Finite Group Databasebeta

← #10827545

Subgroups — #10827545

Order 768.

1286 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×#555C211yesmaximal, characteristic
2C2×#5040C211yesmaximal, characteristic, complemented
3#82349C211yesmaximal, characteristic, complemented
4C2×#5004C211yesmaximal, characteristic, complemented
5#71794C212yesmaximal, complemented
6#71685C212yesmaximal, complemented
7#71682C211yesmaximal, characteristic, complemented
8#71612C212yesmaximal, complemented
9#71611C211yesmaximal, characteristic, complemented
10#71585C212yesmaximal, complemented
11#71575C211yesmaximal, characteristic, complemented
12#23977—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C3×#555—24no—
15C3×#555—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#124—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C2×C3×#117C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C22×#214C2212yes—
28C2×#924C2212yes—
29C22×#208C2211yescharacteristic
30#5220—21no—
31C2×#832C2212yes—
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2212yes—
35#5040—24no—
36#5004—24no—
37#4979—21no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41#4947—21no—
42#4946—21no—
43#4929C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#555D311yescharacteristic
49C2×#448—31no—
50C2×#419—31no—
51#2409—31no—
52#2024—32no—
53#2000—32no—
54#1895—31no—
55#1877—32no—
56#1873—32no—
57#1870—31no—
58#1825—31no—
59C22×C3×Q8—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4D411yescharacteristic
63C22×C3×D4C2311yescharacteristic
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×C4⋊Q8—22no—
75C3×C4⋊Q8—22no—
76C3×#124—28no—
77C3×#124—28no—
78C3×#121—28no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—22no—
82C3×#121D414yes—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×#117—28no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117—22no—
91C3×#117D414yes—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4D411yescharacteristic
103C2×C3×C22⋊C4D411yescharacteristic
104C2×C3×C42—21no—
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—
116C2×#208—22no—
117C2×#208—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—
131C22×M16—31no—
132#555—24no—
133#555—24no—
134C2×C4⋊Q8C2×D311yescharacteristic
135C2×#124—21no—
136C2×#124—21no—
137C2×C22⋊Q8—32no—
138C2×#121—21no—
139C2×#121C2×D311yescharacteristic
140C2×#121C2×D311yescharacteristic
141C2×C4×D4C2×D311yescharacteristic
142C2×C4×D4C2×D311yescharacteristic
143C2×#117—21no—
144C2×#117C2×D311yescharacteristic
145C2×#117C2×D311yescharacteristic
146#448—64no—
147#419—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
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×Q8—24no—
168C2×C3×Q8—82no—
169C2×C3×D4—22no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—41no—
177C2×C3×D4—82no—
178C2×C3×D4—82no—
179C2×C3×D4C2×D412yes—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D411yescharacteristic
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—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4C2×D412yes—
220C3×C4⋊C4C2×D412yes—
221C3×C4⋊C4C2×D412yes—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—22no—
229C3×C42—24no—
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—
239#208—44no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C22×Q8—21no—
247C22×Q8—32no—
248C22×Q8—64no—
249C22×Q8—61no—
250C22×Q8—62no—
251C22×Q8—62no—
252C22×Q8—122no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4#6711yescharacteristic
256C22×D4C22×D311yescharacteristic
257C23×C4—21no—
258C23×C4—61no—
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C2×M16—62no—
265C2×M16—122no—
266C22×C8—32no—
267C22×C8—32no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270#124—28no—
271#124—28no—
272C22⋊Q8—64no—
273C22⋊Q8—122no—
274#121—28no—
275#121—22no—
276#121—22no—
277#121—22no—
278#121#6714yes—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283#117—28no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117#6714yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4#6711yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—62no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4#6711yescharacteristic
303C2×C22⋊C4#6711yescharacteristic
304C2×C42—21no—
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—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—42no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—42no—
330C23×C3#12012yes—
331C23×C3C22×D411yescharacteristic
332C3×Q8—84no—
333C3×D4—24no—
334C3×D4—44no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—84no—
338C3×D4—84no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—48no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4#12012yes—
361C2×C3×C4#13712yes—
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—24no—
384C2×Q8—34no—
385C2×Q8—68no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×Q8—62no—
391C2×Q8—82no—
392C2×Q8—124no—
393C2×Q8—128no—
394C2×Q8—124no—
395C2×Q8—122no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—41no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4C2×#6712yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—62no—
420C22×C4—62no—
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—
431M16—124no—
432C2×C8—62no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8—62no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—62no—
444C4⋊C4—62no—
445C4⋊C4—124no—
446C4⋊C4C2×#6712yes—
447C4⋊C4C2×#6712yes—
448C4⋊C4C2×#6712yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—22no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—124no—
457C22⋊C4—122no—
458C42—24no—
459C42—24no—
460C42—21no—
461C42—21no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—41no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3C2×#12011yescharacteristic
478C22×C3#54612yes—
479C22×C3C2×#13711yescharacteristic
480C22×C3#57511yescharacteristic
481C22×C3#57712yes—
482C3×C4—22no—
483C3×C4—22no—
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—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23#95312yes—
509C23C22×#6711yescharacteristic
510Q8—64no—
511Q8—64no—
512Q8—84no—
513Q8—128no—
514Q8—122no—
515Q8—122no—
516D4—24no—
517D4—44no—
518D4—41no—
519D4—41no—
520D4—84no—
521D4—84no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—64no—
544C2×C4—128no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4#95112yes—
551C2×C4#95312yes—
552C2×C4D3×#4112yes—
553C2×C4#100912yes—
554C2×C4C22×#6711yescharacteristic
555C2×C4C22×#6711yescharacteristic
556C8—122no—
557C8—122no—
558C2×C3—42no—
559C2×C3—81no—
560C2×C3—82no—
561C2×C3#240411yescharacteristic
562C2×C3#300711yescharacteristic
563C2×C3#300911yescharacteristic
564C2×C3#332112yes—
565C2×C3#332912yes—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—82no—
580C22—82no—
581C22#582211yescharacteristic
582C22C2×#95111yescharacteristic
583C22#582912yes—
584C22#583612yes—
585C22C2×#95311yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—124no—
594C4—242no—
595C4#582811yescharacteristic
596C4#582912yes—
597C4#583011yescharacteristic
598C3#2397711yescharacteristic, complemented, Sylow
599C2—42no—
600C2—81no—
601C2—82no—
602C2#8365612yes—
603C2#8365712yes—
604C2#8408411yescharacteristic
605C2#8409611yescharacteristic
606C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#555C2
socleC23×C3C22×D4
hypercenterC2×#555D3
hypocenterC3#23977
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
2. C2×#5040C2semidirect
3. #82349C2semidirect
4. C2×#5004C2semidirect
5. #71794C2semidirect
6. #71685C2semidirect
7. #71682C2semidirect
8. #71612C2semidirect
9. #71611C2semidirect
10. #71585C2semidirect
11. #71575C2semidirect
598. C3#23977semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5552
C2×#50402
#823492
C2×#50042
#717942
#716852
#716822
#716122
#716112
#715852
#715752
#239773