Finite Group Databasebeta

← #10900868

Subgroups — #10900868

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic
2C2×#5239C211yesmaximal, characteristic, complemented
3C2×#5218C211yesmaximal, characteristic, complemented
4#82263C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic, complemented
6#72128C211yesmaximal, characteristic, complemented
7#71768C212yesmaximal, complemented
8#71762C212yesmaximal, complemented
9#71729C212yesmaximal, complemented
10#71722C212yesmaximal, complemented
11#71584C211yesmaximal, characteristic, complemented
12#29632—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×#216C2211yescharacteristic, complemented
24C22×#214C2212yes—
25C22×#214C2211yescharacteristic, complemented
26C2×#924C2211yescharacteristic
27#5239—24no—
28C2×#835C2211yescharacteristic, complemented
29C2×#835C2212yescomplemented
30#5218—24no—
31C2×#832C2212yes—
32C2×#832C2212yes—
33C2×#832C2212yes—
34C2×#832C2211yescharacteristic, complemented
35C2×#832C2212yescomplemented
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39#4985—21no—
40#4983—21no—
41#4952—22no—
42#4929C2212yescomplemented
43#4928C2212yescomplemented
44#4524C2212yescomplemented
45#4524C2212yescomplemented
46C2×#535D311yescharacteristic
47C2×#452—31no—
48C2×#450—31no—
49#2625—31no—
50#2042—32no—
51#2041—32no—
52#2015—32no—
53#2014—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×Dic6C2311yescharacteristic
76C3×#127—24no—
77C3×#127—24no—
78C3×#124—24no—
79C3×#124—24no—
80C3×#121—28no—
81C3×#121—28no—
82C3×#120—216no—
83C3×#117—22no—
84C3×#117—22no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4D412yes—
90C2×C3×C42C2312yes—
91C2×#217—21no—
92C2×#217—21no—
93C2×#216—22no—
94C2×#216—42no—
95C2×#214—24no—
96C2×#214—22no—
97C2×#214—42no—
98C2×#214—42no—
99C2×#214—42no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2312yes—
102#924—22no—
103#924—22no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—22no—
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2312yes—
109#835—22no—
110#835—42no—
111#832—24no—
112#832—22no—
113#832—42no—
114#832—42no—
115#832—42no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119C3⋊#95—21no—
120#830—22no—
121#830—22no—
122C22×#41C2×D311yescharacteristic
123C23×D4C2×D311yescharacteristic
124C22×Q16—31no—
125C22×SD16—32no—
126C22×SD16—31no—
127#535—24no—
128#535—24no—
129C2×#127C2×D312yes—
130C2×#124C2×D312yes—
131C2×C22⋊Q8—31no—
132C2×#121—21no—
133C2×#121—21no—
134C2×#120—22no—
135C2×#117C2×D311yescharacteristic
136#452—64no—
137#450—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#400—61no—
148#398—61no—
149#397—62no—
150#391—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
163C3×#41—44no—
164C3×#41—44no—
165C3×#41—48no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8C2×D412yes—
170C2×C3×D4—24no—
171C2×C3×D4—28no—
172C2×C3×D4—28no—
173C2×C3×D4—24no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—48no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—82no—
188C2×C3×D4—82no—
189C2×C3×D4—82no—
190C2×C3×D4—82no—
191C2×C3×D4C2×D414yes—
192C2×C3×D4C2×D412yes—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—24no—
197C22×C3×C4—44no—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—22no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—22no—
210C2×Dic6—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—28no—
215C3×C22⋊C4—24no—
216C3×C42—22no—
217C3×C42—22no—
218#217—42no—
219#217—81no—
220#216—44no—
221#214—48no—
222#214—44no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227Dic3⋊C4—42no—
228Dic3⋊C4—82no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C25—21no—
237C2×#41—22no—
238C2×#41—42no—
239C2×#41—42no—
240C2×#41#6714yes—
241C22×Q8—31no—
242C22×Q8C22×D311yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4#6714yes—
251C22×D4C22×D312yes—
252C22×D4C22×D311yescharacteristic
253C23×C4—21no—
254C23×C4C22×D311yescharacteristic
255C2×Q16—62no—
256C2×Q16—122no—
257C2×SD16—64no—
258C2×SD16—62no—
259C2×SD16—122no—
260C2×SD16—122no—
261C2×SD16—122no—
262C22×C8—31no—
263C22×C8—32no—
264C22×C8—31no—
265#127—24no—
266#127—24no—
267#124—24no—
268#124—24no—
269C22⋊Q8—62no—
270C22⋊Q8—62no—
271#121—28no—
272#121—28no—
273#120—216no—
274#117—22no—
275#117—22no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4#6711yescharacteristic
281C2×C4⋊C4#6711yescharacteristic
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4#6712yes—
287C2×C42C22×D312yes—
288#103—62no—
289#103—122no—
290#102—64no—
291#102—62no—
292#102—122no—
293#102—122no—
294#102—122no—
295C22⋊C8—62no—
296C22⋊C8—62no—
297#95—61no—
298#95—62no—
299#95—62no—
300#95—62no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—24no—
306C23×C3—22no—
307C23×C3—48no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—41no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—84no—
319C23×C3—84no—
320C23×C3—88no—
321C23×C3—88no—
322C23×C3#12012yes—
323C23×C3C22×D411yescharacteristic
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—42no—
327C3×D4—24no—
328C3×D4—22no—
329C3×D4—24no—
330C3×D4—22no—
331C3×D4—48no—
332C3×D4—48no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—88no—
337C3×D4—88no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—48no—
350C2×C3×C4—42no—
351C2×C3×C4#12012yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4H(5,𝔽2)12yes—
355C2×C3×C4H(5,𝔽2)12yes—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—41no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364Dic6—42no—
365Dic6—41no—
366Dic6—41no—
367C3⋊C8—42no—
368C3⋊C8—41no—
369C3⋊C8—41no—
370C24—22no—
371C24—21no—
372C24—24no—
373C24—44no—
374C24—81no—
375C24—81no—
376C24—82no—
377C24C2×#6711yescharacteristic
378#41—44no—
379#41—44no—
380#41—48no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—62no—
385C2×Q8—62no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8C2×#6712yes—
389C2×D4—24no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—48no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4C2×#6714yes—
411C2×D4C2×#6712yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—44no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4C2×#6712yes—
421C22×C4C2×#6712yes—
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6712yes—
424C22×C4C2×#6712yes—
425C22×C4C23×D311yescharacteristic
426Q16—124no—
427SD16—128no—
428SD16—124no—
429C2×C8—62no—
430C2×C8—62no—
431C2×C8—62no—
432C2×C8—62no—
433C2×C8—62no—
434C2×C8—61no—
435C2×C8—61no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—122no—
443C4⋊C4—242no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—24no—
447C22⋊C4—122no—
448C22⋊C4—241no—
449C42—22no—
450C42—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—44no—
458C22×C3—42no—
459C22×C3—44no—
460C22×C3—42no—
461C22×C3—44no—
462C22×C3—816no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3C2×#12011yescharacteristic
470C22×C3#53512yes—
471C22×C3#57611yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57712yes—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4#53412yes—
479C3×C4#53411yescharacteristic
480C3×C4#53411yescharacteristic
481Dic3—81no—
482Dic3—82no—
483Dic3—81no—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—48no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—84no—
502C23—84no—
503C23—88no—
504C23—88no—
505C23#95312yes—
506C23C22×#6711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—42no—
510Q8—122no—
511Q8—121no—
512Q8—121no—
513D4—24no—
514D4—22no—
515D4—24no—
516D4—22no—
517D4—48no—
518D4—48no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—88no—
523D4—88no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—122no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4#95312yes—
546C2×C4#100412yes—
547C2×C4#100412yes—
548C2×C4C22×#6711yescharacteristic
549C2×C4C22×#6711yescharacteristic
550C8—122no—
551C8—121no—
552C8—121no—
553C2×C3—42no—
554C2×C3—82no—
555C2×C3—84no—
556C2×C3#261311yescharacteristic
557C2×C3#300911yescharacteristic
558C2×C3#301211yescharacteristic
559C2×C3#322212yes—
560C2×C3#322412yes—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—44no—
572C22—816no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22#580512yes—
580C22#583511yescharacteristic
581C22#583611yescharacteristic
582C22#583612yes—
583C22C2×#95311yescharacteristic
584C4—22no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—241no—
589C4—241no—
590C4—242no—
591C4#580412yes—
592C4#580411yescharacteristic
593C4#580411yescharacteristic
594C3#2963211yescharacteristic, complemented, Sylow
595C2—42no—
596C2—84no—
597C2—82no—
598C2#8338912yes—
599C2#8339312yes—
600C2#8405411yescharacteristic
601C2#8410211yescharacteristic
602C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#535C2
socleC23×C3C22×D4
hypercenterC2×#535D3
hypocenterC3#29632
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
2. C2×#5239C2semidirect
3. C2×#5218C2semidirect
4. #82263C2semidirect
5. #72135C2semidirect
6. #72128C2semidirect
7. #71768C2semidirect
8. #71762C2semidirect
9. #71729C2semidirect
10. #71722C2semidirect
11. #71584C2semidirect
23. C22×#216C22semidirect
25. C22×#214C22semidirect
28. C2×#835C22semidirect
29. C2×#835C22semidirect
34. C2×#832C22semidirect
35. C2×#832C22semidirect
42. #4929C22semidirect
43. #4928C22semidirect
44. #4524C22semidirect
45. #4524C22semidirect
594. C3#29632semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C2×#52392
C2×#52182
#822632
#721352
#721282
#717682
#717622
#717292
#717222
#715842
#296323