Finite Group Databasebeta

← #10822309

Subgroups — #10822309

Order 768.

2178 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic, complemented
2C22×#950C211yesmaximal, characteristic, complemented
3C22×#908C211yesmaximal, characteristic, complemented
4C2×C3×#417C211yesmaximal, characteristic, complemented
5#82186C211yesmaximal, characteristic, complemented
6C22×#809C211yesmaximal, characteristic, complemented
7#71980C211yesmaximal, characteristic, complemented
8#70278C212yesmaximal, complemented
9#70277C212yesmaximal, complemented
10#70210C212yesmaximal, complemented
11#70209C212yesmaximal, complemented
12#23426—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18#5830—24no—
19#5830—24no—
20C2×#950—24no—
21C2×#950—24no—
22C2×#950—24no—
23C2×#947C2212yes—
24C22×#216C2211yescharacteristic, complemented
25C22×#215C2211yescharacteristic, complemented
26C2×#940C2212yes—
27C2×#937—22no—
28C22×#214C2211yescharacteristic, complemented
29C22×#213C2211yescharacteristic, complemented
30C2×#930—22no—
31C2×#926C2211yescharacteristic
32C23×C3⋊C8C2211yescharacteristic
33C2×#908—24no—
34C2×#908—24no—
35C2×#908—24no—
36C22×D24C2212yes—
37C22×#204C2212yes—
38C3×#417—24no—
39C2×C3×#103C2212yescomplemented
40C2×C3×#102C2212yescomplemented
41#5213—22no—
42#4988C2212yescomplemented
43#4987—22no—
44#4973C2212yescomplemented
45C2×#809—24no—
46C2×#809C2212yes—
47C2×#809C2212yes—
48#4550C2212yescomplemented
49#4550C2212yescomplemented
50C22×AGL(1,ℤ/8)—31no—
51C22×#135—31no—
52C2×#535—31no—
53C2×#417D311yescharacteristic, complemented
54C22×#102—31no—
55#2311—31no—
56#2015—32no—
57#2014—32no—
58#2012—32no—
59#2011—32no—
60#1729—31no—
61C24×D3—21no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—22no—
66C2×C3×#41—22no—
67C2×C3×#41D414yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—21no—
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C22×#67—21no—
75C22×#67—24no—
76C2×#235—22no—
77C2×#235—22no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235D414yescomplemented
82C22×D12—22no—
83C22×D12—22no—
84C22×D12—22no—
85C22×D12—22no—
86C22×D12—21no—
87C22×D12—22no—
88C22×D12D414yes—
89C22×D12C2312yes—
90C22×D12C2311yescharacteristic
91C22×C4×D3—21no—
92C22×Dic6C2311yescharacteristic
93C2×C3×M16—22no—
94C2×C3×M16—22no—
95C2×C3×M16D414yescomplemented
96C22×C3×C8C2312yes—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2312yes—
101C2×#217D412yes—
102#950—232no—
103#947—24no—
104#947—24no—
105C2×#216—22no—
106C2×#216—22no—
107C2×#216—22no—
108C2×#215—22no—
109C2×#215—22no—
110C2×#215—22no—
111#940—24no—
112#940—24no—
113#937—216no—
114C2×#214—22no—
115C2×#214—22no—
116C2×#214—22no—
117C2×#213—22no—
118C2×#213—22no—
119C2×#213—22no—
120#930—216no—
121C2×(C2×D3)⋊C4—24no—
122C2×(C2×D3)⋊C4—24no—
123#926—22no—
124#926—22no—
125C2×C4⋊Dic3D412yes—
126C2×C4×Dic3C2312yes—
127C22×C3⋊C8—22no—
128C22×C3⋊C8—22no—
129C22×C3⋊C8D414yescomplemented
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132#908—232no—
133C2×D24—24no—
134C2×D24—24no—
135C2×D24—24no—
136C2×#204—24no—
137C2×#204—24no—
138C2×#204—24no—
139C3×#103—24no—
140C3×#102—24no—
141C3⋊#95—24no—
142#809—24no—
143#809—24no—
144C22×#41—31no—
145C22×#41C2×D311yescharacteristic
146C23×D4—31no—
147C2×AGL(1,ℤ/8)—64no—
148C2×AGL(1,ℤ/8)—64no—
149C2×AGL(1,ℤ/8)—64no—
150C2×#135—64no—
151C2×#135—64no—
152C2×#135—64no—
153C22×Q16—31no—
154C22×SD16—31no—
155C22×SD16—31no—
156C22×SD16—32no—
157C22×D8—31no—
158C22×D8—32no—
159C22×M16C2×D311yescharacteristic
160C23×C8—31no—
161#535—64no—
162#535—64no—
163C2×#127—32no—
164C2×#124—32no—
165C2×#121—62no—
166C2×#120—62no—
167C2×#117—31no—
168C22×C4⋊C4C2×D311yescharacteristic
169#417—24no—
170C2×#103C2×D312yescomplemented
171C2×#102—32no—
172C2×#102—32no—
173C2×#102—64no—
174C2×#102C2×D312yescomplemented
175#391—32no—
176#390—32no—
177#386—62no—
178#380—62no—
179#336—32no—
180#336—32no—
181C24×C3—21no—
182C24×C3C2×D411yescharacteristic
183C23×D3—21no—
184C23×D3—22no—
185C23×D3—24no—
186C23×D3—42no—
187C23×D3—42no—
188C23×D3—82no—
189C23×D3—82no—
190C3×#41—28no—
191C3×#41—28no—
192C3×#41—44no—
193C3×#41—44no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8C2×D412yes—
200C2×C3×D4—24no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—42no—
207C2×C3×D4—42no—
208C2×C3×D4C2×D412yes—
209C22×C3×C4—21no—
210C22×C3×C4—24no—
211C22×C3×C4—22no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4—82no—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D412yes—
222C22×C3×C4#4112yes—
223C22×C3×C4C2411yescharacteristic
224C2×#67—24no—
225C2×#67—216no—
226C2×#67—42no—
227C2×#67—42no—
228C2×#67—84no—
229C2×#67—84no—
230C22×Dic3—21no—
231C22×Dic3C2×D414yes—
232#235—28no—
233#235—28no—
234#235—44no—
235#235—44no—
236C2×D12—28no—
237C2×D12—28no—
238C2×D12—22no—
239C2×D12—22no—
240C2×D12—22no—
241C2×D12—21no—
242C2×D12—21no—
243C2×D12—21no—
244C2×D12—21no—
245C2×D12—22no—
246C2×D12—24no—
247C2×D12—22no—
248C2×D12—21no—
249C2×D12—44no—
250C2×D12—42no—
251C2×D12—44no—
252C2×D12—42no—
253C2×D12C2×D414yes—
254C2×D12C2×D412yes—
255C2×C4×D3—24no—
256C2×C4×D3—42no—
257C2×C4×D3—42no—
258C2×Dic6—21no—
259C2×Dic6—21no—
260C2×Dic6—21no—
261C2×Dic6—21no—
262C2×Dic6—21no—
263C2×Dic6C2×D412yes—
264C3×M16—28no—
265C2×C3×C8—22no—
266C2×C3×C8—22no—
267C2×C3×C8C2×D414yes—
268C3×C4⋊C4—22no—
269C3×C4⋊C4—22no—
270C3×C4⋊C4—42no—
271#217—24no—
272#216—28no—
273#215—28no—
274#214—28no—
275#213—28no—
276(C2×D3)⋊C4—216no—
277(C2×D3)⋊C4—84no—
278C4⋊Dic3—24no—
279C4×Dic3—22no—
280C4×Dic3—22no—
281C2×C3⋊C8—28no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8C2×D412yes—
287C2×C3⋊C8C2×D412yes—
288D24—216no—
289#204—216no—
290C25—61no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—34no—
297C2×#41—62no—
298C2×#41—62no—
299C2×#41—62no—
300C2×#41—62no—
301C2×#41—62no—
302C2×#41D1214yes—
303C22×Q8—31no—
304C22×Q8C22×D311yescharacteristic
305C22×D4—21no—
306C22×D4—34no—
307C22×D4—32no—
308C22×D4—31no—
309C22×D4—64no—
310C22×D4—61no—
311C22×D4—62no—
312C22×D4—62no—
313C22×D4—62no—
314C22×D4—62no—
315C22×D4—61no—
316C22×D4—62no—
317C22×D4C22×D311yescharacteristic
318C23×C4—21no—
319C23×C4—21no—
320C23×C4—61no—
321C23×C4C22×D311yescharacteristic
322AGL(1,ℤ/8)—632no—
323#135—632no—
324C2×Q16—62no—
325C2×Q16—62no—
326C2×Q16—62no—
327C2×SD16—64no—
328C2×SD16—64no—
329C2×SD16—62no—
330C2×SD16—62no—
331C2×SD16—62no—
332C2×SD16—62no—
333C2×SD16—62no—
334C2×SD16—62no—
335C2×SD16—64no—
336C2×D8—64no—
337C2×D8—64no—
338C2×D8—62no—
339C2×D8—62no—
340C2×D8—62no—
341C2×D8—64no—
342C2×M16—22no—
343C2×M16—22no—
344C2×M16#6714yescomplemented
345C22×C8—34no—
346C22×C8—31no—
347C22×C8—31no—
348C22×C8—62no—
349C22×C8—62no—
350C22×C8C22×D312yes—
351#127—64no—
352#127—64no—
353#124—64no—
354#124—64no—
355#121—616no—
356#120—616no—
357#117—62no—
358#117—62no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—32no—
363C2×C4⋊C4C22×D312yes—
364C2×C22⋊C4—32no—
365C2×C22⋊C4—64no—
366C2×C22⋊C4—64no—
367C2×C42—32no—
368#103—24no—
369#102—24no—
370#102—64no—
371#102—64no—
372#95—64no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—42no—
379C23×C3—42no—
380C23×C3#12012yes—
381C23×C3#12112yes—
382C23×C3#12712yes—
383C23×C3C22×D411yescharacteristic
384C22×D3—24no—
385C22×D3—22no—
386C22×D3—41no—
387C22×D3—42no—
388C22×D3—41no—
389C22×D3—42no—
390C22×D3—42no—
391C22×D3—44no—
392C22×D3—44no—
393C22×D3—42no—
394C22×D3—48no—
395C22×D3—88no—
396C22×D3—84no—
397C22×D3—84no—
398C22×D3—88no—
399C3×Q8—22no—
400C3×Q8—22no—
401C3×Q8—41no—
402C3×Q8—41no—
403C3×D4—22no—
404C3×D4—22no—
405C3×D4—48no—
406C3×D4—41no—
407C3×D4—41no—
408C2×C3×C4—22no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—48no—
412C2×C3×C4—42no—
413C2×C3×C4—44no—
414C2×C3×C4—84no—
415C2×C3×C4#12014yes—
416C2×C3×C4#12012yes—
417C2×C3×C4#12014yes—
418C2×C3×C4#12114yes—
419C2×C3×C4#12112yes—
420C2×C3×C4#12114yes—
421C2×C3×C4#12712yes—
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C22×D411yescharacteristic
426C2×C3×C4C22×D411yescharacteristic
427C2×C3×C4C2×#4111yescharacteristic
428#67—48no—
429#67—816no—
430C2×Dic3—24no—
431C2×Dic3—22no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436D12—24no—
437D12—22no—
438D12—22no—
439D12—24no—
440D12—42no—
441D12—42no—
442D12—41no—
443D12—41no—
444D12—48no—
445C4×D3—48no—
446Dic6—22no—
447Dic6—22no—
448Dic6—41no—
449Dic6—41no—
450C3×C8—24no—
451C3⋊C8—22no—
452C3⋊C8—22no—
453C24—21no—
454C24—61no—
455C24—62no—
456C24—64no—
457C24—122no—
458C24—122no—
459C24—242no—
460C24—242no—
461C24D3×D411yescharacteristic
462#41—28no—
463#41—28no—
464#41—44no—
465#41—44no—
466#41—68no—
467#41—68no—
468#41—124no—
469#41—124no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—32no—
476C2×Q8—61no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8C2×D1212yes—
482C2×D4—24no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—34no—
489C2×D4—32no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—64no—
493C2×D4—616no—
494C2×D4—64no—
495C2×D4—68no—
496C2×D4—68no—
497C2×D4—61no—
498C2×D4—62no—
499C2×D4—62no—
500C2×D4—61no—
501C2×D4—62no—
502C2×D4—61no—
503C2×D4—62no—
504C2×D4—61no—
505C2×D4—61no—
506C2×D4—62no—
507C2×D4—122no—
508C2×D4—122no—
509C2×D4—124no—
510C2×D4—122no—
511C2×D4—124no—
512C2×D4—122no—
513C2×D4—244no—
514C2×D4—244no—
515C2×D4C2×D1212yes—
516C22×C4—21no—
517C22×C4—24no—
518C22×C4—22no—
519C22×C4—34no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—42no—
523C22×C4—61no—
524C22×C4—64no—
525C22×C4—82no—
526C22×C4—122no—
527C22×C4—122no—
528C22×C4C2×D1212yes—
529C22×C4#23512yes—
530C22×C4D3×D411yescharacteristic
531C22×C4D3×D412yes—
532C22×C4D3×D412yes—
533C22×C4D3×D412yes—
534C22×C4C2×#6712yes—
535C22×C4C23×D311yescharacteristic
536Q16—68no—
537SD16—616no—
538SD16—68no—
539SD16—68no—
540D8—616no—
541D8—68no—
542M16—28no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—32no—
546C2×C8—32no—
547C2×C8—68no—
548C2×C8—61no—
549C2×C8—61no—
550C2×C8—61no—
551C2×C8—61no—
552C2×C8C2×#6714yes—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—42no—
556C4⋊C4—64no—
557C22⋊C4—616no—
558C22⋊C4—64no—
559C22⋊C4—244no—
560C42—62no—
561C42—62no—
562C22×C3—22no—
563C22×C3—22no—
564C22×C3—24no—
565C22×C3—24no—
566C22×C3—42no—
567C22×C3—44no—
568C22×C3—81no—
569C22×C3—81no—
570C22×C3—82no—
571C22×C3#39214yes—
572C22×C3C2×#12011yescharacteristic
573C22×C3C2×#12111yescharacteristic
574C22×C3C2×#12711yescharacteristic
575C22×C3#57612yes—
576C22×C3#57712yes—
577C2×D3—42no—
578C2×D3—44no—
579C2×D3—48no—
580C2×D3—44no—
581C2×D3—816no—
582C2×D3—81no—
583C2×D3—82no—
584C2×D3—81no—
585C2×D3—82no—
586C2×D3—82no—
587C2×D3—84no—
588C3×C4—42no—
589C3×C4—82no—
590C3×C4#39214yes—
591C3×C4C2×#12011yescharacteristic
592C3×C4C2×#12011yescharacteristic
593C3×C4C2×#12111yescharacteristic
594C3×C4C2×#12111yescharacteristic
595Dic3—42no—
596Dic3—44no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—41no—
601C23—42no—
602C23—42no—
603C23—41no—
604C23—64no—
605C23—62no—
606C23—128no—
607C23—121no—
608C23—122no—
609C23—122no—
610C23—124no—
611C23—124no—
612C23—122no—
613C23—121no—
614C23—122no—
615C23—248no—
616C23—244no—
617C23—248no—
618C23—244no—
619C23#88212yes—
620C23#88412yes—
621C23#94012yes—
622C23C2×D3×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—41no—
626Q8—41no—
627Q8—62no—
628Q8—62no—
629Q8—121no—
630Q8—121no—
631D4—22no—
632D4—22no—
633D4—48no—
634D4—41no—
635D4—41no—
636D4—62no—
637D4—64no—
638D4—62no—
639D4—64no—
640D4—128no—
641D4—128no—
642D4—122no—
643D4—121no—
644D4—122no—
645D4—121no—
646D4—2416no—
647C2×C4—22no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—48no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—64no—
654C2×C4—62no—
655C2×C4—64no—
656C2×C4—64no—
657C2×C4—84no—
658C2×C4—121no—
659C2×C4—121no—
660C2×C4—128no—
661C2×C4#88212yes—
662C2×C4#88214yes—
663C2×C4#88412yes—
664C2×C4#88414yes—
665C2×C4#93014yes—
666C2×C4#93714yes—
667C2×C4#94012yes—
668C2×C4C22×D1211yescharacteristic
669C2×C4C2×#23511yescharacteristic
670C2×C4C2×D3×D411yescharacteristic
671C2×C4C2×D3×D411yescharacteristic
672C2×C4C2×D3×D411yescharacteristic
673C2×C4C22×#6711yescharacteristic
674C8—24no—
675C8—62no—
676C8—62no—
677C2×C3—24no—
678C2×C3—82no—
679C2×C3C2×#39211yescharacteristic
680C2×C3#301212yes—
681C2×C3#304512yes—
682C2×C3#315411yescharacteristic
683C2×C3#315511yescharacteristic
684D3—82no—
685D3—84no—
686C22—22no—
687C22—24no—
688C22—24no—
689C22—22no—
690C22—44no—
691C22—42no—
692C22—81no—
693C22—82no—
694C22—81no—
695C22—128no—
696C22—124no—
697C22—122no—
698C22—124no—
699C22—2416no—
700C22—242no—
701C22—241no—
702C22—242no—
703C22—244no—
704C22—242no—
705C22—241no—
706C22#496014yes—
707C22C2×#88211yescharacteristic
708C22C2×#88411yescharacteristic
709C22#577412yes—
710C22#577812yes—
711C22C2×#94011yescharacteristic
712C4—42no—
713C4—82no—
714C4—124no—
715C4—122no—
716C4#496014yes—
717C4C2×#88211yescharacteristic
718C4C2×#88411yescharacteristic
719C4C2×#93011yescharacteristic
720C4C2×#93711yescharacteristic
721C3#2342611yescharacteristic, complemented, Sylow
722C2—24no—
723C2—82no—
724C2—244no—
725C2—242no—
726C2#7888612yes—
727C2#7891112yes—
728C2C2×#496011yescharacteristic
729C2#8394011yescharacteristic
730C2#8397411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#417C2
socleC23×C3C22×D4
hypercenterC2×#417D3
hypocenterC3#23426
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. C2×#5830C2semidirect
2. C22×#950C2semidirect
3. C22×#908C2semidirect
4. C2×C3×#417C2semidirect
5. #82186C2semidirect
6. C22×#809C2semidirect
7. #71980C2semidirect
8. #70278C2semidirect
9. #70277C2semidirect
10. #70210C2semidirect
11. #70209C2semidirect
24. C22×#216C22semidirect
25. C22×#215C22semidirect
28. C22×#214C22semidirect
29. C22×#213C22semidirect
39. C2×C3×#103C22semidirect
40. C2×C3×#102C22semidirect
42. #4988C22semidirect
44. #4973C22semidirect
48. #4550C22semidirect
49. #4550C22semidirect
53. C2×#417D3semidirect
81. C2×#235D4semidirect
95. C2×C3×M16D4semidirect
129. C22×C3⋊C8D4semidirect
170. C2×#103C2×D3semidirect
174. C2×#102C2×D3semidirect
344. C2×M16#67semidirect
721. C3#23426semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
C22×#9502
C22×#9082
C2×C3×#4172
#821862
C22×#8092
#719802
#702782
#702772
#702102
#702092
#234263