Finite Group Databasebeta

← #10944126

Subgroups — #10944126

Order 768.

1588 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic, complemented
2C2×#5218C211yesmaximal, characteristic, complemented
3C2×#5217C211yesmaximal, characteristic
4C2×#5016C212yesmaximal, complemented
5#82287C211yesmaximal, characteristic, complemented
6#71613C212yesmaximal, complemented
7#71612C212yesmaximal, complemented
8#71604C212yesmaximal, complemented
9#71586C211yesmaximal, characteristic, complemented
10#71585C211yesmaximal, characteristic, complemented
11#71583C211yesmaximal, characteristic, complemented
12#32917—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C3×#518—28no—
15C2×C3×C4×D4—22no—
16C2×C3×C4×D4C2212yes—
17C2×C3×C4×D4C2212yes—
18C2×C3×#117C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C3×C22⋊C4—22no—
21C22×#214C2212yes—
22C22×#213C2212yes—
23C2×#930C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C2×#875C2211yescharacteristic, complemented
26#5218—24no—
27#5217—24no—
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32#5016—28no—
33#4994—22no—
34C2×#810C2212yescomplemented
35C2×#809C2212yescomplemented
36#4961—22no—
37#4939C2211yescharacteristic, complemented
38#4935C2211yescharacteristic, complemented
39#4929C2211yescharacteristic, complemented
40C2×C4⋊(C3⋊C8)C2212yescomplemented
41#4529C2212yescomplemented
42#4529C2212yescomplemented
43#4524C2212yescomplemented
44C2×#518D311yescharacteristic, complemented
45C2×#463—32no—
46C2×#450—31no—
47C2×#447—31no—
48#2620—31no—
49#1879—32no—
50#1877—32no—
51#1875—31no—
52#1873—31no—
53#1802—32no—
54#1798—31no—
55C25×C3—21no—
56C22×C3×D4—24no—
57C22×C3×D4—24no—
58C22×C3×D4—21no—
59C22×C3×D4—22no—
60C22×C3×D4D414yes—
61C22×C3×D4C2312yes—
62C23×C3×C4—22no—
63C23×C3×C4—22no—
64C23×C3×C4C2311yescharacteristic
65C22×#67—21no—
66C22×D12C2311yescharacteristic
67C22×Dic6C2311yescharacteristic
68C3×C4×D4—216no—
69C3×C4×D4—24no—
70C3×C4×D4—24no—
71C3×C4×D4—24no—
72C3×C4×D4D418yescomplemented
73C3×#117—24no—
74C3×#117—24no—
75C2×C3×C4⋊C4D412yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—216no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4D412yes—
82C2×C3×C42—21no—
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86C2×#214—24no—
87C2×#214—24no—
88C2×#214—24no—
89C2×#213—24no—
90C2×#213—24no—
91C2×#213—24no—
92#930—22no—
93#930—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100#924—22no—
101#924—22no—
102C2×Dic3⋊C4—21no—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105#875—24no—
106#832—24no—
107#832—24no—
108C3⋊#95—22no—
109#830—22no—
110#830—22no—
111#810—24no—
112#809—24no—
113C4⋊(C3⋊C8)—24no—
114C23×D4C2×D311yescharacteristic
115C22×SD16—32no—
116C22×D8—32no—
117C2×#124—31no—
118C2×C22⋊Q8—31no—
119C2×#121—31no—
120#518—28no—
121C2×C4×D4—22no—
122C2×C4×D4C2×D312yes—
123C2×C4×D4C2×D312yes—
124C2×#117C2×D311yescharacteristic
125C2×#117C2×D311yescharacteristic
126C22×C22⋊C4—22no—
127#463—68no—
128#450—64no—
129#447—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×C22⋊C8—31no—
136C2×C22⋊C8—31no—
137#396—62no—
138#394—62no—
139#391—31no—
140#390—31no—
141#384—31no—
142#340—32no—
143#340—32no—
144#336—32no—
145C24×C3—24no—
146C24×C3—24no—
147C24×C3—22no—
148C24×C3—24no—
149C24×C3—82no—
150C24×C3—82no—
151C24×C3C2×D411yescharacteristic
152C23×D3—21no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—216no—
156C2×C3×D4—216no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4C2×D414yes—
162C2×C3×D4C2×D414yes—
163C2×C3×D4#4114yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—28no—
168C22×C3×C4—28no—
169C22×C3×C4—82no—
170C22×C3×C4—82no—
171C22×C3×C4—82no—
172C22×C3×C4—82no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4#4112yes—
178C22×C3×C4C2411yescharacteristic
179C2×#67—42no—
180C2×#67—82no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C2×D12—22no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12—22no—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—28no—
199C3×C42—24no—
200C3×C42—22no—
201C3×C42C2×D414yes—
202#217—42no—
203#214—48no—
204#213—48no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—42no—
207(C2×D3)⋊C4—82no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—82no—
213Dic3⋊C4—42no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C25—21no—
221C22×Q8—31no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—31no—
227C22×D4—61no—
228C22×D4D1214yes—
229C22×D4C22×D312yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4C22×D311yescharacteristic
233C2×SD16—64no—
234C2×SD16—64no—
235C2×SD16—64no—
236C2×D8—64no—
237C2×D8—64no—
238C2×D8—64no—
239C22×C8—32no—
240C22×C8—32no—
241#124—64no—
242C22⋊Q8—62no—
243C22⋊Q8—62no—
244#121—62no—
245#121—62no—
246C4×D4—216no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4#6718yescomplemented
251#117—24no—
252#117—24no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4#6712yes—
258C2×C4⋊C4C22×D312yes—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4#6712yes—
268C2×C42—21no—
269C2×C42C22×D311yescharacteristic
270C2×C42C22×D311yescharacteristic
271C4⋊C8—64no—
272#103—64no—
273#102—64no—
274#102—64no—
275#102—64no—
276C22⋊C8—62no—
277C22⋊C8—62no—
278#95—62no—
279C23×C3—216no—
280C23×C3—21no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—21no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—84no—
292C23×C3—84no—
293C23×C3—88no—
294C23×C3—88no—
295C23×C3#12112yes—
296C23×C3C22×D411yescharacteristic
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—42no—
301C22×D3—42no—
302C3×D4—216no—
303C3×D4—24no—
304C3×D4—24no—
305C3×D4—24no—
306C3×D4#12118yes—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—88no—
320C2×C3×C4—88no—
321C2×C3×C4#12014yes—
322C2×C3×C4#12112yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D412yes—
325C2×C3×C4C2×#4112yes—
326C2×C3×C4C2×#4111yescharacteristic
327#67—84no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335D12—42no—
336D12—42no—
337Dic6—42no—
338Dic6—42no—
339C3⋊C8—42no—
340C3⋊C8—42no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—61no—
346C24—82no—
347C24—82no—
348C24C2×#6711yescharacteristic
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—122no—
366C2×D4—242no—
367C2×D4C2×D1214yes—
368C2×D4#23514yes—
369C2×D4C2×#6714yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4C2×D1212yes—
383C22×C4#23512yes—
384C22×C4C2×#6712yes—
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6712yes—
387C22×C4C23×D311yescharacteristic
388SD16—128no—
389D8—128no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—122no—
403C4⋊C4—242no—
404C4⋊C4C2×#6714yes—
405C22⋊C4—24no—
406C22⋊C4—28no—
407C22⋊C4—24no—
408C22⋊C4—28no—
409C22⋊C4—122no—
410C22⋊C4—122no—
411C22⋊C4—124no—
412C22⋊C4—242no—
413C42—24no—
414C42—22no—
415C42C2×#6714yes—
416C22×C3—22no—
417C22×C3—28no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—816no—
429C22×C3C2×#12011yescharacteristic
430C22×C3C2×#12111yescharacteristic
431C22×C3#52511yescharacteristic
432C22×C3#57612yes—
433C22×C3#57712yes—
434C2×D3—81no—
435C2×D3—82no—
436C2×D3—82no—
437C2×D3—81no—
438C2×D3—81no—
439C3×C4—22no—
440C3×C4—24no—
441C3×C4—42no—
442C3×C4—44no—
443C3×C4C2×#12112yes—
444C3×C4#52512yes—
445Dic3—81no—
446Dic3—81no—
447Dic3—81no—
448C23—216no—
449C23—24no—
450C23—21no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—84no—
461C23—84no—
462C23—88no—
463C23—88no—
464C23—121no—
465C23—121no—
466C23—122no—
467C23—122no—
468C23—121no—
469C23#93012yes—
470C23C22×#6711yescharacteristic
471Q8—122no—
472Q8—122no—
473D4—216no—
474D4—24no—
475D4—24no—
476D4—24no—
477D4—122no—
478D4—122no—
479D4—244no—
480D4#93018yes—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—88no—
494C2×C4—88no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—121no—
502C2×C4#93012yes—
503C2×C4#95314yes—
504C2×C4C22×D1211yescharacteristic
505C2×C4C2×#23512yes—
506C2×C4C2×#23511yescharacteristic
507C2×C4C22×#6712yes—
508C8—122no—
509C8—122no—
510C2×C3—42no—
511C2×C3—48no—
512C2×C3#284711yescharacteristic
513C2×C3#301212yes—
514C2×C3#304111yescharacteristic
515C2×C3#304211yescharacteristic
516C2×C3#311812yes—
517D3—161no—
518C22—22no—
519C22—28no—
520C22—22no—
521C22—22no—
522C22—44no—
523C22—42no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—42no—
530C22—816no—
531C22—241no—
532C22—241no—
533C22—242no—
534C22—241no—
535C22—242no—
536C22C2×#93011yescharacteristic
537C22#579111yescharacteristic
538C22#583512yes—
539C22#583612yes—
540C22C2×#95311yescharacteristic
541C4—22no—
542C4—24no—
543C4—42no—
544C4—44no—
545C4—241no—
546C4—241no—
547C4—241no—
548C4C2×#93012yes—
549C4#579112yes—
550C3#3291711yescharacteristic, complemented, Sylow
551C2—42no—
552C2—48no—
553C2—481no—
554C2#8254011yescharacteristic
555C2#8254112yes—
556C2#8254411yescharacteristic
557C2#8403211yescharacteristic
558C2#8410212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#518C2
socleC23×C3C22×D4
prime radicalC2×#5217C2
hypercenterC2×#518D3
hypocenterC3#32917
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
1. C2×C3×#518C2semidirect
2. C2×#5218C2semidirect
4. C2×#5016C2semidirect
5. #82287C2semidirect
6. #71613C2semidirect
7. #71612C2semidirect
8. #71604C2semidirect
9. #71586C2semidirect
10. #71585C2semidirect
11. #71583C2semidirect
25. C2×#875C22semidirect
34. C2×#810C22semidirect
35. C2×#809C22semidirect
37. #4939C22semidirect
38. #4935C22semidirect
39. #4929C22semidirect
40. C2×C4⋊(C3⋊C8)C22semidirect
41. #4529C22semidirect
42. #4529C22semidirect
43. #4524C22semidirect
44. C2×#518D3semidirect
72. C3×C4×D4D4semidirect
250. C4×D4#67semidirect
550. C3#32917semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C2×#52182
C2×#52172
C2×#50162
#822872
#716132
#716122
#716042
#715862
#715852
#715832
#329173