Finite Group Databasebeta

← #10848754

Subgroups — #10848754

Order 768.

1178 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×#529C211yesmaximal, characteristic
2C2×#5069C211yesmaximal, characteristic, complemented
3C2×#5060C211yesmaximal, characteristic, complemented
4#82076C211yesmaximal, characteristic, complemented
5#71741C211yesmaximal, characteristic, complemented
6#71726C211yesmaximal, characteristic, complemented
7#71700C212yesmaximal, complemented
8#71685C212yesmaximal
9#71632C212yesmaximal, complemented
10#71612C212yesmaximal, complemented
11#71529C211yesmaximal, characteristic, complemented
12#25604—31nomaximal, Sylow
13C3×#529—24no—
14C3×#529—24no—
15C2×C3×#126—21no—
16C2×C3×#125C2211yescharacteristic
17C2×C3×#124C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C22×C3×C42C2211yescharacteristic
26C22×#216C2212yes—
27C22×#214C2212yes—
28C2×#924C2212yes—
29C2×#869C2211yescharacteristic
30C2×#835C2212yes—
31C2×#832C2212yes—
32#5212—21no—
33#5212—21no—
34C2×#830C2212yes—
35#5069—24no—
36#5060—24no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39#4935C2211yescharacteristic
40#4929C2212yes—
41#4928—21no—
42#4928—21no—
43C2×#803C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#529D311yescharacteristic
49C2×#497—31no—
50C2×#491—31no—
51#2857—31no—
52#2001—32no—
53#2000—32no—
54#1992—31no—
55#1959—31no—
56#1878—32no—
57#1877—32no—
58#1749—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C22×Dic6C2312yes—
67C3×#126—28no—
68C3×#125—22no—
69C3×#125D414yes—
70C3×#124—22no—
71C3×#124D414yes—
72C3×#123—28no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×#121—22no—
76C3×#121—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×D4—28no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×#117—28no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42—21no—
95C2×C3×C42—22no—
96C2×C3×C42—22no—
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—22no—
102C2×#216—24no—
103C2×#216—24no—
104C2×#216—42no—
105C2×#214—24no—
106C2×#214—24no—
107C2×#214—42no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2312yes—
113#924—42no—
114#924—42no—
115C2×Dic3⋊C4—22no—
116C22×C3⋊C8C2312yes—
117C22×C3⋊C8C2312yes—
118#869—22no—
119#869—22no—
120#835—42no—
121#832—42no—
122C3⋊#95—22no—
123C3⋊#95—22no—
124#830—24no—
125#810—24no—
126#810—24no—
127#803—22no—
128#803—22no—
129C22×Q16—32no—
130C22×SD16—32no—
131C2×C4⋊Q8—31no—
132#529—24no—
133#529—24no—
134C2×#126—21no—
135C2×#125C2×D311yescharacteristic
136C2×#124C2×D311yescharacteristic
137C2×#123—21no—
138C2×C22⋊Q8—32no—
139C2×C22⋊Q8C2×D311yescharacteristic
140C2×#121C2×D311yescharacteristic
141C2×C4×Q8C2×D311yescharacteristic
142C2×C4×D4—21no—
143C2×C4×D4C2×D311yescharacteristic
144C2×#117—21no—
145C22×C42C2×D311yescharacteristic
146#497—64no—
147#491—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×#97—31no—
154C2×#97—31no—
155#391—32no—
156#389—61no—
157#389—61no—
158#384—31no—
159#380—61no—
160#380—61no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8C2×D412yes—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C2×C3×D4C2×D412yes—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—41no—
190C22×C3×C4—41no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—22no—
200C22×Dic3—22no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—22no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4C2×D412yes—
214C3×C4⋊C4C2×D412yes—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C42—24no—
221C3×C42—22no—
222C3×C42—22no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42C2×D412yes—
226C3×C42C2×D412yes—
227#217—44no—
228#216—48no—
229#214—48no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—42no—
233C4⋊Dic3—42no—
234C4⋊Dic3—42no—
235C4⋊Dic3—42no—
236Dic3⋊C4—44no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8#4114yes—
243C22×Q8—32no—
244C22×Q8C22×D311yescharacteristic
245C22×D4—21no—
246C22×D4C22×D311yescharacteristic
247C23×C4—21no—
248C23×C4#6711yescharacteristic
249C23×C4#6711yescharacteristic
250C23×C4C22×D311yescharacteristic
251C2×Q16—64no—
252C2×Q16—64no—
253C2×Q16—122no—
254C2×SD16—64no—
255C2×SD16—64no—
256C2×SD16—122no—
257C22×C8—32no—
258C22×C8—32no—
259C4⋊Q8—62no—
260C4⋊Q8—62no—
261#126—28no—
262#125—22no—
263#125#6714yes—
264#124—22no—
265#124#6714yes—
266#123—28no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—122no—
270C22⋊Q8—122no—
271#121—22no—
272#121—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×D4—28no—
276C4×D4—22no—
277C4×D4—22no—
278#117—28no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—62no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42#6711yescharacteristic
301C2×C42#6711yescharacteristic
302C2×C42C22×D311yescharacteristic
303C2×C42C22×D311yescharacteristic
304#103—64no—
305#103—64no—
306#103—122no—
307#102—122no—
308C22⋊C8—64no—
309#97—62no—
310#97—62no—
311#95—62no—
312#95—62no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3#13712yes—
327C23×C3C22×D411yescharacteristic
328C3×Q8—24no—
329C3×Q8—42no—
330C3×D4—24no—
331C3×D4—48no—
332C3×D4—42no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—44no—
349C2×C3×C4—44no—
350C2×C3×C4—48no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4#12012yes—
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—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372Dic6—42no—
373Dic6—42no—
374Dic6—42no—
375Dic6—42no—
376C3⋊C8—24no—
377C3⋊C8—42no—
378C24—21no—
379C24C2×#6711yescharacteristic
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—62no—
384C2×Q8—62no—
385C2×Q8—62no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×Q8C2×#6712yes—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4C2×#6712yes—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C23×D311yescharacteristic
421Q16—128no—
422SD16—128no—
423C2×C8—34no—
424C2×C8—62no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—124no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4—122no—
442C4⋊C4C2×#6712yes—
443C4⋊C4C2×#6712yes—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—124no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—21no—
454C42—21no—
455C42C2×#6712yes—
456C42C2×#6712yes—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3C2×#12011yescharacteristic
472C22×C3#53211yescharacteristic
473C22×C3#53511yescharacteristic
474C22×C3C2×#13711yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3#57511yescharacteristic
477C22×C3#57511yescharacteristic
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—41no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4#54612yes—
487C3×C4#55012yes—
488Dic3—82no—
489Dic3—82no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23#95112yes—
504C23C22×#6711yescharacteristic
505Q8—24no—
506Q8—42no—
507Q8—122no—
508Q8—122no—
509Q8—122no—
510Q8—122no—
511D4—24no—
512D4—48no—
513D4—42no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4#95112yes—
547C2×C4#95312yes—
548C2×C4#95312yes—
549C2×C4D3×#4112yes—
550C2×C4#100912yes—
551C2×C4C22×#6711yescharacteristic
552C2×C4C22×#6711yescharacteristic
553C8—64no—
554C8—122no—
555C2×C3—42no—
556C2×C3—82no—
557C2×C3#243111yescharacteristic
558C2×C3#300711yescharacteristic
559C2×C3#300711yescharacteristic
560C2×C3#317211yescharacteristic
561C2×C3#318111yescharacteristic
562C2×C3#321811yescharacteristic
563C2×C3#321911yescharacteristic
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22#582211yescharacteristic
579C22#582211yescharacteristic
580C22C2×#95111yescharacteristic
581C22C2×#95111yescharacteristic
582C22#583011yescharacteristic
583C22#583211yescharacteristic
584C22C2×#95311yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—41no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—242no—
594C4—242no—
595C4#582912yes—
596C4#583412yes—
597C3#2560411yescharacteristic, complemented, Sylow
598C2—42no—
599C2—82no—
600C2#8313611yescharacteristic
601C2#8313811yescharacteristic
602C2#8314711yescharacteristic
603C2#8315411yescharacteristic
604C2#8408411yescharacteristic
605C2#8408411yescharacteristic
606C2#8409211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#529C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#529D3
hypocenterC3#25604
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#5069C2semidirect
3. C2×#5060C2semidirect
4. #82076C2semidirect
5. #71741C2semidirect
6. #71726C2semidirect
7. #71700C2semidirect
9. #71632C2semidirect
10. #71612C2semidirect
11. #71529C2semidirect
597. C3#25604semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×#50692
C2×#50602
#820762
#717412
#717262
#717002
#716852
#716322
#716122
#715292
#256043