Finite Group Databasebeta

← #10822343

Subgroups — #10822343

Order 768.

2706 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#940C211yesmaximal, characteristic, complemented
2C22×#932C211yesmaximal, characteristic, complemented
3C22×#908C211yesmaximal, characteristic, complemented
4C2×C3×#418C211yesmaximal, characteristic, complemented
5#82228C211yesmaximal, characteristic, complemented
6C2×#5001C211yesmaximal, characteristic, complemented
7#71979C211yesmaximal, characteristic, complemented
8#70210C214yesmaximal, complemented
9#70209C214yesmaximal, complemented
10#23430—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C23×#67—22no—
13C22×#235C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#940—24no—
18C2×#940—24no—
19C2×#940—22no—
20C2×#940C2214yes—
21C22×#214C2212yescomplemented
22C2×#932—24no—
23C2×#932—24no—
24C2×#932—24no—
25C22×#213C2212yescomplemented
26C22×C4×Dic3C2211yescharacteristic
27C22×#208C2211yescharacteristic
28C2×#908—24no—
29C2×#908—24no—
30C2×#908—24no—
31C22×D24C2212yescomplemented
32C22×#204C2212yescomplemented
33C3×#418—24no—
34C2×C3×#102C2214yescomplemented
35#5001—24no—
36#4988C2212yescomplemented
37#4973C2212yescomplemented
38C2×#809C2214yescomplemented
39#4958—22no—
40C4×(C2×D3)⋊C4—22no—
41#4550C2214yescomplemented
42C22×AGL(1,ℤ/8)—31no—
43C22×AGL(1,ℤ/8)—31no—
44C22×#127—31no—
45C2×#418—31no—
46C2×#418D311yescharacteristic, complemented
47#2297—31no—
48#2014—34no—
49#2011—34no—
50#1728—31no—
51C25×C3—21no—
52C24×D3—21no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4D414yes—
60C22×C3×D4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C22×#67—216no—
63C22×#67—21no—
64C22×#67—28no—
65C22×#67—416no—
66C23×Dic3D412yes—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235D414yescomplemented
73C22×D12—22no—
74C22×D12—22no—
75C22×D12—22no—
76C22×D12—22no—
77C22×D12—22no—
78C22×D12—21no—
79C22×D12D414yes—
80C22×D12C2312yes—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×Dic6C2311yescharacteristic
84C2×C3×M16—22no—
85C2×C3×M16—22no—
86C2×C3×M16D414yescomplemented
87C22×C3×C8C2312yes—
88C3×#117—22no—
89C3×#117—22no—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93#940—216no—
94#940—28no—
95#940—28no—
96C2×#214—24no—
97C2×#214—24no—
98C2×#214—24no—
99#932—232no—
100C2×#213—24no—
101C2×#213—24no—
102C2×#213—24no—
103C2×(C2×D3)⋊C4—24no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3—22no—
106C2×C4×Dic3D412yes—
107C2×C4×Dic3C2312yes—
108C2×#208—22no—
109C2×#208—22no—
110C2×#208D414yescomplemented
111C22×C3⋊C8C2312yes—
112#908—232no—
113C2×D24—24no—
114C2×D24—24no—
115C2×D24—24no—
116C2×#204—24no—
117C2×#204—24no—
118C2×#204—24no—
119C3×#102—28no—
120C3⋊#95—24no—
121#809—28no—
122C22×#41—31no—
123C23×D4—31no—
124C23×D4—62no—
125C23×D4C2×D311yescharacteristic
126C2×AGL(1,ℤ/8)—64no—
127C2×AGL(1,ℤ/8)—64no—
128C2×AGL(1,ℤ/8)—64no—
129C2×AGL(1,ℤ/8)—64no—
130C2×AGL(1,ℤ/8)—64no—
131C2×AGL(1,ℤ/8)—64no—
132C22×SD16—32no—
133C22×SD16—32no—
134C22×D8—32no—
135C22×D8—32no—
136C22×M16—31no—
137C22×M16C2×D311yescharacteristic
138C2×#127—34no—
139C2×#127—62no—
140C2×#127—64no—
141C2×#127—64no—
142C2×#117C2×D311yescharacteristic
143C22×C42—31no—
144#418—24no—
145#418—64no—
146C2×#102—34no—
147C2×#102C2×D314yescomplemented
148#391—32no—
149#390—32no—
150#386—62no—
151C4×C22⋊C4—62no—
152#336—34no—
153C24×C3—24no—
154C24×C3—22no—
155C24×C3—48no—
156C24×C3—42no—
157C24×C3—42no—
158C24×C3C2×D411yescharacteristic
159C23×D3—24no—
160C23×D3—21no—
161C23×D3—22no—
162C23×D3—48no—
163C23×D3—42no—
164C23×D3—42no—
165C2×C3×D4—28no—
166C2×C3×D4—24no—
167C2×C3×D4—28no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—44no—
174C2×C3×D4—44no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4C2×D414yes—
178C22×C3×C4—22no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4#4112yes—
186C22×C3×C4C2411yescharacteristic
187C2×#67—232no—
188C2×#67—24no—
189C2×#67—464no—
190C2×#67—416no—
191C2×#67—42no—
192C2×#67—42no—
193C2×#67—416no—
194C22×Dic3—24no—
195C22×Dic3—28no—
196C22×Dic3—21no—
197C22×Dic3C2×D414yes—
198#235—28no—
199#235—28no—
200#235—44no—
201#235—44no—
202C2×D12—28no—
203C2×D12—28no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—22no—
212C2×D12—24no—
213C2×D12—22no—
214C2×D12—21no—
215C2×D12—44no—
216C2×D12—42no—
217C2×D12—44no—
218C2×D12—42no—
219C2×D12C2×D414yes—
220C2×D12C2×D412yescomplemented
221C2×C4×D3—24no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6C2×D412yescomplemented
230C3×M16—28no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C2×C3×C8C2×D414yescomplemented
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C22⋊C4—42no—
237C3×C42—42no—
238#214—216no—
239#213—216no—
240(C2×D3)⋊C4—48no—
241C4×Dic3—24no—
242C4×Dic3—22no—
243C4×Dic3—22no—
244#208—28no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8C2×D414yescomplemented
248D24—216no—
249#204—216no—
250C25—21no—
251C25—61no—
252C2×#41—34no—
253C2×#41—62no—
254C2×#41—62no—
255C2×#41—62no—
256C2×#41—62no—
257C2×#41—62no—
258C22×Q8—31no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—34no—
266C22×D4—32no—
267C22×D4—31no—
268C22×D4—68no—
269C22×D4—616no—
270C22×D4—61no—
271C22×D4—62no—
272C22×D4—62no—
273C22×D4—62no—
274C22×D4—62no—
275C22×D4—61no—
276C22×D4—62no—
277C22×D4—1216no—
278C22×D4D1214yes—
279C22×D4C22×D312yes—
280C23×C4—32no—
281C23×C4—61no—
282C23×C4C22×D311yescharacteristic
283AGL(1,ℤ/8)—632no—
284AGL(1,ℤ/8)—632no—
285C2×SD16—64no—
286C2×SD16—64no—
287C2×SD16—64no—
288C2×SD16—64no—
289C2×SD16—64no—
290C2×SD16—64no—
291C2×D8—64no—
292C2×D8—64no—
293C2×D8—64no—
294C2×D8—64no—
295C2×D8—64no—
296C2×D8—64no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—34no—
300C2×M16—62no—
301C2×M16—62no—
302C2×M16#6714yescomplemented
303C22×C8—32no—
304C22×C8C22×D312yes—
305#127—68no—
306#127—616no—
307#127—68no—
308#117—22no—
309#117—22no—
310C2×C4⋊C4C22×D312yes—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—64no—
313C2×C42—21no—
314C2×C42—32no—
315C2×C42—32no—
316C2×C42—62no—
317C2×C42—62no—
318#102—28no—
319#102—68no—
320#95—64no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—416no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—48no—
327C23×C3—48no—
328C23×C3—88no—
329C23×C3—88no—
330C23×C3#12012yes—
331C23×C3#12112yes—
332C23×C3#12712yes—
333C23×C3C22×D411yescharacteristic
334C22×D3—24no—
335C22×D3—22no—
336C22×D3—416no—
337C22×D3—41no—
338C22×D3—42no—
339C22×D3—41no—
340C22×D3—42no—
341C22×D3—44no—
342C22×D3—48no—
343C22×D3—48no—
344C22×D3—88no—
345C22×D3—88no—
346C3×D4—24no—
347C3×D4—24no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—48no—
351C2×C3×C4—42no—
352C2×C3×C4—44no—
353C2×C3×C4#12014yes—
354C2×C3×C4#12012yes—
355C2×C3×C4#12014yes—
356C2×C3×C4#12114yes—
357C2×C3×C4#12112yes—
358C2×C3×C4#12114yes—
359C2×C3×C4#12712yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366#67—464no—
367#67—48no—
368C2×Dic3—216no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—22no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3AGL(1,ℤ/8)18yescomplemented
375D12—24no—
376D12—22no—
377D12—22no—
378D12—24no—
379D12—42no—
380D12—42no—
381D12—41no—
382D12—41no—
383D12—48no—
384C4×D3—48no—
385Dic6—22no—
386Dic6—22no—
387Dic6—41no—
388Dic6—41no—
389C3×C8—24no—
390C3⋊C8—24no—
391C24—24no—
392C24—22no—
393C24—48no—
394C24—42no—
395C24—42no—
396C24—64no—
397C24—61no—
398C24—62no—
399C24—128no—
400C24—122no—
401C24—122no—
402C24D3×D411yescharacteristic
403#41—68no—
404#41—68no—
405#41—124no—
406#41—124no—
407C2×Q8—32no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×D4—28no—
414C2×D4—24no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—34no—
422C2×D4—32no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—632no—
428C2×D4—64no—
429C2×D4—64no—
430C2×D4—68no—
431C2×D4—68no—
432C2×D4—61no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—61no—
436C2×D4—62no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—62no—
440C2×D4—61no—
441C2×D4—62no—
442C2×D4—1216no—
443C2×D4—1264no—
444C2×D4—1216no—
445C2×D4—122no—
446C2×D4—122no—
447C2×D4—124no—
448C2×D4—122no—
449C2×D4—124no—
450C2×D4—122no—
451C2×D4C2×D1214yes—
452C22×C4—22no—
453C22×C4—34no—
454C22×C4—68no—
455C22×C4—64no—
456C22×C4—61no—
457C22×C4—64no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4C2×D1212yes—
461C22×C4#23512yes—
462C22×C4D3×D411yescharacteristic
463C22×C4D3×D412yes—
464C22×C4D3×D412yes—
465C22×C4D3×D412yes—
466C22×C4C2×#6712yes—
467C22×C4C23×D311yescharacteristic
468SD16—616no—
469SD16—616no—
470D8—616no—
471D8—616no—
472M16—28no—
473M16—68no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—34no—
477C2×C8—62no—
478C2×C8—62no—
479C2×C8C2×#6714yescomplemented
480C4⋊C4—22no—
481C4⋊C4—22no—
482C22⋊C4—42no—
483C22⋊C4—128no—
484C42—42no—
485C42—64no—
486C42—62no—
487C42—62no—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—44no—
492C22×C3—48no—
493C22×C3—816no—
494C22×C3—84no—
495C22×C3—84no—
496C22×C3#39214yes—
497C22×C3C2×#12011yescharacteristic
498C22×C3C2×#12111yescharacteristic
499C22×C3C2×#12711yescharacteristic
500C22×C3C2×AGL(1,ℤ/8)14yes—
501C2×D3—42no—
502C2×D3—44no—
503C2×D3—48no—
504C2×D3—44no—
505C2×D3—816no—
506C2×D3—82no—
507C2×D3—84no—
508C2×D3—82no—
509C2×D3—84no—
510C3×C4—82no—
511C3×C4#39214yes—
512C3×C4C2×#12011yescharacteristic
513C3×C4C2×#12011yescharacteristic
514C3×C4C2×#12111yescharacteristic
515C3×C4C2×#12111yescharacteristic
516Dic3—28no—
517Dic3—42no—
518C23—24no—
519C23—24no—
520C23—416no—
521C23—48no—
522C23—42no—
523C23—48no—
524C23—42no—
525C23—64no—
526C23—62no—
527C23—88no—
528C23—88no—
529C23—1216no—
530C23—128no—
531C23—121no—
532C23—122no—
533C23—124no—
534C23—128no—
535C23—121no—
536C23—122no—
537C23—248no—
538C23—248no—
539C23#88212yes—
540C23#88412yes—
541C23#94012yes—
542C23C2×D3×D411yescharacteristic
543Q8—62no—
544Q8—62no—
545Q8—121no—
546Q8—121no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—42no—
551D4—42no—
552D4—62no—
553D4—64no—
554D4—62no—
555D4—64no—
556D4—1264no—
557D4—128no—
558D4—128no—
559D4—122no—
560D4—121no—
561D4—122no—
562D4—121no—
563C2×C4—38no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—616no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—62no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—128no—
573C2×C4#88212yes—
574C2×C4#88214yes—
575C2×C4#88412yes—
576C2×C4#88414yes—
577C2×C4#93014yes—
578C2×C4#93714yes—
579C2×C4#94012yes—
580C2×C4C22×D1211yescharacteristic
581C2×C4C2×#23511yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×D3×D411yescharacteristic
584C2×C4C2×D3×D411yescharacteristic
585C2×C4C22×#6711yescharacteristic
586C8—24no—
587C8—64no—
588C2×C3—24no—
589C2×C3—84no—
590C2×C3C2×#39211yescharacteristic
591C2×C3#300312yes—
592C2×C3#303812yes—
593C2×C3#315112yes—
594D3—82no—
595D3—84no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—48no—
600C22—44no—
601C22—816no—
602C22—84no—
603C22—84no—
604C22—128no—
605C22—124no—
606C22—122no—
607C22—124no—
608C22—2416no—
609C22—242no—
610C22—244no—
611C22—244no—
612C22—242no—
613C22#496014yes—
614C22C2×#88211yescharacteristic
615C22C2×#88411yescharacteristic
616C22D3×AGL(1,ℤ/8)14yes—
617C22C2×#94011yescharacteristic
618C4—68no—
619C4—82no—
620C4—122no—
621C4#496014yes—
622C4C2×#88211yescharacteristic
623C4C2×#88411yescharacteristic
624C4C2×#93011yescharacteristic
625C4C2×#93711yescharacteristic
626C3#2343011yescharacteristic, complemented, Sylow
627C2—24no—
628C2—84no—
629C2—244no—
630C2—242no—
631C2#7892612yes—
632C2#7894212yes—
633C2C2×#496011yescharacteristic
634C2#8393712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#418C2
socleC23×C3C22×D4
hypercenterC2×#418D3
hypocenterC3#23430
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
1. C22×#940C2semidirect
2. C22×#932C2semidirect
3. C22×#908C2semidirect
4. C2×C3×#418C2semidirect
5. #82228C2semidirect
6. C2×#5001C2semidirect
7. #71979C2semidirect
8. #70210C2semidirect
9. #70209C2semidirect
21. C22×#214C22semidirect
25. C22×#213C22semidirect
31. C22×D24C22semidirect
32. C22×#204C22semidirect
34. C2×C3×#102C22semidirect
36. #4988C22semidirect
37. #4973C22semidirect
38. C2×#809C22semidirect
41. #4550C22semidirect
46. C2×#418D3semidirect
72. C2×#235D4semidirect
86. C2×C3×M16D4semidirect
110. C2×#208D4semidirect
147. C2×#102C2×D3semidirect
220. C2×D12C2×D4semidirect
229. C2×Dic6C2×D4semidirect
233. C2×C3×C8C2×D4semidirect
247. C2×C3⋊C8C2×D4semidirect
302. C2×M16#67semidirect
374. C2×Dic3AGL(1,ℤ/8)semidirect
479. C2×C8C2×#67semidirect
626. C3#23430semidirect

Maximal subgroups

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

subgroupindex
C22×#9402
C22×#9322
C22×#9082
C2×C3×#4182
#822282
C2×#50012
#719792
#702102
#702092
#234303