Finite Group Databasebeta

← #10822304

Subgroups — #10822304

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
1C22×#949C211yesmaximal, characteristic, complemented
2C22×#950C211yesmaximal, characteristic, complemented
3C2×#5525C211yesmaximal, characteristic, complemented
4C3×#2311C211yesmaximal, characteristic, complemented
5C2×#4966C211yesmaximal, characteristic, complemented
6C22×#809C211yesmaximal, characteristic, complemented
7#71579C211yesmaximal, characteristic, complemented
8#69689C212yesmaximal, complemented
9#69688C212yesmaximal, complemented
10#69659C212yesmaximal, complemented
11#69658C212yesmaximal, complemented
12#23426—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#950—24no—
19C2×#950—24no—
20C2×#950—24no—
21C2×#949—24no—
22C2×#949—24no—
23C2×#949—24no—
24C22×#216C2211yescharacteristic, complemented
25C22×#215C2212yes—
26C22×#215C2211yescharacteristic, complemented
27C22×#214C2211yescharacteristic, complemented
28C22×#213C2212yes—
29C22×#213C2211yescharacteristic, complemented
30C22×#208C2211yescharacteristic
31C23×C3⋊C8C2211yescharacteristic
32#5525—24no—
33#5525—24no—
34C2×#895—22no—
35C2×#882—22no—
36C2×#875C2212yes—
37C2×#874C2212yes—
38C3×#391C2212yescomplemented
39C3×#390C2212yescomplemented
40C3×#386—22no—
41C3×#380—22no—
42C2×#810C2212yescomplemented
43#4966—24no—
44C2×#809—24no—
45C2×#809C2212yes—
46C2×#809C2212yes—
47C2×#809C2212yescomplemented
48#4524C2212yescomplemented
49#4524C2212yescomplemented
50C22×AGL(1,ℤ/8)—31no—
51C22×#135—31no—
52C2×#535—31no—
53C2×#417—31no—
54C22×#102—31no—
55#2311D311yescharacteristic, complemented
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×#41D414yescomplemented
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—21no—
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C22×#67—21no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235—22no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235D414yes—
81C22×D12—22no—
82C22×D12—22no—
83C22×D12—22no—
84C22×D12—22no—
85C22×D12—21no—
86C22×D12—22no—
87C22×D12—24no—
88C22×D12D414yes—
89C22×D12C2312yes—
90C22×D12C2311yescharacteristic
91C22×C4×D3—21no—
92C22×Dic6C2311yescharacteristic
93C3×#117—22no—
94C3×#117—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4D412yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4—24no—
101C2×C3×C22⋊C4D412yes—
102C2×C3×C42C2312yes—
103#950—232no—
104#949—232no—
105C2×#216—22no—
106C2×#216—22no—
107C2×#216—22no—
108C2×#215—24no—
109C2×#215—22no—
110C2×#215—24no—
111C2×#215—22no—
112C2×#215—24no—
113C2×#215—22no—
114C2×#214—22no—
115C2×#214—22no—
116C2×#214—22no—
117C2×#213—24no—
118C2×#213—22no—
119C2×#213—24no—
120C2×#213—22no—
121C2×#213—24no—
122C2×#213—22no—
123C2×(C2×D3)⋊C4—24no—
124C2×#208—22no—
125C2×#208—22no—
126C2×#208D414yescomplemented
127C22×C3⋊C8—22no—
128C22×C3⋊C8—22no—
129C22×C3⋊C8D414yescomplemented
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2312yes—
133#895—216no—
134#882—216no—
135#875—24no—
136#875—24no—
137#874—24no—
138#874—24no—
139C3×#95—24no—
140#810—24no—
141#809—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—32no—
155C22×SD16—31no—
156C22×SD16—31no—
157C22×D8—32no—
158C22×D8—31no—
159C22×M16—31no—
160C23×C8—31no—
161#535—64no—
162#535—64no—
163C2×#127—32no—
164C2×#124—32no—
165C2×#121—62no—
166C2×#120—62no—
167C2×#117C2×D311yescharacteristic
168C22×C4⋊C4C2×D311yescharacteristic
169#417—64no—
170C2×#103—32no—
171C2×#102—32no—
172C2×#102—32no—
173C2×#102—32no—
174C2×#102—64no—
175#391C2×D312yescomplemented
176#390C2×D312yescomplemented
177#386—22no—
178#380—22no—
179#336—32no—
180#336—32no—
181C24×C3—21no—
182C24×C3C2×D411yescharacteristic
183C23×D3—22no—
184C23×D3—21no—
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×D412yes—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D414yes—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4#4112yes—
224C22×C3×C4C2411yescharacteristic
225C2×#67—24no—
226C2×#67—42no—
227C2×#67—42no—
228C22×Dic3—21no—
229#235—28no—
230#235—28no—
231#235—44no—
232#235—44no—
233C2×D12—28no—
234C2×D12—28no—
235C2×D12—22no—
236C2×D12—21no—
237C2×D12—22no—
238C2×D12—22no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12—22no—
242C2×D12—21no—
243C2×D12—24no—
244C2×D12—21no—
245C2×D12—22no—
246C2×D12—216no—
247C2×D12—44no—
248C2×D12—44no—
249C2×D12—42no—
250C2×D12—42no—
251C2×D12—84no—
252C2×D12—84no—
253C2×D12C2×D412yes—
254C2×D12C2×D414yes—
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×C4⋊C4—24no—
265C3×C4⋊C4—22no—
266C3×C4⋊C4—22no—
267C3×C4⋊C4—42no—
268C3×C22⋊C4—24no—
269C3×C22⋊C4—84no—
270C3×C42—22no—
271C3×C42—22no—
272#216—28no—
273#215—216no—
274#215—28no—
275#214—28no—
276#213—216no—
277#213—28no—
278(C2×D3)⋊C4—216no—
279#208—28no—
280C2×C3⋊C8—28no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—22no—
286C2×C3⋊C8—22no—
287C2×C3⋊C8C2×D412yes—
288C2×C3⋊C8C2×D412yes—
289C2×C3⋊C8C2×D414yes—
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×#41D1214yescomplemented
303C22×Q8—31no—
304C22×Q8C22×D311yescharacteristic
305C22×D4—21no—
306C22×D4—34no—
307C22×D4—32no—
308C22×D4—31no—
309C22×D4—61no—
310C22×D4—62no—
311C22×D4—62no—
312C22×D4—62no—
313C22×D4—62no—
314C22×D4—62no—
315C22×D4—61no—
316C22×D4—64no—
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—62no—
328C2×SD16—62no—
329C2×SD16—64no—
330C2×SD16—62no—
331C2×SD16—62no—
332C2×SD16—64no—
333C2×SD16—62no—
334C2×SD16—64no—
335C2×SD16—62no—
336C2×D8—62no—
337C2×D8—64no—
338C2×D8—62no—
339C2×D8—64no—
340C2×D8—62no—
341C2×D8—64no—
342C2×M16—34no—
343C2×M16—62no—
344C2×M16—62no—
345C22×C8—34no—
346C22×C8—31no—
347C22×C8—31no—
348C22×C8—32no—
349C22×C8—62no—
350C22×C8—62no—
351#127—64no—
352#127—64no—
353#124—64no—
354#124—64no—
355#121—616no—
356#120—616no—
357#117—22no—
358#117—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4#6712yes—
363C2×C4⋊C4C22×D312yes—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—64no—
366C2×C22⋊C4#6712yes—
367C2×C42C22×D312yes—
368#103—64no—
369#102—64no—
370#102—64no—
371#102—64no—
372#95—24no—
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—42no—
393C22×D3—44no—
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—24no—
409C2×C3×C4—22no—
410C2×C3×C4—24no—
411C2×C3×C4—24no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—48no—
415C2×C3×C4—42no—
416C2×C3×C4—44no—
417C2×C3×C4—84no—
418C2×C3×C4#12014yes—
419C2×C3×C4#12014yes—
420C2×C3×C4#12012yes—
421C2×C3×C4#12114yes—
422C2×C3×C4#12114yes—
423C2×C3×C4#12112yes—
424C2×C3×C4#12712yes—
425C2×C3×C4C22×D411yescharacteristic
426C2×C3×C4C22×D411yescharacteristic
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4C2×#4111yescharacteristic
431#67—48no—
432C2×Dic3—22no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435D12—24no—
436D12—22no—
437D12—24no—
438D12—22no—
439D12—42no—
440D12—41no—
441D12—42no—
442D12—41no—
443D12—48no—
444D12—816no—
445C4×D3—48no—
446Dic6—22no—
447Dic6—22no—
448Dic6—41no—
449Dic6—41no—
450C3⋊C8—22no—
451C3⋊C8—22no—
452C3⋊C8—24no—
453C24—21no—
454C24—62no—
455C24—61no—
456C24—64no—
457C24—122no—
458C24—122no—
459C24—242no—
460C24—242no—
461C24C2×#6711yescharacteristic
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—616no—
493C2×D4—64no—
494C2×D4—68no—
495C2×D4—68no—
496C2×D4—64no—
497C2×D4—62no—
498C2×D4—61no—
499C2×D4—62no—
500C2×D4—61no—
501C2×D4—62no—
502C2×D4—61no—
503C2×D4—62no—
504C2×D4—61no—
505C2×D4—62no—
506C2×D4—61no—
507C2×D4—122no—
508C2×D4—122no—
509C2×D4—124no—
510C2×D4—124no—
511C2×D4—122no—
512C2×D4—122no—
513C2×D4—244no—
514C2×D4—244no—
515C2×D4C2×D1212yes—
516C22×C4—21no—
517C22×C4—24no—
518C22×C4—22no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—61no—
523C22×C4—64no—
524C22×C4—82no—
525C22×C4—122no—
526C22×C4—122no—
527C22×C4C2×D1212yes—
528C22×C4#23512yes—
529C22×C4D3×D412yes—
530C22×C4D3×D412yes—
531C22×C4D3×D412yes—
532C22×C4D3×D412yes—
533C22×C4C2×#6714yes—
534C22×C4C2×#6711yescharacteristic
535C22×C4C23×D311yescharacteristic
536Q16—68no—
537SD16—68no—
538SD16—68no—
539SD16—616no—
540D8—68no—
541D8—616no—
542M16—68no—
543C2×C8—32no—
544C2×C8—32no—
545C2×C8—34no—
546C2×C8—68no—
547C2×C8—61no—
548C2×C8—61no—
549C2×C8—62no—
550C2×C8—61no—
551C2×C8—61no—
552C2×C8—62no—
553C4⋊C4—24no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—42no—
557C22⋊C4—24no—
558C22⋊C4—616no—
559C22⋊C4—84no—
560C42—22no—
561C42—22no—
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—44no—
580C2×D3—48no—
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—44no—
590C3×C4—82no—
591C3×C4#39214yes—
592C3×C4C2×#12011yescharacteristic
593C3×C4C2×#12011yescharacteristic
594C3×C4C2×#12111yescharacteristic
595C3×C4C2×#12111yescharacteristic
596Dic3—42no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—41no—
601C23—41no—
602C23—42no—
603C23—42no—
604C23—64no—
605C23—62no—
606C23—128no—
607C23—121no—
608C23—122no—
609C23—121no—
610C23—122no—
611C23—122no—
612C23—124no—
613C23—122no—
614C23—124no—
615C23—248no—
616C23—248no—
617C23—244no—
618C23—244no—
619C23#93012yes—
620C23#93712yes—
621C23#94012yes—
622C23C22×#6711yescharacteristic
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—64no—
637D4—62no—
638D4—64no—
639D4—62no—
640D4—128no—
641D4—128no—
642D4—122no—
643D4—121no—
644D4—122no—
645D4—121no—
646D4—2416no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—48no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—62no—
657C2×C4—84no—
658C2×C4—121no—
659C2×C4—121no—
660C2×C4—128no—
661C2×C4#88214yes—
662C2×C4#88214yes—
663C2×C4#88414yes—
664C2×C4#88414yes—
665C2×C4#93012yes—
666C2×C4#93712yes—
667C2×C4#94012yes—
668C2×C4C22×D1211yescharacteristic
669C2×C4C2×#23511yescharacteristic
670C2×C4C2×D3×D411yescharacteristic
671C2×C4C2×D3×D411yescharacteristic
672C2×C4C2×D3×D411yescharacteristic
673C2×C4C2×D3×D411yescharacteristic
674C8—62no—
675C8—62no—
676C8—64no—
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—24no—
687C22—22no—
688C22—22no—
689C22—24no—
690C22—44no—
691C22—42no—
692C22—81no—
693C22—81no—
694C22—82no—
695C22—122no—
696C22—124no—
697C22—124no—
698C22—128no—
699C22—2416no—
700C22—244no—
701C22—242no—
702C22—242no—
703C22—241no—
704C22—242no—
705C22—241no—
706C22#496014yes—
707C22C2×#93011yescharacteristic
708C22C2×#93711yescharacteristic
709C22C2×#94011yescharacteristic
710C22#583512yes—
711C22#583712yes—
712C4—42no—
713C4—44no—
714C4—82no—
715C4—122no—
716C4#496014yes—
717C4C2×#88211yescharacteristic
718C4C2×#88211yescharacteristic
719C4C2×#88411yescharacteristic
720C4C2×#88411yescharacteristic
721C3#2342611yescharacteristic, complemented, Sylow
722C2—24no—
723C2—82no—
724C2—244no—
725C2—242no—
726C2C2×#496011yescharacteristic
727C2#8256512yes—
728C2#8338211yescharacteristic
729C2#8340211yescharacteristic
730C2#8342412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2311C2
socleC23×C3C22×D4
hypercenter#2311D3
hypocenterC3#23426
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
1. C22×#949C2semidirect
2. C22×#950C2semidirect
3. C2×#5525C2semidirect
4. C3×#2311C2semidirect
5. C2×#4966C2semidirect
6. C22×#809C2semidirect
7. #71579C2semidirect
8. #69689C2semidirect
9. #69688C2semidirect
10. #69659C2semidirect
11. #69658C2semidirect
24. C22×#216C22semidirect
26. C22×#215C22semidirect
27. C22×#214C22semidirect
29. C22×#213C22semidirect
38. C3×#391C22semidirect
39. C3×#390C22semidirect
42. C2×#810C22semidirect
47. C2×#809C22semidirect
48. #4524C22semidirect
49. #4524C22semidirect
55. #2311D3semidirect
67. C2×C3×#41D4semidirect
126. C2×#208D4semidirect
129. C22×C3⋊C8D4semidirect
175. #391C2×D3semidirect
176. #390C2×D3semidirect
302. C2×#41D12semidirect
721. C3#23426semidirect

Maximal subgroups

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

subgroupindex
C22×#9492
C22×#9502
C2×#55252
C3×#23112
C2×#49662
C22×#8092
#715792
#696892
#696882
#696592
#696582
#234263