Finite Group Databasebeta

← #10944229

Subgroups — #10944229

Order 768.

1708 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D3×C8C211yesmaximal, characteristic, complemented
2C2×C3×#465C211yesmaximal, characteristic, complemented
3C2×#5171C212yesmaximal, complemented
4#82834C211yesmaximal, characteristic, complemented
5#82723C211yesmaximal, characteristic
6#71970C211yesmaximal, characteristic, complemented
7#71696C211yesmaximal, characteristic, complemented
8#71681C211yesmaximal, characteristic, complemented
9#70577C212yesmaximal, complemented
10#70270C212yesmaximal, complemented
11#70202C212yesmaximal, complemented
12#32929—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic, complemented
15C23×C3×C8C2211yescharacteristic
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#121C2211yescharacteristic
18C2×#946C2212yes—
19C2×#938C2212yes—
20C2×#937—21no—
21C23×C3⋊C8C2211yescharacteristic
22C22×D3×C8—24no—
23C22×D3×C8—28no—
24C22×D3×C8C2212yes—
25C22×D3×C8C2212yes—
26C3×#465—24no—
27C2×C3×#106C2211yescharacteristic, complemented
28C2×C3×#103C2211yescharacteristic, complemented
29C2×C3×#102C2211yescharacteristic, complemented
30C2×#835C2212yescomplemented
31#5220—21no—
32C2×#832C2212yescomplemented
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35#5171—28no—
36C2×#817C2212yescomplemented
37#4989—22no—
38#4987C2212yes—
39#4987C2212yes—
40#4956—22no—
41#4955—21no—
42#4946—21no—
43#4551C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46C24×C8—31no—
47C2×#465—32no—
48C2×#465D311yescharacteristic, complemented
49#2848—31no—
50#2847—31no—
51#1893—32no—
52#1893—31no—
53#1892—32no—
54#1892—31no—
55#1807—32no—
56#1807—31no—
57C24×D3D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—22no—
63C23×Dic3D411yescharacteristic
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—24no—
67C22×C4×D3—24no—
68C22×C4×D3D412yes—
69C22×C4×D3C2312yes—
70C2×C3×SD16—24no—
71C2×C3×SD16—22no—
72C22×C3×C8—24no—
73C22×C3×C8D414yescomplemented
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#121—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×#217—21no—
86C2×#217—21no—
87C2×#217—22no—
88#946—24no—
89#946—24no—
90#938—24no—
91#938—24no—
92#937—44no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×C4⋊Dic3C2312yes—
97C2×C4⋊Dic3C2312yes—
98C2×Dic3⋊C4—22no—
99C22×C3⋊C8—22no—
100C22×C3⋊C8—24no—
101C22×C3⋊C8C2312yes—
102C2×D3×C8—232no—
103C2×D3×C8—24no—
104C2×D3×C8—24no—
105C2×D3×C8—24no—
106C2×D3×C8—24no—
107C2×D3×C8—24no—
108C2×D3×C8D418yescomplemented
109C3×#106—22no—
110C3×#103—22no—
111C3×#102—22no—
112#835—24no—
113#832—24no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#817—24no—
119C24×C4—31no—
120C22×SD16C2×D311yescharacteristic, complemented
121C23×C8—32no—
122C23×C8—31no—
123C23×C8—32no—
124C23×C8—68no—
125C23×C8—64no—
126C23×C8C2×D311yescharacteristic
127C2×C22⋊Q8—32no—
128C2×C22⋊Q8C2×D311yescharacteristic
129C2×#121—32no—
130C2×#121C2×D311yescharacteristic
131C2×#120—61no—
132#465—24no—
133#465—68no—
134C2×#106—32no—
135C2×#106C2×D311yescharacteristic, complemented
136C2×#103—32no—
137C2×#103C2×D311yescharacteristic, complemented
138C2×#102—32no—
139C2×#102C2×D311yescharacteristic, complemented
140#386—62no—
141#386—61no—
142#385—61no—
143#385—62no—
144#380—31no—
145#380—31no—
146#380—32no—
147#380—32no—
148#379—61no—
149#340—32no—
150#340—32no—
151#340—32no—
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C23×D3—22no—
155C23×D3—22no—
156C23×D3—24no—
157C23×D3—24no—
158C23×D3SD1614yescomplemented
159C23×D3C2×D412yes—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—21no—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D412yes—
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—44no—
181C2×#67—44no—
182C2×#67—44no—
183C22×Dic3—22no—
184C22×Dic3—24no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3C2×D412yes—
188C2×C4×D3—28no—
189C2×C4×D3—216no—
190C2×C4×D3—28no—
191C2×C4×D3—24no—
192C2×C4×D3—24no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—24no—
198C2×C4×D3C2×D414yes—
199C2×C4×D3#4114yes—
200C3×SD16—44no—
201C2×C3×C8—22no—
202C2×C3×C8—28no—
203C2×C3×C8C2×D414yes—
204C2×C3×C8C2×D412yes—
205C2×C3×C8C2×D412yes—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—42no—
211C3×C22⋊C4—42no—
212C3×C22⋊C4—42no—
213#217—42no—
214#217—42no—
215#217—44no—
216(C2×D3)⋊C4—44no—
217(C2×D3)⋊C4—44no—
218(C2×D3)⋊C4—44no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223Dic3⋊C4—44no—
224C2×C3⋊C8—28no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8C2×D414yes—
228D3×C8—216no—
229D3×C8—216no—
230C25—31no—
231C22×Q8C22×D311yescharacteristic
232C22×D4—21no—
233C22×D4—62no—
234C22×D4C22×D311yescharacteristic
235C23×C4—32no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—64no—
239C23×C4—64no—
240C23×C4—62no—
241C23×C4—62no—
242C23×C4C22×D311yescharacteristic
243C2×SD16—24no—
244C2×SD16—22no—
245C22×C8—24no—
246C22×C8—32no—
247C22×C8—38no—
248C22×C8—64no—
249C22×C8—64no—
250C22×C8—632no—
251C22×C8—64no—
252C22×C8—62no—
253C22×C8—64no—
254C22×C8—64no—
255C22×C8—64no—
256C22×C8#6714yescomplemented
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—64no—
262C22⋊Q8—64no—
263#121—22no—
264#121—22no—
265#121—64no—
266#121—64no—
267#120—124no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282#106—22no—
283#106—64no—
284#103—22no—
285#103—64no—
286#102—22no—
287#102—64no—
288#95—62no—
289#95—62no—
290#95—61no—
291#95—61no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3#12112yes—
300C23×C3C2×SD1612yes—
301C23×C3#13512yes—
302C23×C3C22×D411yescharacteristic
303C22×D3—28no—
304C22×D3—216no—
305C22×D3—28no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—28no—
311C22×D3—216no—
312C22×D3#12114yes—
313C22×D3C2×SD1614yes—
314C22×D3#13514yes—
315C3×Q8—41no—
316C3×Q8—41no—
317C3×Q8—42no—
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—42no—
321C3×D4—84no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4#12112yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4112yes—
338#67—88no—
339C2×Dic3—28no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3#12114yes—
349C4×D3—216no—
350C4×D3—24no—
351C4×D3—24no—
352C4×D3—24no—
353C4×D3—24no—
354C3×C8—22no—
355C3×C8#12014yescomplemented
356C3⋊C8—24no—
357C24—21no—
358C24—32no—
359C24—34no—
360C24—64no—
361C24—64no—
362C24—62no—
363C24—62no—
364C24D3×D411yescharacteristic
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—21no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—124no—
377C2×D4—124no—
378C2×D4—124no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—32no—
385C22×C4—34no—
386C22×C4—34no—
387C22×C4—64no—
388C22×C4—64no—
389C22×C4—68no—
390C22×C4—68no—
391C22×C4—616no—
392C22×C4—64no—
393C22×C4—62no—
394C22×C4—64no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4D3×D411yescharacteristic
402C22×C4#23712yes—
403C22×C4C2×#6712yes—
404C22×C4C23×D311yescharacteristic
405SD16—44no—
406C2×C8—22no—
407C2×C8—28no—
408C2×C8—34no—
409C2×C8—616no—
410C2×C8—68no—
411C2×C8—62no—
412C2×C8—62no—
413C2×C8—616no—
414C2×C8D3×D414yes—
415C2×C8C2×#6712yes—
416C2×C8C2×#6712yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—42no—
422C4⋊C4—62no—
423C4⋊C4—62no—
424C4⋊C4—62no—
425C4⋊C4—62no—
426C4⋊C4—124no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—124no—
430C22⋊C4—124no—
431C22⋊C4—124no—
432C22⋊C4—124no—
433C22⋊C4—122no—
434C22⋊C4—122no—
435C22×C3—22no—
436C22×C3—24no—
437C22×C3—22no—
438C22×C3—24no—
439C22×C3—81no—
440C22×C3—82no—
441C22×C3—81no—
442C22×C3—82no—
443C22×C3—81no—
444C22×C3#46514yescomplemented
445C22×C3C2×#12111yescharacteristic
446C22×C3C2×#12112yes—
447C22×C3C22×SD1611yescharacteristic
448C22×C3C2×#13511yescharacteristic
449C22×C3C2×#13512yes—
450C2×D3—232no—
451C2×D3—24no—
452C2×D3—24no—
453C2×D3—24no—
454C2×D3—24no—
455C2×D3—24no—
456C2×D3—24no—
457C2×D3#46518yescomplemented
458C3×C4—22no—
459C3×C4—81no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4C2×#12011yescharacteristic
463C3×C4#52511yescharacteristic
464C3×C4#52512yes—
465Dic3—24no—
466Dic3—82no—
467Dic3—82no—
468C23—22no—
469C23—22no—
470C23—34no—
471C23—34no—
472C23—34no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—41no—
477C23—42no—
478C23—616no—
479C23—68no—
480C23—616no—
481C23—68no—
482C23—68no—
483C23—62no—
484C23—62no—
485C23—62no—
486C23—62no—
487C23D3×SD1612yes—
488C23#91612yes—
489C23#93812yes—
490C23C2×D3×D411yescharacteristic
491Q8—41no—
492Q8—41no—
493Q8—42no—
494D4—41no—
495D4—41no—
496D4—42no—
497D4—84no—
498D4—248no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—34no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—616no—
512C2×C4—68no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4#93812yes—
526C2×C4C2×D3×D412yes—
527C2×C4C2×#23711yescharacteristic
528C2×C4D3×#4112yes—
529C2×C4C22×#6711yescharacteristic
530C8—22no—
531C8—64no—
532C8#93714yescomplemented
533C2×C3—24no—
534C2×C3—161no—
535C2×C3#284711yescharacteristic
536C2×C3C2×#46512yes—
537C2×C3C2×#46511yescharacteristic
538C2×C3#305012yes—
539C2×C3#305011yescharacteristic
540D3—28no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—38no—
546C22—632no—
547C22—64no—
548C22—64no—
549C22—64no—
550C22—64no—
551C22—64no—
552C22—64no—
553C22—82no—
554C22—82no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22#517114yescomplemented
559C22D3×#12112yes—
560C22C2×D3×SD1611yescharacteristic
561C22C2×#91611yescharacteristic
562C22D3×#13512yes—
563C22C2×#93811yescharacteristic
564C4—22no—
565C4—64no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—242no—
570C4—242no—
571C4#560612yes—
572C4C2×#93711yescharacteristic
573C4#582811yescharacteristic
574C3#3292911yescharacteristic, complemented, Sylow
575C2—24no—
576C2—68no—
577C2—161no—
578C2D3×#46512yes—
579C2#8065612yes—
580C2#8272311yescharacteristic
581C2C2×#517111yescharacteristic
582C2#8391411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#465C2
socleC23×C3C22×D4
prime radical#82723C2
hypercenterC2×#465D3
hypocenterC3#32929
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
1. C23×D3×C8C2semidirect
2. C2×C3×#465C2semidirect
3. C2×#5171C2semidirect
4. #82834C2semidirect
6. #71970C2semidirect
7. #71696C2semidirect
8. #71681C2semidirect
9. #70577C2semidirect
10. #70270C2semidirect
11. #70202C2semidirect
14. C22×C3×SD16C22semidirect
27. C2×C3×#106C22semidirect
28. C2×C3×#103C22semidirect
29. C2×C3×#102C22semidirect
30. C2×#835C22semidirect
32. C2×#832C22semidirect
36. C2×#817C22semidirect
43. #4551C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
48. C2×#465D3semidirect
73. C22×C3×C8D4semidirect
108. C2×D3×C8D4semidirect
120. C22×SD16C2×D3semidirect
135. C2×#106C2×D3semidirect
137. C2×#103C2×D3semidirect
139. C2×#102C2×D3semidirect
158. C23×D3SD16semidirect
256. C22×C8#67semidirect
355. C3×C8#120semidirect
444. C22×C3#465semidirect
457. C2×D3#465semidirect
532. C8#937semidirect
558. C22#5171semidirect
574. C3#32929semidirect

Maximal subgroups

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

subgroupindex
C23×D3×C82
C2×C3×#4652
C2×#51712
#828342
#827232
#719702
#716962
#716812
#705772
#702702
#702022
#329293