Finite Group Databasebeta

← #10967606

Subgroups — #10967606

Order 768.

2474 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5624C211yesmaximal, characteristic
2C22×#887C211yesmaximal, characteristic, complemented
3C3×#2503C211yesmaximal, characteristic, complemented
4#82760C211yesmaximal, characteristic, complemented
5#82177C211yesmaximal, characteristic
6#78204C214yesmaximal, complemented
7#78130C214yesmaximal, complemented
8#76534C212yesmaximal, complemented
9#76481C212yesmaximal, complemented
10#74183C212yesmaximal, complemented
11#74149C212yesmaximal, complemented
12#74064C214yesmaximal, complemented
13#74052C214yesmaximal, complemented
14#73491C211yesmaximal, characteristic
15C4×#877C211yesmaximal, characteristic, complemented
16#34972—31nomaximal, Sylow
17C23×Dic6C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#947C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×#944C2212yes—
23C2×#930C2211yescharacteristic
24C2×#929C2214yes—
25C22×Dic3⋊C4C2212yes—
26C22×C4×Dic3C2212yes—
27C22×C4×Dic3C2211yescharacteristic
28#5624—28no—
29C2×#898C2212yes—
30C2×#893C2211yescharacteristic
31C2×#892C2212yes—
32C2×#890C2211yescharacteristic
33C2×#887—28no—
34C2×#887C2212yes—
35C2×#887C2212yes—
36C2×#887C2212yes—
37C2×#887C2212yes—
38C2×#886C2212yes—
39C2×#67⋊C4C2212yes—
40C2×#877C2212yes—
41C2×#875C2212yes—
42C3×#400C2214yes—
43C3×#391C2212yes—
44C3×#382C2212yes—
45C3×#381C2214yes—
46C3×C4×C22⋊C4C2211yescharacteristic
47C4×#217C2211yescharacteristic
48#4993C2214yes—
49#4988C2212yes—
50#4979C2214yes—
51#4975C2212yes—
52#4973C2212yes—
53#4958C2214yes—
54#4946C2214yes—
55Dic3×C22⋊C4C2212yes—
56C4×Dic3⋊C4C2212yes—
57C42×Dic3C2212yes—
58#4929C2212yes—
59#4676C2218yes—
60#4673C2214yes—
61#4669C2218yes—
62#4669C2218yes—
63#4666C2218yes—
64#4657C2218yes—
65#4652C2218yes—
66#4649C2212yes—
67#4649C2218yes—
68#4648C2212yes—
69#4647C2218yes—
70C2×#529—31no—
71C22×C4×Q8—31no—
72#2604—34no—
73#2565—32no—
74#2545—32no—
75#2503D311yescharacteristic, complemented
76#2501—31no—
77#2436—31no—
78#2434—31no—
79#2413—34no—
80#2396—34no—
81C4×C4⋊Q8—32no—
82C4×#124—32no—
83C4×#123—34no—
84C4×#117—31no—
85C22×C3×Q8C2311yescharacteristic
86C23×C3×C4C2312yes—
87C23×C3×C4C2311yescharacteristic
88C22×#67C2312yes—
89C23×Dic3C2312yes—
90C23×Dic3C2311yescharacteristic
91C23×Dic3C2311yescharacteristic
92C22×D12C2311yescharacteristic
93C22×C4×D3C2312yes—
94C22×Dic6—24no—
95C22×Dic6—28no—
96C22×Dic6C2312yes—
97C22×Dic6C2312yes—
98C3×C22⋊Q8—24no—
99C2×C3×C4⋊C4—24no—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C22⋊C4C2312yes—
105C2×C3×C22⋊C4C2312yes—
106C2×C3×C42C2312yes—
107C2×#217C2312yes—
108#947—24no—
109Q8×Dic3—24no—
110#944—28no—
111#930—24no—
112#929—216no—
113C2×(C2×D3)⋊C4C2318yes—
114C2×(C2×D3)⋊C4C2318yes—
115C2×(C2×D3)⋊C4C2312yes—
116C2×(C2×D3)⋊C4C2312yes—
117C2×(C2×D3)⋊C4C2312yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2311yescharacteristic
122C2×Dic3⋊C4—28no—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2318yes—
125C2×Dic3⋊C4C2318yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2312yes—
128C2×Dic3⋊C4C2314yes—
129C2×C4×Dic3—28no—
130C2×C4×Dic3D418yes—
131C2×C4×Dic3C2314yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2314yes—
134C2×C4×Dic3C2314yes—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2312yes—
137C2×C4×Dic3C2311yescharacteristic
138C2×C4×Dic3C2311yescharacteristic
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2311yescharacteristic
141#898—28no—
142#893—24no—
143#892—28no—
144#890—24no—
145#887—28no—
146#887—28no—
147#887—28no—
148#887—28no—
149#886—28no—
150#67⋊C4—28no—
151#877—28no—
152#875—28no—
153C3×#95C2318yes—
154C3×#95C2312yes—
155C3×#95C2318yes—
156C3⋊#95C2318yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2318yes—
160C23×Q8—31no—
161C2×C4⋊Q8—32no—
162#529—68no—
163C2×#126—32no—
164C2×#125—31no—
165C2×#124—31no—
166C2×#124—32no—
167C2×#123—34no—
168C2×#123—32no—
169C2×C22⋊Q8C2×D311yescharacteristic
170C2×#121—31no—
171C2×C4×Q8—31no—
172C2×C4×Q8—32no—
173C2×C4×Q8—32no—
174C2×C4×Q8—32no—
175C2×C4×Q8—32no—
176C2×C4×Q8—68no—
177C2×C4×D4—31no—
178C2×C4×D4—32no—
179C2×#117—32no—
180C2×#117—32no—
181C22×C4⋊C4—32no—
182C22×C4⋊C4C2×D311yescharacteristic
183C22×C42—31no—
184C22×C42—32no—
185#400C2×D314yes—
186#399—34no—
187#398—38no—
188#396—38no—
189#395—34no—
190#393—34no—
191#391—32no—
192#391—32no—
193#391C2×D312yes—
194#390—32no—
195#389—38no—
196#388—38no—
197#388—38no—
198#387—38no—
199#386—34no—
200#385—34no—
201#384—32no—
202#382C2×D312yes—
203#381C2×D314yes—
204C4×C4⋊C4—32no—
205C4×C4⋊C4—38no—
206C4×C4⋊C4—32no—
207C4×C22⋊C4—31no—
208C4×C22⋊C4—32no—
209C4×C22⋊C4—38no—
210C4×C22⋊C4C2×D311yescharacteristic
211#376—32no—
212C43—32no—
213C24×C3C2411yescharacteristic
214C23×D3C2412yes—
215C2×C3×Q8—22no—
216C2×C3×Q8—24no—
217C22×C3×C4—24no—
218C22×C3×C4—24no—
219C22×C3×C4—28no—
220C22×C3×C4C2×D414yes—
221C22×C3×C4C2412yes—
222C22×C3×C4C2414yes—
223C22×C3×C4C2414yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C2×#67—24no—
228C2×#67—28no—
229C22×Dic3—24no—
230C22×Dic3—24no—
231C22×Dic3—28no—
232C22×Dic3—24no—
233C22×Dic3C2×D414yes—
234C22×Dic3C2×D414yes—
235C22×Dic3C2412yes—
236C22×Dic3C2411yescharacteristic
237C22×Dic3C2411yescharacteristic
238C22×Dic3C2411yescharacteristic
239C22×Dic3C2414yes—
240C22×Dic3C2414yes—
241C22×Dic3C2411yescharacteristic
242C2×D12—22no—
243C2×D12—24no—
244C2×C4×D3—24no—
245C2×C4×D3—28no—
246C2×Dic6—232no—
247C2×Dic6—24no—
248C2×Dic6—24no—
249C2×Dic6—24no—
250C2×Dic6—28no—
251C2×Dic6#4118yes—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—22no—
255C3×C4⋊C4—24no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—24no—
258C3×C42—24no—
259#217#4118yes—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—216no—
265(C2×D3)⋊C4—216no—
266C4⋊Dic3—22no—
267C4⋊Dic3—24no—
268C4⋊Dic3—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—28no—
271Dic3⋊C4—28no—
272Dic3⋊C4—28no—
273Dic3⋊C4—216no—
274Dic3⋊C4—216no—
275C4×Dic3—24no—
276C4×Dic3—24no—
277C4×Dic3—28no—
278C4×Dic3—28no—
279C4×Dic3—28no—
280C4×Dic3—28no—
281C4×Dic3C2×D414yes—
282C4×Dic3C2×D414yes—
283C4×Dic3C2×D414yes—
284C4×Dic3C2×D414yes—
285C22×Q8—32no—
286C22×Q8—32no—
287C22×Q8—68no—
288C22×Q8—64no—
289C22×Q8C22×D311yescharacteristic
290C22×D4—32no—
291C22×D4—31no—
292C23×C4—31no—
293C23×C4—31no—
294C23×C4—32no—
295C23×C4—32no—
296C23×C4C22×D312yes—
297C23×C4C22×D311yescharacteristic
298C4⋊Q8—68no—
299#126—68no—
300#125—64no—
301#124—68no—
302#124—64no—
303#123—68no—
304#123—616no—
305C22⋊Q8—24no—
306#121—64no—
307C4×Q8—68no—
308C4×Q8—68no—
309C4×Q8—68no—
310C4×Q8—68no—
311C4×Q8—64no—
312C4×D4—64no—
313C4×D4—68no—
314#117—68no—
315#117—68no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—34no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—38no—
325C2×C4⋊C4—38no—
326C2×C4⋊C4—68no—
327C2×C4⋊C4C22×D312yes—
328C2×C4⋊C4C22×D312yes—
329C2×C4⋊C4C22×D312yes—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C22⋊C4—32no—
332C2×C22⋊C4—32no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—38no—
337C2×C22⋊C4—38no—
338C2×C22⋊C4C22×D312yes—
339C2×C22⋊C4C22×D312yes—
340C2×C42—38no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—31no—
344C2×C42—31no—
345C2×C42—32no—
346C2×C42—34no—
347C2×C42—34no—
348C2×C42—34no—
349C2×C42—32no—
350C2×C42—34no—
351C2×C42—68no—
352C2×C42C22×D312yes—
353#95—34no—
354#95—32no—
355#95—38no—
356#95—38no—
357#95C22×D312yes—
358#95C22×D318yes—
359#95C22×D318yes—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3C22×D412yes—
363C23×C3C2×#4114yes—
364C23×C3C2511yescharacteristic
365C22×D3—22no—
366C22×D3—24no—
367C22×D3—24no—
368C22×D3—24no—
369C3×Q8—44no—
370C2×C3×C4—22no—
371C2×C3×C4—28no—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—28no—
378C2×C3×C4—24no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—48no—
382C2×C3×C4C22×D412yes—
383C2×C3×C4C22×D412yes—
384C2×C3×C4C2×#4114yes—
385#67—48no—
386C2×Dic3—216no—
387C2×Dic3—28no—
388C2×Dic3—24no—
389C2×Dic3—22no—
390C2×Dic3—24no—
391C2×Dic3—22no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—22no—
395C2×Dic3—28no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3C22×D412yes—
399C2×Dic3C22×D412yes—
400C2×Dic3C22×D412yes—
401C2×Dic3C22×D412yes—
402C2×Dic3C2×#4118yes—
403C2×Dic3C2×#4118yes—
404C2×Dic32−1+414yes—
405D12—44no—
406C4×D3—48no—
407Dic6—216no—
408Dic6—216no—
409C24—32no—
410C24C23×D311yescharacteristic
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—38no—
414C2×Q8—68no—
415C2×Q8—632no—
416C2×Q8—64no—
417C2×Q8—64no—
418C2×Q8—64no—
419C2×D4—64no—
420C2×D4—68no—
421C2×D4—62no—
422C2×D4—64no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—31no—
427C22×C4—34no—
428C22×C4—34no—
429C22×C4—31no—
430C22×C4—34no—
431C22×C4—34no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—32no—
435C22×C4—64no—
436C22×C4—68no—
437C22×C4—68no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4D3×D414yes—
442C22×C4C23×D312yes—
443C22×C4C23×D314yes—
444C22×C4C23×D314yes—
445C22×C4C23×D312yes—
446C22×C4C23×D311yescharacteristic
447C22×C4C23×D311yescharacteristic
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—68no—
453C4⋊C4—68no—
454C4⋊C4—64no—
455C4⋊C4—68no—
456C4⋊C4—64no—
457C4⋊C4—64no—
458C4⋊C4—616no—
459C4⋊C4—616no—
460C4⋊C4—62no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—38no—
464C22⋊C4—64no—
465C22⋊C4—64no—
466C22⋊C4—64no—
467C22⋊C4—616no—
468C22⋊C4—616no—
469C22⋊C4—64no—
470C42—24no—
471C42—34no—
472C42—34no—
473C42—34no—
474C42—34no—
475C42—68no—
476C42—68no—
477C42—64no—
478C42—68no—
479C42—68no—
480C42—64no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—28no—
484C22×C3#53214yes—
485C22×C3C23×D411yescharacteristic
486C22×C3C22×#4112yes—
487C22×C3C22×#4112yes—
488C22×C3C2×2−1+412yes—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—48no—
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4—42no—
497C3×C4—44no—
498C3×C4#52514yes—
499Dic3—28no—
500Dic3—22no—
501Dic3—22no—
502Dic3—22no—
503Dic3—28no—
504Dic3—42no—
505Dic3#53614yes—
506C23—22no—
507C23—22no—
508C23—62no—
509C23—64no—
510C23—64no—
511C23—64no—
512C23C2×D3×D412yes—
513C23D3×#4114yes—
514C23C24×D311yescharacteristic
515Q8—44no—
516Q8—616no—
517Q8—616no—
518D4—128no—
519D4—124no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—32no—
532C2×C4—34no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—32no—
536C2×C4—38no—
537C2×C4—38no—
538C2×C4—48no—
539C2×C4—68no—
540C2×C4—616no—
541C2×C4—68no—
542C2×C4—64no—
543C2×C4—62no—
544C2×C4—64no—
545C2×C4—62no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—64no—
551C2×C4—128no—
552C2×C4C2×D3×D412yes—
553C2×C4C2×D3×D412yes—
554C2×C4D3×#4114yes—
555C2×C3—24no—
556C2×C3C2×#52511yescharacteristic
557C2×C3C2×#53211yescharacteristic
558C2×C3C2×#53611yescharacteristic
559C2×C3#348214yes—
560D3—82no—
561C22—28no—
562C22—22no—
563C22—22no—
564C22—122no—
565C22—128no—
566C22—122no—
567C22—122no—
568C22#566814yes—
569C22C22×D3×D411yescharacteristic
570C22C2×D3×#4112yes—
571C22C2×D3×#4112yes—
572C22D3×2−1+412yes—
573C4—22no—
574C4—34no—
575C4—44no—
576C4—44no—
577C4—42no—
578C4—42no—
579C4—62no—
580C4—62no—
581C4—68no—
582C4—62no—
583C4—68no—
584C4—122no—
585C4#565614yes—
586C3#3497211yescharacteristic, complemented, Sylow
587C2—24no—
588C2—242no—
589C2C2×#565611yescharacteristic
590C2C2×#566811yescharacteristic
591C2D3×#53611yescharacteristic
592C2#8682814yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2503C2
prime radicalC2×#930C22
hypercenter#2503D3
hypocenterC3#34972
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2503

Complements

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

NH ≅ G/Ntype
2. C22×#887C2semidirect
3. C3×#2503C2semidirect
4. #82760C2semidirect
6. #78204C2semidirect
7. #78130C2semidirect
8. #76534C2semidirect
9. #76481C2semidirect
10. #74183C2semidirect
11. #74149C2semidirect
12. #74064C2semidirect
13. #74052C2semidirect
15. C4×#877C2semidirect
75. #2503D3semidirect
586. C3#34972semidirect

Maximal subgroups

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

subgroupindex
C2×#56242
C22×#8872
C3×#25032
#827602
#821772
#782042
#781302
#765342
#764812
#741832
#741492
#740642
#740522
#734912
C4×#8772
#349723