #4928C22semidirect
44. #4524C22semidirect
45. #4524C22semidirect
46. C2×#535D3semidirect
615. C3#29630semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C2×#52382
C2×#52172
#822882
#721352
#721282
#717672
#717622
#717282
#717222
#715842
#296303