Finite Group Databasebeta

← #10943988

Subgroups — #10943988

Order 768.

1576 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic, complemented
2#82700C211yesmaximal, characteristic
3C2×#5040C211yesmaximal, characteristic, complemented
4C2×#4762C212yesmaximal, complemented
5C2×#4738C211yesmaximal, characteristic, complemented
6C3×#1874C211yesmaximal, characteristic, complemented
7#71560C211yesmaximal, characteristic, complemented
8#70213C212yesmaximal, complemented
9#69908C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68904C212yesmaximal, complemented
12#32903—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×#121C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#940C2211yescharacteristic
17C2×#939—21no—
18C2×#939—21no—
19C22×#214C2212yes—
20C2×#930C2211yescharacteristic
21C22×Dic3⋊C4—21no—
22C22×C4×Dic3—21no—
23C22×C4×Dic3C2211yescharacteristic
24C22×#204C2212yes—
25C2×#887—21no—
26C2×#887—21no—
27C2×#887—28no—
28C2×#887C2212yes—
29C2×#887C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×C22⋊C8C2211yescharacteristic
32#5221C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#816C2212yes—
35C2×Dic3⋊C8C2212yes—
36#5040—24no—
37#4988C2211yescharacteristic
38#4975C2211yescharacteristic
39C2×#810C2212yes—
40C2×#809C2212yes—
41#4957—21no—
42#4957—21no—
43#4762—28no—
44#4738—24no—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C22×C4×Q8—31no—
49C2×#460—32no—
50C2×#448—31no—
51C2×#448—31no—
52#2856—31no—
53#1880—32no—
54#1880—32no—
55#1874—31no—
56#1874D311yescharacteristic, complemented
57#1803—32no—
58#1797—31no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×#67—21no—
65C22×#67—21no—
66C23×Dic3—21no—
67C23×Dic3D411yescharacteristic
68C23×Dic3D411yescharacteristic
69C22×D12C2311yescharacteristic
70C22×Dic6—21no—
71C22×Dic6—22no—
72C22×Dic6—24no—
73C22×Dic6—24no—
74C22×Dic6D414yescomplemented
75C22×Dic6C2312yes—
76C22×C3×C8C2312yes—
77C3×#121—22no—
78C3×#121—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—21no—
84C2×#217—21no—
85C2×#217—21no—
86#940—24no—
87#939—44no—
88#939—44no—
89C2×#214—24no—
90C2×#214—24no—
91C2×#214—24no—
92#930—22no—
93#930—22no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—28no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—28no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3—22no—
106C2×C4×Dic3—24no—
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2312yes—
112C2×#204—24no—
113C2×#204—24no—
114C2×#204—24no—
115#887—28no—
116#887—28no—
117#887—24no—
118#887—24no—
119#887—24no—
120#887D418yescomplemented
121C3×#102—24no—
122C3×C22⋊C8—22no—
123#830—22no—
124#816—24no—
125Dic3⋊C8—24no—
126#810—24no—
127#809—24no—
128C23×Q8—31no—
129C22×SD16—32no—
130C22×SD16—32no—
131C2×#127—31no—
132C2×#121—31no—
133C2×#121—61no—
134C2×#121—61no—
135C2×#121C2×D311yescharacteristic
136C2×C4×Q8—32no—
137C2×C4×Q8—32no—
138C2×C4×Q8—61no—
139C2×C4×Q8—61no—
140C2×C4×Q8—68no—
141C22×C4⋊C4—61no—
142C22×C4⋊C4C2×D311yescharacteristic
143C22×C42—31no—
144C22×C42—61no—
145#460—68no—
146#448—64no—
147#448—64no—
148C2×C4⋊C8—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#102—32no—
152C2×#102C2×D312yes—
153C2×C22⋊C8—31no—
154C2×C22⋊C8C2×D311yescharacteristic
155#390—31no—
156#390—31no—
157#390—61no—
158#390—61no—
159#384—31no—
160#340—32no—
161#340C2×D312yes—
162#336—32no—
163C24×C3—21no—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4#4112yes—
183C22×C3×C4C2411yescharacteristic
184C2×#67—42no—
185C2×#67—42no—
186C2×#67—42no—
187C2×#67—42no—
188C2×#67—42no—
189C2×#67—42no—
190C2×#67—42no—
191C2×#67—42no—
192C2×#67—42no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—21no—
202C22×Dic3—41no—
203C22×Dic3SD1614yescomplemented
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×D12—22no—
209C2×D12—21no—
210C2×D12—22no—
211C2×D12—21no—
212C2×Dic6—24no—
213C2×Dic6—24no—
214C2×Dic6—216no—
215C2×Dic6—24no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—216no—
220C2×Dic6C2×D414yes—
221C2×Dic6C2×D414yes—
222C2×Dic6#4114yes—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4C2×D414yes—
231C3×C22⋊C4—42no—
232#217—42no—
233#217—42no—
234#214—48no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240Dic3⋊C4—28no—
241Dic3⋊C4—28no—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244C4×Dic3—28no—
245C4×Dic3—22no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—28no—
249C4×Dic3C2×D414yes—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253#204—48no—
254C22×Q8—34no—
255C22×Q8—32no—
256C22×Q8—64no—
257C22×Q8—64no—
258C22×Q8—62no—
259C22×Q8—61no—
260C22×D4—21no—
261C22×D4—31no—
262C22×D4—61no—
263C22×D4—61no—
264C22×D4—61no—
265C22×D4C22×D311yescharacteristic
266C23×C4—21no—
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271C2×SD16—64no—
272C2×SD16—64no—
273C2×SD16—64no—
274C2×SD16—64no—
275C2×SD16—64no—
276C2×SD16—64no—
277C22×C8—32no—
278C22×C8C22×D312yes—
279#127—64no—
280#121—22no—
281#121—22no—
282#121—62no—
283#121—62no—
284#121—124no—
285#121—124no—
286C4×Q8—38no—
287C4×Q8—64no—
288C4×Q8—64no—
289C4×Q8—68no—
290C4×Q8—68no—
291C4×Q8—64no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—62no—
296C2×C4⋊C4—62no—
297C2×C4⋊C4—68no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D312yes—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—64no—
313C2×C42—62no—
314C2×C42—62no—
315C2×C42—68no—
316C4⋊C8—64no—
317#103—64no—
318#103—64no—
319#102—24no—
320#102—64no—
321C22⋊C8—22no—
322C22⋊C8—62no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3#12112yes—
335C23×C3C2×SD1612yes—
336C23×C3AGL(1,ℤ/8)12yes—
337C23×C3C22×D411yescharacteristic
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C3×D4—42no—
346C3×D4—42no—
347C3×D4—84no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—48no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4#12112yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D412yes—
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4112yes—
364#67—84no—
365#67—84no—
366#67—84no—
367C2×Dic3—216no—
368C2×Dic3—21no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—28no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—44no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3#12012yes—
384C2×Dic3#12012yes—
385C2×Dic3C2×SD1612yes—
386C2×Dic3C2×SD1612yes—
387C2×Dic3#13712yes—
388D12—42no—
389D12—42no—
390Dic6—216no—
391Dic6—24no—
392Dic6—24no—
393Dic6—24no—
394Dic6#12118yescomplemented
395C3×C8—42no—
396C3⋊C8—42no—
397C24—21no—
398C24—61no—
399C24D3×D411yescharacteristic
400C2×Q8—34no—
401C2×Q8—34no—
402C2×Q8—34no—
403C2×Q8—616no—
404C2×Q8—64no—
405C2×Q8—64no—
406C2×Q8—616no—
407C2×Q8—64no—
408C2×Q8—62no—
409C2×Q8—62no—
410C2×Q8—62no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—61no—
419C2×D4—62no—
420C2×D4—61no—
421C2×D4—62no—
422C2×D4—122no—
423C2×D4—122no—
424C2×D4—122no—
425C2×D4—122no—
426C2×D4—122no—
427C2×D4—122no—
428C2×D4—122no—
429C2×D4—122no—
430C2×D4—122no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—34no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—64no—
447C22×C4—64no—
448C22×C4—64no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—61no—
452C22×C4—121no—
453C22×C4#23512yes—
454C22×C4D3×D411yescharacteristic
455C22×C4D3×D412yes—
456C22×C4C23×D311yescharacteristic
457SD16—128no—
458SD16—128no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—62no—
463C2×C8—62no—
464C2×C8—62no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—68no—
470C4⋊C4—68no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4D3×D414yes—
476C22⋊C4—42no—
477C22⋊C4—122no—
478C22⋊C4—122no—
479C22⋊C4—122no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C42—34no—
483C42—62no—
484C42—62no—
485C42—68no—
486C42—68no—
487C42—62no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3#46014yescomplemented
502C22×C3C2×#12011yescharacteristic
503C22×C3C2×#12111yescharacteristic
504C22×C3#52511yescharacteristic
505C22×C3C22×SD1611yescharacteristic
506C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
507C22×C3C2×#13711yescharacteristic
508C22×C3#57511yescharacteristic
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C3×C4—22no—
517C3×C4—44no—
518C3×C4—81no—
519C3×C4C2×#12112yes—
520C3×C4#52512yes—
521Dic3—28no—
522Dic3—22no—
523Dic3—22no—
524Dic3—41no—
525Dic3—81no—
526Dic3#44814yescomplemented
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23#88412yes—
546C23#91112yes—
547C23D3×SD1612yes—
548C23C2×D3×D411yescharacteristic
549Q8—38no—
550Q8—616no—
551Q8—64no—
552Q8—64no—
553Q8—64no—
554D4—42no—
555D4—42no—
556D4—84no—
557D4—122no—
558D4—122no—
559D4—244no—
560D4—244no—
561D4—244no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—32no—
568C2×C4—32no—
569C2×C4—32no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—68no—
579C2×C4—616no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—64no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—124no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4#88412yes—
595C2×C4C2×#23511yescharacteristic
596C2×C4C2×D3×D412yes—
597C2×C4C2×D3×D411yescharacteristic
598C2×C4D3×#4112yes—
599C8—42no—
600C8—122no—
601C2×C3—24no—
602C2×C3—161no—
603C2×C3#284711yescharacteristic
604C2×C3C2×#44811yescharacteristic
605C2×C3#300611yescharacteristic
606C2×C3C2×#46011yescharacteristic
607C2×C3#303711yescharacteristic
608C2×C3#310311yescharacteristic
609C2×C3#311211yescharacteristic
610D3—161no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—241no—
625C22—241no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22—241no—
630C22—241no—
631C22#476214yescomplemented
632C22C2×#88411yescharacteristic
633C22D3×#12011yescharacteristic
634C22#565611yescharacteristic
635C22C2×#91111yescharacteristic
636C22C2×D3×SD1611yescharacteristic
637C22#577311yescharacteristic
638C22D3×#13711yescharacteristic
639C4—22no—
640C4—34no—
641C4—44no—
642C4—62no—
643C4—62no—
644C4—68no—
645C4—81no—
646C4—121no—
647C4—241no—
648C4#554712yes—
649C4D3×#12112yes—
650C3#3290311yescharacteristic, complemented, Sylow
651C2—24no—
652C2—161no—
653C2—481no—
654C2#7493511yescharacteristic
655C2C2×#476211yescharacteristic
656C2#7891311yescharacteristic
657C2#8000311yescharacteristic
658C2D3×#44811yescharacteristic
659C2#8127011yescharacteristic
660C2#8130811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1874C2
socleC23×C3C22×D4
prime radical#82700C2
hypercenter#1874D3
hypocenterC3#32903
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#887C2semidirect
3. C2×#5040C2semidirect
4. C2×#4762C2semidirect
5. C2×#4738C2semidirect
6. C3×#1874C2semidirect
7. #71560C2semidirect
8. #70213C2semidirect
9. #69908C2semidirect
10. #68909C2semidirect
11. #68904C2semidirect
56. #1874D3semidirect
74. C22×Dic6D4semidirect
120. #887D4semidirect
203. C22×Dic3SD16semidirect
394. Dic6#121semidirect
501. C22×C3#460semidirect
526. Dic3#448semidirect
631. C22#4762semidirect
650. C3#32903semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
#827002
C2×#50402
C2×#47622
C2×#47382
C3×#18742
#715602
#702132
#699082
#689092
#689042
#329033