Finite Group Databasebeta

← #10826936

Subgroups — #10826936

Order 768.

2252 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#930C211yesmaximal, characteristic
2C22×#924C211yesmaximal, characteristic, complemented
3C2×#4738C212yesmaximal, complemented
4C2×#4731C212yesmaximal, complemented
5C3×#1779C211yesmaximal, characteristic, complemented
6#69962C214yesmaximal, complemented
7#69932C212yesmaximal, complemented
8#69930C212yesmaximal, complemented
9#23919—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C23×#67—21no—
12C23×D12C2211yescharacteristic
13C23×Dic6C2211yescharacteristic, complemented
14C22×#217—21no—
15C2×#930—21no—
16C2×#930—21no—
17C2×#930—28no—
18C2×#930C2212yes—
19C2×#930C2212yes—
20C22×(C2×D3)⋊C4—21no—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic, complemented
23C2×#924—21no—
24C2×#924—21no—
25C2×#924—28no—
26C2×#924C2212yes—
27C2×#924C2212yescomplemented
28C22×Dic3⋊C4—21no—
29C22×#204C2214yescomplemented
30C2×C3×C22⋊C8C2212yes—
31C2×C3×C22⋊C8C2212yescomplemented
32C2×C3×C22⋊C8C2212yescomplemented
33C2×#821C2214yescomplemented
34C2×#816C2214yescomplemented
35#4738—28no—
36#4731—28no—
37#4551C2214yescomplemented
38C22×C22⋊Q8—31no—
39C22×#121—31no—
40C2×#448—32no—
41C2×#450—32no—
42#2000—34no—
43#1874—32no—
44#1872—32no—
45#1779D311yescharacteristic, complemented
46C25×C3D411yescharacteristic
47C24×D3—21no—
48C23×C3×C4—24no—
49C23×C3×C4—24no—
50C23×C3×C4—42no—
51C23×C3×C4D411yescharacteristic
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C22×#67—28no—
58C22×#67—22no—
59C22×#67—22no—
60C22×#67—44no—
61C22×#67—44no—
62C23×Dic3—21no—
63C23×Dic3—21no—
64C23×Dic3—21no—
65C22×D12—21no—
66C22×D12—24no—
67C22×D12—24no—
68C22×D12—42no—
69C22×D12C2312yes—
70C22×Dic6—21no—
71C22×Dic6—24no—
72C22×Dic6—24no—
73C22×Dic6—42no—
74C22×Dic6C2312yes—
75C22×C3×C8C2314yes—
76C2×#217—28no—
77C2×#217—22no—
78C2×#217—22no—
79#930—28no—
80#930—28no—
81#930—24no—
82#930—24no—
83#930—24no—
84#930D418yes—
85C2×(C2×D3)⋊C4—28no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—24no—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2312yes—
94#924—28no—
95#924—28no—
96#924—24no—
97#924—24no—
98#924—24no—
99#924D418yescomplemented
100C2×Dic3⋊C4—28no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×#204—28no—
104C2×#204—28no—
105C2×#204—44no—
106C3×C22⋊C8—24no—
107C3×C22⋊C8—24no—
108C3×C22⋊C8—42no—
109#821—28no—
110#816—28no—
111C23×Q8—31no—
112C23×D4—31no—
113C23×D4—61no—
114C24×C4C2×D311yescharacteristic
115C22×SD16—34no—
116C2×C22⋊Q8—32no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—68no—
119C2×C22⋊Q8—61no—
120C2×C22⋊Q8—61no—
121C2×#121—32no—
122C2×#121—32no—
123C2×#121—68no—
124C2×#121—61no—
125C2×#121—61no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—61no—
129C22×C22⋊C4—61no—
130C22×C22⋊C4—61no—
131#450—68no—
132#448—68no—
133C2×#103—34no—
134C2×#102—34no—
135C2×C22⋊C8C2×D312yes—
136C2×C22⋊C8C2×D312yescomplemented
137C2×C22⋊C8C2×D312yescomplemented
138#340—34no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—41no—
144C24×C3—41no—
145C24×C3SD1614yescomplemented
146C24×C3C2×D411yescharacteristic
147C24×C3C2×D411yescharacteristic
148C24×C3C2×D411yescharacteristic
149C23×D3—24no—
150C23×D3—22no—
151C23×D3—42no—
152C23×D3—42no—
153C23×D3—42no—
154C23×D3—42no—
155C23×D3—42no—
156C23×D3—42no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—41no—
171C22×C3×C4—41no—
172C22×C3×C4—41no—
173C22×C3×C4—416no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D414yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—28no—
181C2×#67—28no—
182C2×#67—432no—
183C2×#67—44no—
184C2×#67—44no—
185C2×#67—44no—
186C2×#67—44no—
187C22×Dic3—24no—
188C22×Dic3—24no—
189C22×Dic3—24no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—42no—
194C22×Dic3—42no—
195C22×Dic3—42no—
196C22×Dic3—42no—
197C22×Dic3—42no—
198C22×Dic3—42no—
199C2×D12—24no—
200C2×D12—24no—
201C2×D12—24no—
202C2×D12—22no—
203C2×D12—416no—
204C2×D12—42no—
205C2×D12—42no—
206C2×D12C2×D414yes—
207C2×Dic6—24no—
208C2×Dic6—24no—
209C2×Dic6—24no—
210C2×Dic6—22no—
211C2×Dic6—416no—
212C2×Dic6—42no—
213C2×Dic6—42no—
214C2×Dic6C2×D414yescomplemented
215C2×C3×C8—24no—
216C2×C3×C8—24no—
217C2×C3×C8—24no—
218#217—28no—
219#217—28no—
220(C2×D3)⋊C4—28no—
221(C2×D3)⋊C4—28no—
222C4⋊Dic3—24no—
223C4⋊Dic3—24no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3C2×D414yes—
227C4⋊Dic3C2×D414yescomplemented
228Dic3⋊C4—28no—
229Dic3⋊C4—28no—
230#204—416no—
231C25—61no—
232C25D1211yescharacteristic
233C22×Q8—32no—
234C22×Q8—64no—
235C22×Q8—61no—
236C22×Q8—64no—
237C22×Q8—122no—
238C22×D4—32no—
239C22×D4—64no—
240C22×D4—68no—
241C22×D4—62no—
242C22×D4—62no—
243C22×D4—61no—
244C22×D4—64no—
245C22×D4—122no—
246C22×D4—124no—
247C22×D4—124no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—42no—
251C23×C4—61no—
252C23×C4—61no—
253C23×C4—61no—
254C23×C4D1211yescharacteristic
255C23×C4D1211yescharacteristic
256C23×C4D1211yescharacteristic
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D311yescharacteristic
260C2×SD16—68no—
261C2×SD16—68no—
262C2×SD16—124no—
263C22×C8C22×D314yes—
264C22⋊Q8—38no—
265C22⋊Q8—64no—
266C22⋊Q8—64no—
267C22⋊Q8—68no—
268C22⋊Q8—68no—
269C22⋊Q8—64no—
270#121—38no—
271#121—64no—
272#121—64no—
273#121—68no—
274#121—68no—
275#121—64no—
276C2×C4⋊C4—32no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—64no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—68no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C22⋊C4—68no—
286C2×C22⋊C4—68no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291#103—68no—
292#102—68no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295C22⋊C8—42no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—28no—
311C23×C3—28no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3#12012yes—
318C23×C3#12012yes—
319C23×C3C2×SD1612yes—
320C23×C3C2×SD1612yes—
321C23×C3C2×SD1612yes—
322C23×C3AGL(1,ℤ/8)12yes—
323C23×C3#13712yes—
324C23×C3C22×D411yescharacteristic
325C22×D3—24no—
326C22×D3—48no—
327C22×D3—48no—
328C22×D3—48no—
329C22×D3—42no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—84no—
336C22×D3—84no—
337C22×D3—84no—
338C22×D3—84no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—28no—
344C2×C3×C4—28no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—44no—
349C2×C3×C4—44no—
350C2×C3×C4—44no—
351C2×C3×C4—44no—
352C2×C3×C4#12012yes—
353C2×C3×C4#12012yes—
354C2×C3×C4C22×D412yes—
355C2×C3×C4C22×D412yes—
356C2×C3×C4H(5,𝔽2)12yes—
357#67—416no—
358#67—416no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—48no—
363C2×Dic3—48no—
364C2×Dic3—48no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371D12—28no—
372D12—48no—
373D12—44no—
374Dic6—28no—
375Dic6—48no—
376Dic6—44no—
377C3×C8—44no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—64no—
385C24—62no—
386C24—122no—
387C24—122no—
388C24—122no—
389C24—122no—
390C24—122no—
391C24—122no—
392C24#20414yescomplemented
393C24C2×D1211yescharacteristic
394C24C2×D1211yescharacteristic
395C24C2×D1211yescharacteristic
396C2×Q8—34no—
397C2×Q8—64no—
398C2×Q8—62no—
399C2×Q8—64no—
400C2×Q8—64no—
401C2×Q8—1216no—
402C2×Q8—122no—
403C2×Q8—122no—
404C2×D4—34no—
405C2×D4—64no—
406C2×D4—68no—
407C2×D4—68no—
408C2×D4—64no—
409C2×D4—62no—
410C2×D4—64no—
411C2×D4—1216no—
412C2×D4—122no—
413C2×D4—1232no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—124no—
418C2×D4—122no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—416no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4C2×D1211yescharacteristic
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1211yescharacteristic
451C22×C4D3×D414yes—
452C22×C4D3×D414yes—
453C22×C4C23×D311yescharacteristic
454SD16—1216no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C4⋊C4—34no—
459C4⋊C4—34no—
460C4⋊C4—64no—
461C4⋊C4—68no—
462C4⋊C4—64no—
463C4⋊C4—68no—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C22⋊C4—68no—
467C22⋊C4—68no—
468C22⋊C4—68no—
469C22⋊C4—68no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—416no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3#44814yescomplemented
491C22×C3#45014yescomplemented
492C22×C3C2×#12011yescharacteristic
493C22×C3C2×#12011yescharacteristic
494C22×C3#53411yescharacteristic
495C22×C3C22×SD1611yescharacteristic
496C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
497C22×C3C2×#13711yescharacteristic
498C22×C3#57511yescharacteristic
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—44no—
502C2×D3—816no—
503C2×D3—82no—
504C2×D3—82no—
505C2×D3—82no—
506C2×D3—82no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—41no—
510C3×C4D4214yes—
511Dic3—44no—
512Dic3—44no—
513Dic3—44no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—28no—
529C23—28no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—64no—
536C23—128no—
537C23—128no—
538C23—128no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23—244no—
546C23—244no—
547C23—244no—
548C23—244no—
549C23#88212yes—
550C23#88212yes—
551C23C2×#20412yes—
552C23C2×#20412yes—
553C23C2×#20412yes—
554C23#90812yes—
555C23#90912yes—
556C23C22×D1211yescharacteristic
557Q8—68no—
558Q8—128no—
559Q8—124no—
560D4—68no—
561D4—128no—
562D4—1216no—
563D4—1216no—
564D4—124no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—128no—
582C2×C4—128no—
583C2×C4—128no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4#88212yes—
591C2×C4#88212yes—
592C2×C4C2×D3×D412yes—
593C2×C4C2×D3×D412yes—
594C2×C4#100412yes—
595C8—44no—
596C2×C3—24no—
597C2×C3—24no—
598C2×C3—42no—
599C2×C3#240311yescharacteristic
600C2×C3C2×#45011yescharacteristic
601C2×C3C2×#44811yescharacteristic
602C2×C3#300611yescharacteristic
603C2×C3#300711yescharacteristic
604C2×C3#318711yescharacteristic
605C2×C3#319311yescharacteristic
606D3—84no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—416no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—124no—
628C22—124no—
629C22—124no—
630C22—2416no—
631C22—242no—
632C22—242no—
633C22—242no—
634C22—242no—
635C22#473114yescomplemented
636C22#473814yescomplemented
637C22C2×#88211yescharacteristic
638C22C2×#88211yescharacteristic
639C22#549311yescharacteristic
640C22C22×#20411yescharacteristic
641C22C2×#90811yescharacteristic
642C22C2×#90911yescharacteristic
643C22#574811yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—124no—
648C4—124no—
649C4—124no—
650C4#559014yes—
651C3#2391911yescharacteristic, complemented, Sylow
652C2—24no—
653C2—24no—
654C2—42no—
655C2—244no—
656C2#7385211yescharacteristic
657C2C2×#473111yescharacteristic
658C2#7861211yescharacteristic
659C2C2×#473811yescharacteristic
660C2#7863911yescharacteristic
661C2#7864011yescharacteristic
662C2#7864211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1779C2
socleC23×C3C22×D4
prime radicalC22×#930C2
hypercenter#1779D3
hypocenterC3#23919
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C22×#924C2semidirect
3. C2×#4738C2semidirect
4. C2×#4731C2semidirect
5. C3×#1779C2semidirect
6. #69962C2semidirect
7. #69932C2semidirect
8. #69930C2semidirect
13. C23×Dic6C22semidirect
22. C22×C4⋊Dic3C22semidirect
27. C2×#924C22semidirect
29. C22×#204C22semidirect
31. C2×C3×C22⋊C8C22semidirect
32. C2×C3×C22⋊C8C22semidirect
33. C2×#821C22semidirect
34. C2×#816C22semidirect
37. #4551C22semidirect
45. #1779D3semidirect
99. #924D4semidirect
136. C2×C22⋊C8C2×D3semidirect
137. C2×C22⋊C8C2×D3semidirect
145. C24×C3SD16semidirect
214. C2×Dic6C2×D4semidirect
227. C4⋊Dic3C2×D4semidirect
392. C24#204semidirect
490. C22×C3#448semidirect
491. C22×C3#450semidirect
635. C22#4731semidirect
636. C22#4738semidirect
651. C3#23919semidirect

Maximal subgroups

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

subgroupindex
C22×#9302
C22×#9242
C2×#47382
C2×#47312
C3×#17792
#699622
#699322
#699302
#239193