Finite Group Databasebeta

← #10827382

Subgroups — #10827382

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#553C211yesmaximal, characteristic
2C2×#5040C211yesmaximal, characteristic, complemented
3#82349C211yesmaximal, characteristic, complemented
4C22×#810C211yesmaximal, characteristic, complemented
5#71744C212yesmaximal, complemented
6#71685C212yesmaximal, complemented
7#71681C211yesmaximal, characteristic, complemented
8#71612C212yesmaximal, complemented
9#71610C211yesmaximal, characteristic, complemented
10#71585C212yesmaximal, complemented
11#71554C211yesmaximal, characteristic, complemented
12#23964—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C3×#553—24no—
15C3×#553—24no—
16C2×C3×#126—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C22×#214C2212yes—
28C2×#924C2212yes—
29C23×C3⋊C8C2211yescharacteristic
30#5220—21no—
31C2×#832C2212yes—
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2212yes—
35#5040—24no—
36#4979—21no—
37C2×#810—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41#4947—21no—
42#4946—21no—
43#4929C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#553D311yescharacteristic
49C2×#448—31no—
50C22×#103—31no—
51#2409—31no—
52#2026—32no—
53#2000—32no—
54#1893—31no—
55#1877—32no—
56#1873—32no—
57#1869—31no—
58#1824—31no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×Dic3—21no—
68C22×Dic6—22no—
69C22×Dic6—24no—
70C22×Dic6—22no—
71C22×Dic6—21no—
72C22×Dic6—42no—
73C22×Dic6C2312yes—
74C3×#126—28no—
75C3×#125—22no—
76C3×#125—22no—
77C3×#123—28no—
78C3×#121—28no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—22no—
82C3×#121D414yes—
83C3×C4×D4—28no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×#117—22no—
89C3×#117—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D414yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4D411yescharacteristic
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C42—21no—
106C2×C3×C42C2311yescharacteristic
107C2×#217—21no—
108C2×#217—22no—
109C2×#214—24no—
110C2×#214—24no—
111C2×#214—42no—
112C2×C4⋊Dic3C2312yes—
113#924—24no—
114#924—42no—
115C2×Dic3⋊C4—22no—
116C22×C3⋊C8—22no—
117C22×C3⋊C8—42no—
118C22×C3⋊C8C2312yes—
119C22×C3⋊C8C2312yes—
120#832—42no—
121C3⋊#95—22no—
122C3⋊#95—22no—
123C3⋊#95—21no—
124#830—22no—
125#830—42no—
126#810—24no—
127#810—24no—
128#810—24no—
129C23×Q8—31no—
130C22×SD16—32no—
131C23×C8—31no—
132#553—24no—
133#553—24no—
134C2×#126—21no—
135C2×#125C2×D311yescharacteristic
136C2×#123—21no—
137C2×C22⋊Q8—32no—
138C2×#121—21no—
139C2×#121C2×D311yescharacteristic
140C2×#121C2×D311yescharacteristic
141C2×C4×D4—21no—
142C2×C4×D4C2×D311yescharacteristic
143C2×C4×D4C2×D311yescharacteristic
144C2×#117C2×D311yescharacteristic
145C22×C4⋊C4C2×D311yescharacteristic
146#448—64no—
147C2×#103—32no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—64no—
151C2×#102—32no—
152C2×C22⋊C8—31no—
153C2×C22⋊C8—32no—
154#395—61no—
155#393—61no—
156#391—32no—
157#386—61no—
158#385—61no—
159#380—31no—
160#340—32no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C24×C3C2×D411yescharacteristic
167C2×C3×D4—22no—
168C2×C3×D4—24no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—24no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—82no—
178C2×C3×D4C2×D412yes—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—82no—
191C22×C3×C4—82no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—44no—
202C22×Dic3—42no—
203C2×Dic6—28no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—22no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—48no—
210C2×Dic6—44no—
211C2×Dic6—44no—
212C2×Dic6—42no—
213C2×Dic6#4114yes—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4C2×D412yes—
222C3×C4⋊C4C2×D412yes—
223C3×C4⋊C4C2×D412yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—22no—
230C3×C42—24no—
231C3×C42—21no—
232C3×C42—21no—
233#217—44no—
234#217—42no—
235#214—48no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238Dic3⋊C4—44no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—44no—
246C22×Q8—32no—
247C22×Q8—64no—
248C22×Q8—61no—
249C22×Q8—62no—
250C22×Q8—62no—
251C22×Q8—122no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4#6711yescharacteristic
255C22×D4C22×D311yescharacteristic
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—61no—
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C22×C8—32no—
265C22×C8—32no—
266C22×C8—62no—
267C22×C8—122no—
268#126—28no—
269#125—22no—
270#125—22no—
271#123—28no—
272C22⋊Q8—64no—
273C22⋊Q8—122no—
274#121—28no—
275#121—22no—
276#121—22no—
277#121—22no—
278#121#6714yes—
279C4×D4—28no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284#117—22no—
285#117—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4#6711yescharacteristic
293C2×C4⋊C4#6714yes—
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—62no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4#6711yescharacteristic
304C2×C22⋊C4#6711yescharacteristic
305C2×C42—21no—
306C2×C42C22×D311yescharacteristic
307#103—64no—
308#103—64no—
309#103—64no—
310#102—122no—
311C22⋊C8—62no—
312C22⋊C8—122no—
313#95—61no—
314#95—62no—
315#95—62no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—42no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—42no—
328C23×C3#12012yes—
329C23×C3#13512yes—
330C23×C3#13712yes—
331C23×C3C22×D411yescharacteristic
332C3×D4—24no—
333C3×D4—44no—
334C3×D4—48no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—84no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—84no—
360C2×C3×C4—84no—
361C2×C3×C4#12012yes—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4112yes—
365C2×C3×C4H(5,𝔽2)12yes—
366C2×Dic3—24no—
367C2×Dic3—48no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373Dic6—24no—
374Dic6—24no—
375Dic6—48no—
376Dic6—42no—
377Dic6—42no—
378C3⋊C8—42no—
379C3⋊C8—42no—
380C24—21no—
381C24C2×#6711yescharacteristic
382C24C2×#6711yescharacteristic
383C2×Q8—34no—
384C2×Q8—68no—
385C2×Q8—62no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×Q8—124no—
391C2×Q8—128no—
392C2×Q8—124no—
393C2×Q8—122no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—82no—
405C2×D4C2×#6712yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—124no—
422C22×C4—122no—
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C2×#6712yes—
428C22×C4C2×#6712yes—
429C22×C4C23×D311yescharacteristic
430SD16—128no—
431C2×C8—62no—
432C2×C8—62no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8—124no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—62no—
446C4⋊C4—62no—
447C4⋊C4—124no—
448C4⋊C4C2×#6712yes—
449C4⋊C4C2×#6712yes—
450C4⋊C4C2×#6712yes—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—124no—
458C22⋊C4—122no—
459C42—24no—
460C42—21no—
461C42—21no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—41no—
467C22×C3—42no—
468C22×C3—41no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—41no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3#45214yes—
477C22×C3C2×#12011yescharacteristic
478C22×C3#54612yes—
479C22×C3C2×#13511yescharacteristic
480C22×C3C2×#13711yescharacteristic
481C22×C3#57712yes—
482C3×C4—22no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—42no—
486C3×C4—41no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4#52511yescharacteristic
490C3×C4#53511yescharacteristic
491C3×C4#54612yes—
492Dic3—44no—
493Dic3—82no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—21no—
498C23—22no—
499C23—22no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23#95012yes—
507C23#95112yes—
508C23#95312yes—
509C23C22×#6711yescharacteristic
510Q8—64no—
511Q8—64no—
512Q8—128no—
513Q8—122no—
514Q8—122no—
515D4—24no—
516D4—44no—
517D4—48no—
518D4—41no—
519D4—41no—
520D4—84no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—64no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—128no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4#95312yes—
552C2×C4D3×#4112yes—
553C2×C4#100912yes—
554C2×C4C22×#6711yescharacteristic
555C2×C4C22×#6711yescharacteristic
556C8—122no—
557C8—122no—
558C2×C3—24no—
559C2×C3—81no—
560C2×C3—82no—
561C2×C3#240411yescharacteristic
562C2×C3C2×#45211yescharacteristic
563C2×C3#300911yescharacteristic
564C2×C3#330112yes—
565C2×C3#332012yes—
566C22—22no—
567C22—22no—
568C22—24no—
569C22—24no—
570C22—41no—
571C22—42no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—82no—
579C22—82no—
580C22#523914yes—
581C22C2×#95011yescharacteristic
582C22C2×#95111yescharacteristic
583C22#582912yes—
584C22#583612yes—
585C22C2×#95311yescharacteristic
586C4—22no—
587C4—41no—
588C4—42no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—124no—
594C4—242no—
595C4#582811yescharacteristic
596C4#582912yes—
597C4#583011yescharacteristic
598C3#2396411yescharacteristic, complemented, Sylow
599C2—24no—
600C2—81no—
601C2—82no—
602C2#8361812yes—
603C2#8362012yes—
604C2C2×#523911yescharacteristic
605C2#8409611yescharacteristic
606C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#553C2
socleC23×C3C22×D4
hypercenterC2×#553D3
hypocenterC3#23964
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
2. C2×#5040C2semidirect
3. #82349C2semidirect
4. C22×#810C2semidirect
5. #71744C2semidirect
6. #71685C2semidirect
7. #71681C2semidirect
8. #71612C2semidirect
9. #71610C2semidirect
10. #71585C2semidirect
11. #71554C2semidirect
598. C3#23964semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5532
C2×#50402
#823492
C22×#8102
#717442
#716852
#716812
#716122
#716102
#715852
#715542
#239643