Finite Group Databasebeta

← #10826925

Subgroups — #10826925

Order 768.

2596 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#121C212yesmaximal, complemented
2C2×#5156C212yesmaximal, complemented
3C2×#4773C212yesmaximal, complemented
4C3×#1999C211yesmaximal, characteristic, complemented
5#71684C212yesmaximal, complemented
6#70200C212yesmaximal, complemented
7#69961C211yesmaximal, characteristic, complemented
8#69955C211yesmaximal, characteristic, complemented
9#68880C212yesmaximal, complemented
10#23918—31nomaximal, Sylow
11C22×D3×D4—22no—
12C22×D3×D4C2212yescomplemented
13C23×C4×D3C2211yescharacteristic
14C22×C3×D8C2211yescharacteristic, complemented
15C2×C3×#121C2212yescomplemented
16C2×#939—22no—
17C2×#938C2212yes—
18C22×#213C2212yescomplemented
19C2×#930C2212yescomplemented
20C22×D24C2211yescharacteristic, complemented
21D3×#121—28no—
22D3×#121—28no—
23C2×#895C2212yes—
24C2×D3×C4⋊C4C2212yescomplemented
25C2×#884—22no—
26C2×D3×C22⋊C4—22no—
27C2×C3×#102C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic, complemented
29C2×#832C2212yescomplemented
30C2×#830C2211yescharacteristic, complemented
31#5156—28no—
32C2×#821C2212yescomplemented
33C2×(C2×D3)⋊C8C2212yescomplemented
34C2×(C2×D3)⋊C8C2212yescomplemented
35C2×#809C2212yescomplemented
36#4773—28no—
37C3×#340C2211yescharacteristic, complemented
38#4551C2211yescharacteristic, complemented
39#4529C2212yescomplemented
40C22×#121—32no—
41C2×#447—32no—
42C2×#447—32no—
43#1999—31no—
44#1999—32no—
45#1999D311yescharacteristic, complemented
46#1874—32no—
47#1874—32no—
48#1779—31no—
49C24×D3—22no—
50C24×D3D411yescharacteristic
51C22×C3×D4—22no—
52C22×C3×D4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C22×#67—22no—
56C22×#67—22no—
57C23×Dic3D411yescharacteristic
58C2×D3×D4—216no—
59C2×D3×D4—24no—
60C2×D3×D4—24no—
61C2×D3×D4—24no—
62C2×D3×D4—24no—
63C2×D3×D4—42no—
64C2×D3×D4—42no—
65C2×D3×D4—48no—
66C2×D3×D4—48no—
67C22×D12—22no—
68C22×D12C2312yes—
69C22×C4×D3—22no—
70C22×C4×D3—24no—
71C22×C4×D3—24no—
72C22×C4×D3—41no—
73C22×C4×D3—41no—
74C22×C4×D3D412yes—
75C22×C4×D3C2312yes—
76C2×C3×D8—24no—
77C2×C3×D8—41no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×#121—24no—
81C3×#121—24no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×#217—22no—
85#939—216no—
86#938—24no—
87#938D418yescomplemented
88C2×#213—24no—
89C2×#213—24no—
90C2×#213—42no—
91#930—24no—
92#930—24no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3C2312yes—
96C2×Dic3⋊C4—22no—
97C22×C3⋊C8C2312yes—
98C2×D24—24no—
99C2×D24—41no—
100#895—24no—
101#895D418yescomplemented
102D3×C4⋊C4—24no—
103D3×C4⋊C4—24no—
104#884—216no—
105D3×C22⋊C4—216no—
106C3×#102—24no—
107C3×C22⋊C8—41no—
108#832—24no—
109#830—41no—
110#821—24no—
111(C2×D3)⋊C8—24no—
112(C2×D3)⋊C8—24no—
113#809—24no—
114C23×D4—32no—
115C23×D4—62no—
116C24×C4—31no—
117C22×D8—32no—
118C22×D8—31no—
119C22×D8C2×D311yescharacteristic, complemented
120C2×#121—32no—
121C2×#121—32no—
122C2×#121—32no—
123C2×#121—62no—
124C2×#121—68no—
125C2×#121—68no—
126C2×#121—62no—
127C2×#121C2×D312yescomplemented
128C22×C4⋊C4—32no—
129C22×C22⋊C4—62no—
130#447—68no—
131#447—68no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102C2×D312yescomplemented
136C2×C22⋊C8—32no—
137C2×C22⋊C8—32no—
138C2×C22⋊C8—31no—
139C2×C22⋊C8C2×D311yescharacteristic, complemented
140#340—31no—
141#340—32no—
142#340C2×D311yescharacteristic, complemented
143C24×C3—22no—
144C24×C3C2×D411yescharacteristic
145C23×D3—24no—
146C23×D3—24no—
147C23×D3—28no—
148C23×D3—22no—
149C23×D3—41no—
150C23×D3—41no—
151C23×D3—44no—
152C23×D3—44no—
153C23×D3—44no—
154C23×D3—44no—
155C23×D3—44no—
156C23×D3—44no—
157C23×D3D814yescomplemented
158C23×D3C2×D412yes—
159C2×C3×D4—28no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—44no—
166C2×C3×D4—44no—
167C2×C3×D4C2×D414yescomplemented
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—28no—
176C2×#67—28no—
177C2×#67—28no—
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—44no—
181C2×#67—44no—
182C2×#67—44no—
183C2×#67—44no—
184C22×Dic3—22no—
185C22×Dic3—24no—
186C22×Dic3—41no—
187C22×Dic3C2×D412yes—
188D3×D4—48no—
189D3×D4—48no—
190D3×D4—48no—
191D3×D4—48no—
192D3×D4—464no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—28no—
198C2×D12—22no—
199C2×D12—44no—
200C2×D12—44no—
201C2×D12C2×D414yescomplemented
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—28no—
207C2×C4×D3—24no—
208C2×C4×D3—24no—
209C2×C4×D3—44no—
210C2×C4×D3—44no—
211C2×C4×D3—48no—
212C2×C4×D3—42no—
213C2×C4×D3—44no—
214C2×C4×D3—44no—
215C2×C4×D3C2×D414yes—
216C2×C4×D3C2×D414yes—
217C3×D8—44no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4C2×D414yescomplemented
224C3×C22⋊C4—28no—
225#217—28no—
226#213—48no—
227(C2×D3)⋊C4—28no—
228(C2×D3)⋊C4—28no—
229C4⋊Dic3—22no—
230C4⋊Dic3C2×D414yescomplemented
231Dic3⋊C4—28no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235D24—44no—
236C25—31no—
237C25—62no—
238C22×D4—22no—
239C22×D4—32no—
240C22×D4—62no—
241C22×D4—64no—
242C22×D4—64no—
243C22×D4—62no—
244C22×D4—64no—
245C22×D4—616no—
246C22×D4—62no—
247C22×D4—62no—
248C22×D4—64no—
249C22×D4—122no—
250C22×D4—122no—
251C22×D4—128no—
252C22×D4—128no—
253C22×D4C22×D312yes—
254C23×C4—32no—
255C23×C4—31no—
256C23×C4—32no—
257C23×C4—64no—
258C23×C4—64no—
259C23×C4—62no—
260C23×C4—121no—
261C23×C4—121no—
262C23×C4C22×D311yescharacteristic
263C2×D8—24no—
264C2×D8—41no—
265C2×D8—64no—
266C2×D8—64no—
267C2×D8—64no—
268C2×D8—121no—
269C2×D8—122no—
270C22×C8—32no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273#121—24no—
274#121—24no—
275#121—38no—
276#121—38no—
277#121—616no—
278#121—64no—
279#121—64no—
280#121—64no—
281#121—64no—
282#121—616no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—64no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—64no—
287C2×C4⋊C4C22×D312yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—616no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—62no—
292C2×C22⋊C4—62no—
293#102—24no—
294#102—64no—
295#102—64no—
296#102—64no—
297C22⋊C8—41no—
298C22⋊C8—64no—
299C22⋊C8—64no—
300C22⋊C8—121no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—24no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3C2×D812yes—
313C23×C3C22×D411yescharacteristic
314C22×D3—22no—
315C22×D3—22no—
316C22×D3—22no—
317C22×D3—22no—
318C22×D3—216no—
319C22×D3—24no—
320C22×D3—44no—
321C22×D3—44no—
322C22×D3—44no—
323C22×D3—48no—
324C22×D3—416no—
325C22×D3—416no—
326C22×D3—416no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—42no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—88no—
334C22×D3—88no—
335C22×D3—88no—
336C22×D3—88no—
337C22×D3#12014yes—
338C22×D3C2×D814yes—
339C22×D3AGL(1,ℤ/8)14yes—
340C3×D4—24no—
341C3×D4—24no—
342C3×D4—416no—
343C3×D4—42no—
344C3×D4—42no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4C22×D412yes—
354C2×C3×C4C22×D412yes—
355C2×C3×C4H(5,𝔽2)11yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357#67—416no—
358#67—416no—
359#67—416no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—24no—
363C2×Dic3—44no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3#12014yes—
367D12—24no—
368D12—24no—
369D12—416no—
370D12—42no—
371D12—42no—
372C4×D3—24no—
373C4×D3—24no—
374C4×D3—24no—
375C4×D3—24no—
376C4×D3—48no—
377C4×D3—416no—
378C3×C8—41no—
379C3×C8—41no—
380C3⋊C8—42no—
381C24—22no—
382C24—32no—
383C24—34no—
384C24—64no—
385C24—64no—
386C24—68no—
387C24—62no—
388C24—121no—
389C24—121no—
390C24—124no—
391C24—124no—
392C24—124no—
393C24—124no—
394C24—124no—
395C24—124no—
396C24D3×D411yescharacteristic
397C2×D4—28no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—34no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—68no—
407C2×D4—68no—
408C2×D4—68no—
409C2×D4—68no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—124no—
418C2×D4—128no—
419C2×D4—128no—
420C2×D4—128no—
421C2×D4—128no—
422C2×D4—124no—
423C2×D4—1264no—
424C2×D4—124no—
425C2×D4—124no—
426C2×D4—124no—
427C2×D4—124no—
428C2×D4D3×D414yescomplemented
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—32no—
435C22×C4—34no—
436C22×C4—34no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—68no—
440C22×C4—64no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—122no—
447C22×C4—124no—
448C22×C4—124no—
449C22×C4—128no—
450C22×C4—124no—
451C22×C4—124no—
452C22×C4—121no—
453C22×C4D3×D411yescharacteristic
454C22×C4C23×D311yescharacteristic
455D8—44no—
456D8—124no—
457D8—128no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—62no—
463C2×C8—62no—
464C2×C8—62no—
465C4⋊C4—22no—
466C4⋊C4—34no—
467C4⋊C4—68no—
468C4⋊C4—62no—
469C4⋊C4D3×D414yescomplemented
470C22⋊C4—28no—
471C22⋊C4—68no—
472C22⋊C4—68no—
473C22⋊C4—68no—
474C22×C3—22no—
475C22×C3—24no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—42no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—44no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3C2×#12012yes—
487C22×C3#53411yescharacteristic
488C22×C3C22×D811yescharacteristic
489C22×C3C2×AGL(1,ℤ/8)12yes—
490C22×C3#57511yescharacteristic
491C2×D3—24no—
492C2×D3—24no—
493C2×D3—24no—
494C2×D3—24no—
495C2×D3—24no—
496C2×D3—24no—
497C2×D3—416no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—832no—
502C2×D3—82no—
503C2×D3—82no—
504C2×D3—82no—
505C2×D3—82no—
506C2×D3#44718yescomplemented
507C3×C4—41no—
508C3×C4—44no—
509C3×C4D4211yescharacteristic
510C3×C4D4211yescharacteristic
511C3×C4D4212yes—
512Dic3—24no—
513Dic3—44no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—24no—
519C23—34no—
520C23—34no—
521C23—34no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—616no—
529C23—62no—
530C23—62no—
531C23—62no—
532C23—62no—
533C23—64no—
534C23—124no—
535C23—124no—
536C23—128no—
537C23—124no—
538C23—1216no—
539C23—1216no—
540C23—1216no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—248no—
548C23—248no—
549C23—248no—
550C23—248no—
551C23D3×D812yes—
552C23C2×D3×D411yescharacteristic
553D4—24no—
554D4—24no—
555D4—416no—
556D4—42no—
557D4—42no—
558D4—64no—
559D4—64no—
560D4—1216no—
561D4—1216no—
562D4—1216no—
563D4—1216no—
564D4—122no—
565D4—122no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—21no—
569C2×C4—34no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—128no—
583C2×C4—1216no—
584C2×C4—124no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4C2×D3×D412yes—
588C2×C4C2×D3×D412yes—
589C2×C4#100411yescharacteristic
590C2×C4#100911yescharacteristic
591C8—41no—
592C8—41no—
593C8—122no—
594C2×C3—42no—
595C2×C3—84no—
596C2×C3#240311yescharacteristic
597C2×C3C2×#44712yes—
598C2×C3#300612yes—
599C2×C3#318611yescharacteristic
600C2×C3#319211yescharacteristic
601D3—28no—
602D3—84no—
603C22—22no—
604C22—24no—
605C22—38no—
606C22—41no—
607C22—42no—
608C22—41no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—64no—
613C22—64no—
614C22—64no—
615C22—64no—
616C22—64no—
617C22—64no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—1216no—
623C22—124no—
624C22—124no—
625C22—124no—
626C22—2432no—
627C22—242no—
628C22—242no—
629C22—242no—
630C22—242no—
631C22D3×#12012yes—
632C22#560511yescharacteristic
633C22C2×D3×D811yescharacteristic
634C22#576811yescharacteristic
635C22D3×AGL(1,ℤ/8)12yes—
636C4—44no—
637C4—41no—
638C4—64no—
639C4—124no—
640C4#559011yescharacteristic
641C4#560912yes—
642C4#571711yescharacteristic
643C3#2391811yescharacteristic, complemented, Sylow
644C2—42no—
645C2—68no—
646C2—84no—
647C2—244no—
648C2#7675311yescharacteristic
649C2D3×#44712yes—
650C2#8008412yes—
651C2#8071911yescharacteristic
652C2#8086811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1999C2
socleC23×C3C22×D4
hypercenter#1999D3
hypocenterC3#23918
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. C2×D3×#121C2semidirect
2. C2×#5156C2semidirect
3. C2×#4773C2semidirect
4. C3×#1999C2semidirect
5. #71684C2semidirect
6. #70200C2semidirect
7. #69961C2semidirect
8. #69955C2semidirect
9. #68880C2semidirect
12. C22×D3×D4C22semidirect
14. C22×C3×D8C22semidirect
15. C2×C3×#121C22semidirect
18. C22×#213C22semidirect
19. C2×#930C22semidirect
20. C22×D24C22semidirect
24. C2×D3×C4⋊C4C22semidirect
27. C2×C3×#102C22semidirect
28. C2×C3×C22⋊C8C22semidirect
29. C2×#832C22semidirect
30. C2×#830C22semidirect
32. C2×#821C22semidirect
33. C2×(C2×D3)⋊C8C22semidirect
34. C2×(C2×D3)⋊C8C22semidirect
35. C2×#809C22semidirect
37. C3×#340C22semidirect
38. #4551C22semidirect
39. #4529C22semidirect
45. #1999D3semidirect
87. #938D4semidirect
101. #895D4semidirect
119. C22×D8C2×D3semidirect
127. C2×#121C2×D3semidirect
135. C2×#102C2×D3semidirect
139. C2×C22⋊C8C2×D3semidirect
142. #340C2×D3semidirect
157. C23×D3D8semidirect
167. C2×C3×D4C2×D4semidirect
201. C2×D12C2×D4semidirect
223. C3×C4⋊C4C2×D4semidirect
230. C4⋊Dic3C2×D4semidirect
428. C2×D4D3×D4semidirect
469. C4⋊C4D3×D4semidirect
506. C2×D3#447semidirect
643. C3#23918semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1212
C2×#51562
C2×#47732
C3×#19992
#716842
#702002
#699612
#699552
#688802
#239183