Finite Group Databasebeta

← #11025110

Subgroups — #11025110

Order 768.

2012 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C2×D3×#125C211yesmaximal, characteristic, complemented
3#83269C211yesmaximal, characteristic, complemented
4#83245C211yesmaximal, characteristic, complemented
5#83101C214yesmaximal, complemented
6#83082C214yesmaximal
7#83050C214yesmaximal, complemented
8#83028C214yesmaximal, complemented
9#83023C211yesmaximal, characteristic, complemented
10#82772C212yesmaximal, complemented
11#82752C212yesmaximal
12#82490C212yesmaximal, complemented
13#82471C212yesmaximal, complemented
14#82268C211yesmaximal, characteristic
15#82265C211yesmaximal, characteristic, complemented
16#37705—31nomaximal, Sylow
17C3×#541—28no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#126C2214yes—
20C2×C3×#125C2214yes—
21C2×C3×#125C2211yescharacteristic
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×C4×Q8C2212yes—
24C2×C3×#117C2211yescharacteristic
25C2×#946C2212yes—
26C2×C4×#67C2211yescharacteristic
27D3×#125—28no—
28C2×D3×C4⋊C4C2212yes—
29C2×D3×C4⋊C4C2214yes—
30C2×#890C2212yes—
31C2×#889C2214yes—
32C2×C42×D3C2211yescharacteristic
33C2×#870C2211yescharacteristic
34#5229C2212yes—
35#5229C2212yes—
36#5229C2212yes—
37#5212C2212yes—
38C4×#217C2211yescharacteristic
39#4989C2214yes—
40#4989C2212yes—
41#4987C2214yes—
42#4987C2214yes—
43#4986C2214yes—
44#4985C2214yes—
45#4985C2214yes—
46#4985C2214yes—
47#4984C2212yes—
48#4984C2214yes—
49#4983C2214yes—
50#4982C2214yes—
51#4982C2214yes—
52#4980C2214yes—
53#4978C2212yes—
54#4977C2214yes—
55#4976C2214yes—
56Dic3×C4⋊C4C2212yes—
57#4972C2214yes—
58#4956C2214yes—
59#4956C2212yes—
60#4953C2214yes—
61#4948C2214yes—
62#4948C2214yes—
63#4945C2214yes—
64#4944C2212yes—
65C4×(C2×D3)⋊C4C2211yescharacteristic
66C4×(C2×D3)⋊C4C2214yes—
67#4935C2211yescharacteristic
68#4933C2214yes—
69#4933C2214yes—
70#4932C2211yescharacteristic
71C4×Dic3⋊C4C2211yescharacteristic
72#4928C2212yes—
73#4928C2214yes—
74C2×#541D311yescharacteristic, complemented
75C22×#125—31no—
76#2857—31no—
77#2725—34no—
78#2719—34no—
79#2642—31no—
80#2629—31no—
81#2489—32no—
82#2481—34no—
83#2453—32no—
84#2442—34no—
85#2417—32no—
86#2360—32no—
87#2330—31no—
88#2290—31no—
89C22×C3×Q8C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67C2311yescharacteristic
92C22×C4×D3C2314yes—
93C22×C4×D3C2312yes—
94C22×C4×D3C2311yescharacteristic
95C3×C4⋊Q8—24no—
96C3×#126—216no—
97C3×#125—216no—
98C3×#125D418yescomplemented
99C3×C22⋊Q8—28no—
100C3×C4×Q8—28no—
101C3×#117—24no—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2314yes—
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2314yes—
112C2×C3×C42C2311yescharacteristic
113C2×#217C2311yescharacteristic
114C2×#217C2312yes—
115C2×#217C2314yes—
116#946—28no—
117C4×#67—24no—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2311yescharacteristic
121C2×C4⋊Dic3C2312yes—
122C2×C4⋊Dic3C2314yes—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2314yes—
125C2×C4⋊Dic3C2314yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2311yescharacteristic
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2311yescharacteristic
136C2×C4×Dic3C2312yes—
137C2×C4×Dic3C2314yes—
138D3×C4⋊C4—216no—
139D3×C4⋊C4—28no—
140#890—28no—
141#889—216no—
142C42×D3—24no—
143#870—24no—
144C3⋊#95C2314yes—
145C3⋊#95C2314yes—
146C3⋊#95C2314yes—
147C3⋊#95C2312yes—
148C3⋊#95C2312yes—
149C3⋊#95C2312yes—
150C3⋊#95C2312yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159#541—28no—
160C2×C4⋊Q8C2×D311yescharacteristic
161C2×#126C2×D314yes—
162C2×#125—34no—
163C2×#125—32no—
164C2×#125—31no—
165C2×#125—68no—
166C2×#125C2×D314yes—
167C2×#125C2×D311yescharacteristic
168C2×C22⋊Q8—32no—
169C2×C22⋊Q8C2×D312yes—
170C2×C4×Q8C2×D312yes—
171C2×C4×D4—31no—
172C2×#117C2×D311yescharacteristic
173C22×C4⋊C4—32no—
174C22×C4⋊C4—34no—
175C22×C42—31no—
176#400—34no—
177#400—34no—
178#400—34no—
179#400—34no—
180#398—34no—
181#398—34no—
182#398—34no—
183#397—34no—
184#396—34no—
185#396—34no—
186#391—32no—
187#391—32no—
188#391—32no—
189#389—32no—
190#389—34no—
191#389—34no—
192#389—34no—
193#389—32no—
194#389—34no—
195#388—32no—
196#388—34no—
197#387—34no—
198#387—32no—
199#387—34no—
200#387—34no—
201#386—34no—
202#386—32no—
203#385—34no—
204#385—32no—
205#384—31no—
206#382—34no—
207#380—32no—
208#380—34no—
209#380—34no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4—31no—
212C4×C22⋊C4—31no—
213C4×C22⋊C4—31no—
214C4×C22⋊C4—34no—
215#376—31no—
216C24×C3C2411yescharacteristic
217C23×D3C2411yescharacteristic
218C2×C3×Q8—24no—
219C2×C3×Q8—24no—
220C2×C3×Q8—24no—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4—22no—
224C22×C3×C4C2414yes—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2412yes—
227C22×C3×C4C2414yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C2×#67—22no—
231C2×#67—24no—
232C22×Dic3C2411yescharacteristic
233C22×Dic3C2412yes—
234C22×Dic3C2412yes—
235C22×Dic3C2411yescharacteristic
236C22×Dic3C2414yes—
237C22×Dic3C2414yes—
238C22×Dic3C2411yescharacteristic
239C2×C4×D3—22no—
240C2×C4×D3—22no—
241C2×C4×D3—28no—
242C2×C4×D3—24no—
243C2×C4×D3—24no—
244C2×C4×D3—28no—
245C2×C4×D3—44no—
246C2×C4×D3—42no—
247C2×C4×D3#4114yes—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—28no—
250C3×C4⋊C4—28no—
251C3×C4⋊C4—28no—
252C3×C4⋊C4C2×D4116yes—
253C3×C4⋊C4C2×D418yes—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—28no—
256C3×C42—22no—
257C3×C42—28no—
258C3×C42C2×D414yes—
259#217—22no—
260#217—42no—
261#217—44no—
262(C2×D3)⋊C4—22no—
263(C2×D3)⋊C4—28no—
264(C2×D3)⋊C4—24no—
265C4⋊Dic3—24no—
266C4⋊Dic3—28no—
267C4⋊Dic3—24no—
268C4⋊Dic3—44no—
269C4⋊Dic3—44no—
270Dic3⋊C4—28no—
271Dic3⋊C4—28no—
272Dic3⋊C4—28no—
273Dic3⋊C4—22no—
274Dic3⋊C4—28no—
275Dic3⋊C4—44no—
276Dic3⋊C4—42no—
277C4×Dic3—24no—
278C4×Dic3—22no—
279C4×Dic3—22no—
280C4×Dic3—22no—
281C4×Dic3—44no—
282C22×Q8C22×D312yes—
283C22×D4—31no—
284C23×C4—31no—
285C23×C4—34no—
286C23×C4—32no—
287C23×C4C22×D311yescharacteristic
288C4⋊Q8—24no—
289#126—216no—
290#125—216no—
291#125—616no—
292#125—68no—
293#125—64no—
294#125#6718yescomplemented
295C22⋊Q8—28no—
296C22⋊Q8—68no—
297C4×Q8—28no—
298C4×D4—64no—
299#117—24no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—34no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—34no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—68no—
313C2×C4⋊C4—616no—
314C2×C4⋊C4C22×D312yes—
315C2×C4⋊C4C22×D314yes—
316C2×C4⋊C4C22×D314yes—
317C2×C4⋊C4C22×D314yes—
318C2×C4⋊C4C22×D312yes—
319C2×C4⋊C4C22×D314yes—
320C2×C22⋊C4—31no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4—32no—
325C2×C22⋊C4—34no—
326C2×C22⋊C4C22×D312yes—
327C2×C22⋊C4C22×D314yes—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—32no—
331C2×C42—31no—
332C2×C42—34no—
333C2×C42—64no—
334C2×C42C22×D311yescharacteristic
335C2×C42C22×D314yes—
336C2×C42C22×D311yescharacteristic
337#95—32no—
338#95—32no—
339#95—32no—
340#95—32no—
341#95—34no—
342#95—34no—
343#95—34no—
344#95—32no—
345#95—34no—
346#95—34no—
347#95—34no—
348#95—34no—
349#95—34no—
350#95—34no—
351#95—34no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—22no—
357C23×C3C2511yescharacteristic
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—22no—
361C22×D3C2×#4112yes—
362C22×D32−1+414yes—
363C3×Q8—48no—
364C2×C3×C4—24no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—24no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—24no—
376C2×C3×C4—22no—
377C2×C3×C4—44no—
378C2×C3×C4C22×D414yes—
379C2×C3×C4C22×D418yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C42−1+412yes—
383#67—44no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—24no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—24no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—24no—
401C2×Dic3—24no—
402C2×Dic3C2×#4112yes—
403C4×D3—28no—
404C4×D3—48no—
405C4×D3—416no—
406C24—31no—
407C24C23×D311yescharacteristic
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×D4—62no—
412C2×D4—64no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—31no—
417C22×C4—34no—
418C22×C4—31no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—31no—
422C22×C4—34no—
423C22×C4—34no—
424C22×C4—68no—
425C22×C4—64no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—64no—
429C22×C4—68no—
430C22×C4—124no—
431C22×C4—122no—
432C22×C4C23×D314yes—
433C22×C4C23×D311yescharacteristic
434C22×C4C23×D312yes—
435C22×C4C23×D314yes—
436C22×C4C23×D312yes—
437C22×C4C23×D311yescharacteristic
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—62no—
443C4⋊C4—68no—
444C4⋊C4—64no—
445C4⋊C4—68no—
446C4⋊C4—68no—
447C4⋊C4—68no—
448C4⋊C4—68no—
449C4⋊C4—64no—
450C4⋊C4—124no—
451C4⋊C4—122no—
452C4⋊C4—124no—
453C4⋊C4—124no—
454C4⋊C4C2×#6718yes—
455C4⋊C4C2×#67116yes—
456C22⋊C4—24no—
457C22⋊C4—28no—
458C22⋊C4—62no—
459C22⋊C4—68no—
460C22⋊C4—64no—
461C22⋊C4—62no—
462C22⋊C4—122no—
463C22⋊C4—124no—
464C42—22no—
465C42—28no—
466C42—62no—
467C42—62no—
468C42—64no—
469C42—62no—
470C42—124no—
471C42C2×#6714yes—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—42no—
475C22×C3—41no—
476C22×C3—42no—
477C22×C3C23×D411yescharacteristic
478C22×C3C22×#4111yescharacteristic
479C22×C3C2×2−1+411yescharacteristic
480C22×C3C2×2−1+412yes—
481C22×C3#58512yes—
482C2×D3—22no—
483C2×D3—24no—
484C2×D3—22no—
485C2×D3—24no—
486C2×D3#54114yescomplemented
487C3×C4—28no—
488C3×C4—24no—
489C3×C4—41no—
490C3×C4—44no—
491C3×C4—42no—
492C3×C4#54814yes—
493Dic3—22no—
494Dic3—41no—
495Dic3—42no—
496Dic3—44no—
497Dic3—41no—
498Dic3—44no—
499Dic3—42no—
500C23—21no—
501C23—21no—
502C23—22no—
503C23—22no—
504C23—21no—
505C23—32no—
506C23—34no—
507C23—61no—
508C23—62no—
509C23—61no—
510C23C24×D311yescharacteristic
511Q8—48no—
512D4—124no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—32no—
527C2×C4—44no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—68no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—64no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—64no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—64no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—64no—
547C2×C4—128no—
548C2×C4—1216no—
549C2×C4D3×#4112yes—
550C2×C4#101012yes—
551C2×C4C22×#6718yes—
552C2×C4C22×#6714yes—
553C2×C4C22×#6712yes—
554C2×C3—81no—
555C2×C3C2×#54111yescharacteristic
556C2×C3C2×#54811yescharacteristic
557C2×C3#348912yes—
558C2×C3#351211yescharacteristic
559C2×C3#356312yes—
560D3—24no—
561C22—34no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—42no—
566C22—42no—
567C22—62no—
568C22—62no—
569C22—64no—
570C22—64no—
571C22C2×D3×#4111yescharacteristic
572C22C2×#101011yescharacteristic
573C22D3×2−1+412yes—
574C22#596812yes—
575C22C23×#6711yescharacteristic
576C4—28no—
577C4—24no—
578C4—41no—
579C4—44no—
580C4—42no—
581C4—62no—
582C4—121no—
583C4—124no—
584C4—121no—
585C4—122no—
586C4—122no—
587C4—124no—
588C4#583114yes—
589C3#3770511yescharacteristic, complemented, Sylow
590C2—64no—
591C2—81no—
592C2#8649711yescharacteristic
593C2D3×#54111yescharacteristic
594C2#8652012yes—
595C2C2×#583111yescharacteristic
596C2#8729512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#541C2
prime radicalC22×#67C23
hypercenterC2×#541D3
hypocenterC3#37705
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×C3×#541C2semidirect
2. C2×D3×#125C2semidirect
3. #83269C2semidirect
4. #83245C2semidirect
5. #83101C2semidirect
7. #83050C2semidirect
8. #83028C2semidirect
9. #83023C2semidirect
10. #82772C2semidirect
12. #82490C2semidirect
13. #82471C2semidirect
15. #82265C2semidirect
74. C2×#541D3semidirect
98. C3×#125D4semidirect
294. #125#67semidirect
486. C2×D3#541semidirect
589. C3#37705semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×D3×#1252
#832692
#832452
#831012
#830822
#830502
#830282
#830232
#827722
#827522
#824902
#824712
#822682
#822652
#377053