Finite Group Databasebeta

← #10822328

Subgroups — #10822328

Order 768.

2282 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic, complemented
2C22×#949C211yesmaximal, characteristic, complemented
3C22×#908C211yesmaximal, characteristic, complemented
4C2×C3×#417C211yesmaximal, characteristic, complemented
5C2×#4988C211yesmaximal, characteristic, complemented
6C2×#4966C211yesmaximal, characteristic, complemented
7#71979C211yesmaximal, characteristic, complemented
8#70282C212yesmaximal, complemented
9#70277C212yesmaximal, complemented
10#70214C212yesmaximal, complemented
11#70209C212yesmaximal, complemented
12#23428—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18#5832—24no—
19#5832—24no—
20C2×#949—24no—
21C2×#949—24no—
22C2×#949—24no—
23C2×#947C2211yescharacteristic, complemented
24C2×#944C2211yescharacteristic, complemented
25C22×#215C2212yes—
26C2×#940C2211yescharacteristic, complemented
27C2×#939—22no—
28C2×#936C2211yescharacteristic, complemented
29C22×#213C2212yes—
30C2×C4×#67—22no—
31C22×(C2×D3)⋊C4—22no—
32C22×C4×Dic3C2211yescharacteristic
33C22×#208C2211yescharacteristic
34C2×#908—24no—
35C2×#908—24no—
36C2×#908—24no—
37C22×D24C2212yes—
38C22×#204C2212yes—
39C3×#417—24no—
40C2×C3×#103C2212yescomplemented
41C2×C3×#102C2212yescomplemented
42#4988—22no—
43#4988C2212yes—
44#4988C2212yes—
45C2×#810C2212yescomplemented
46#4966—24no—
47C2×#809C2212yescomplemented
48#4550C2212yescomplemented
49#4550C2212yescomplemented
50C22×AGL(1,ℤ/8)—31no—
51C22×AGL(1,ℤ/8)—31no—
52C2×#532—31no—
53C2×#417—31no—
54C2×#417D311yescharacteristic, complemented
55C2×#390—31no—
56#2013—32no—
57#2012—32no—
58#2012—32no—
59#2011—32no—
60#1728—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—22no—
76C23×Dic3D412yes—
77C2×#235—22no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235D414yes—
83C22×D12—22no—
84C22×D12—22no—
85C22×D12—22no—
86C22×D12—22no—
87C22×D12—21no—
88C22×D12—22no—
89C22×D12D414yescomplemented
90C22×D12C2312yes—
91C22×D12C2311yescharacteristic
92C22×C4×D3—21no—
93C22×Dic6C2311yescharacteristic
94C2×C3×M16—22no—
95C2×C3×M16—22no—
96C2×C3×M16D414yescomplemented
97C22×C3×C8C2312yes—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2312yes—
102C2×#217—22no—
103#949—232no—
104#947—22no—
105#947—22no—
106#944—22no—
107#944—22no—
108C2×#215—24no—
109C2×#215—24no—
110C2×#215—24no—
111#940—22no—
112#940—22no—
113#939—216no—
114#936—22no—
115#936—22no—
116C2×#213—24no—
117C2×#213—24no—
118C2×#213—24no—
119C4×#67—216no—
120C2×(C2×D3)⋊C4—22no—
121C2×(C2×D3)⋊C4—24no—
122C2×(C2×D3)⋊C4—24no—
123C2×(C2×D3)⋊C4—48no—
124C2×Dic3⋊C4—22no—
125C2×C4×Dic3—22no—
126C2×C4×Dic3—22no—
127C2×C4×Dic3D412yes—
128C2×C4×Dic3C2311yescharacteristic
129C2×C4×Dic3C2311yescharacteristic
130C2×#208—22no—
131C2×#208—22no—
132C2×#208D414yescomplemented
133C22×C3⋊C8C2312yes—
134#908—232no—
135C2×D24—24no—
136C2×D24—24no—
137C2×D24—24no—
138C2×#204—24no—
139C2×#204—24no—
140C2×#204—24no—
141C3×#103—24no—
142C3×#102—24no—
143#810—24no—
144#809—24no—
145C22×#41—31no—
146C22×#41C2×D311yescharacteristic
147C23×D4—31no—
148C2×AGL(1,ℤ/8)—64no—
149C2×AGL(1,ℤ/8)—64no—
150C2×AGL(1,ℤ/8)—64no—
151C2×AGL(1,ℤ/8)—64no—
152C2×AGL(1,ℤ/8)—64no—
153C2×AGL(1,ℤ/8)—64no—
154C22×SD16—32no—
155C22×SD16—32no—
156C22×D8—32no—
157C22×D8—32no—
158C22×M16—31no—
159C22×M16C2×D311yescharacteristic
160#532—64no—
161#532—64no—
162C2×C4⋊Q8—31no—
163C2×#127—31no—
164C2×#124—31no—
165C2×#124—31no—
166C2×#121—62no—
167C2×C4×D4—62no—
168C22×C4⋊C4C2×D311yescharacteristic
169C22×C22⋊C4—62no—
170C22×C42—31no—
171#417—24no—
172#417—64no—
173C2×#103—32no—
174C2×#103C2×D312yescomplemented
175C2×#102—32no—
176C2×#102C2×D312yescomplemented
177#390—32no—
178#390—32no—
179#390—62no—
180#336—32no—
181#336—32no—
182C24×C3—21no—
183C24×C3C2×D411yescharacteristic
184C23×D3—24no—
185C23×D3—22no—
186C23×D3—21no—
187C23×D3—44no—
188C23×D3—44no—
189C23×D3—42no—
190C23×D3—42no—
191C3×#41—28no—
192C3×#41—28no—
193C3×#41—44no—
194C3×#41—44no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8C2×D412yes—
201C2×C3×D4—24no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—42no—
208C2×C3×D4—42no—
209C2×C3×D4C2×D412yes—
210C22×C3×C4—21no—
211C22×C3×C4—24no—
212C22×C3×C4—22no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4—44no—
216C22×C3×C4—42no—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D412yes—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D412yes—
222C22×C3×C4C2×D412yes—
223C22×C3×C4#4112yes—
224C22×C3×C4C2411yescharacteristic
225C2×#67—28no—
226C2×#67—24no—
227C2×#67—44no—
228C2×#67—42no—
229C2×#67—42no—
230C2×#67—44no—
231C22×Dic3—24no—
232C22×Dic3—21no—
233C22×Dic3—24no—
234C22×Dic3—42no—
235C22×Dic3C2×D412yes—
236C22×Dic3C2×D412yes—
237#235—28no—
238#235—28no—
239#235—44no—
240#235—44no—
241C2×D12—28no—
242C2×D12—28no—
243C2×D12—21no—
244C2×D12—22no—
245C2×D12—22no—
246C2×D12—21no—
247C2×D12—21no—
248C2×D12—22no—
249C2×D12—22no—
250C2×D12—21no—
251C2×D12—24no—
252C2×D12—21no—
253C2×D12—22no—
254C2×D12—42no—
255C2×D12—44no—
256C2×D12—44no—
257C2×D12—42no—
258C2×D12C2×D414yes—
259C2×D12C2×D412yes—
260C2×C4×D3—24no—
261C2×C4×D3—42no—
262C2×C4×D3—42no—
263C2×Dic6—21no—
264C2×Dic6—21no—
265C2×Dic6—21no—
266C2×Dic6—21no—
267C2×Dic6—21no—
268C2×Dic6C2×D412yes—
269C3×M16—28no—
270C2×C3×C8—22no—
271C2×C3×C8—22no—
272C2×C3×C8C2×D414yes—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4—42no—
276#217—28no—
277#215—216no—
278#213—216no—
279(C2×D3)⋊C4—28no—
280(C2×D3)⋊C4—48no—
281(C2×D3)⋊C4—48no—
282Dic3⋊C4—28no—
283C4×Dic3—24no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288#208—28no—
289C2×C3⋊C8—22no—
290C2×C3⋊C8—22no—
291C2×C3⋊C8C2×D414yes—
292D24—216no—
293#204—216no—
294C25—61no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41—22no—
300C2×#41—34no—
301C2×#41—62no—
302C2×#41—62no—
303C2×#41—62no—
304C2×#41—62no—
305C2×#41—62no—
306C2×#41D1214yes—
307C22×Q8—31no—
308C22×Q8C22×D311yescharacteristic
309C22×D4—21no—
310C22×D4—34no—
311C22×D4—31no—
312C22×D4—32no—
313C22×D4—62no—
314C22×D4—61no—
315C22×D4—62no—
316C22×D4—62no—
317C22×D4—62no—
318C22×D4—62no—
319C22×D4—61no—
320C22×D4—62no—
321C22×D4C22×D311yescharacteristic
322C23×C4—21no—
323C23×C4—21no—
324C23×C4—32no—
325C23×C4—61no—
326C23×C4C22×D311yescharacteristic
327AGL(1,ℤ/8)—632no—
328AGL(1,ℤ/8)—632no—
329C2×SD16—64no—
330C2×SD16—64no—
331C2×SD16—64no—
332C2×SD16—64no—
333C2×SD16—64no—
334C2×SD16—64no—
335C2×D8—64no—
336C2×D8—64no—
337C2×D8—64no—
338C2×D8—64no—
339C2×D8—64no—
340C2×D8—64no—
341C2×M16—22no—
342C2×M16—22no—
343C2×M16—34no—
344C2×M16—62no—
345C2×M16—62no—
346C2×M16#6714yescomplemented
347C22×C8—32no—
348C22×C8C22×D312yes—
349C4⋊Q8—62no—
350C4⋊Q8—62no—
351#127—62no—
352#127—62no—
353#124—62no—
354#124—62no—
355#124—62no—
356#124—62no—
357#121—616no—
358C4×D4—616no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—62no—
363C2×C4⋊C4C22×D312yes—
364C2×C22⋊C4—62no—
365C2×C22⋊C4—62no—
366C2×C22⋊C4—64no—
367C2×C22⋊C4—64no—
368C2×C22⋊C4—128no—
369C2×C42—32no—
370C2×C42—31no—
371C2×C42—31no—
372C2×C42—62no—
373C2×C42—62no—
374#103—24no—
375#103—64no—
376#102—24no—
377#102—64no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—22no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3—42no—
384C23×C3—42no—
385C23×C3#12012yes—
386C23×C3#12112yes—
387C23×C3#12712yes—
388C23×C3C22×D411yescharacteristic
389C22×D3—22no—
390C22×D3—24no—
391C22×D3—416no—
392C22×D3—42no—
393C22×D3—41no—
394C22×D3—41no—
395C22×D3—42no—
396C22×D3—44no—
397C22×D3—42no—
398C22×D3—42no—
399C22×D3—44no—
400C22×D3—48no—
401C22×D3—84no—
402C22×D3—84no—
403C22×D3—88no—
404C3×Q8—22no—
405C3×Q8—22no—
406C3×Q8—41no—
407C3×Q8—41no—
408C3×D4—22no—
409C3×D4—22no—
410C3×D4—48no—
411C3×D4—41no—
412C3×D4—41no—
413C2×C3×C4—22no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—48no—
417C2×C3×C4—42no—
418C2×C3×C4—44no—
419C2×C3×C4—84no—
420C2×C3×C4#12014yes—
421C2×C3×C4#12012yes—
422C2×C3×C4#12014yes—
423C2×C3×C4#12114yes—
424C2×C3×C4#12112yes—
425C2×C3×C4#12114yes—
426C2×C3×C4#12712yes—
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4C22×D411yescharacteristic
431C2×C3×C4C22×D411yescharacteristic
432C2×C3×C4C2×#4111yescharacteristic
433#67—416no—
434#67—48no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—22no—
441C2×Dic3—48no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3AGL(1,ℤ/8)14yescomplemented
445D12—24no—
446D12—22no—
447D12—24no—
448D12—22no—
449D12—41no—
450D12—42no—
451D12—42no—
452D12—41no—
453D12—48no—
454C4×D3—48no—
455Dic6—22no—
456Dic6—22no—
457Dic6—41no—
458Dic6—41no—
459C3×C8—24no—
460C3⋊C8—24no—
461C24—21no—
462C24—64no—
463C24—62no—
464C24—61no—
465C24—124no—
466C24—124no—
467C24—122no—
468C24—122no—
469C24D3×D411yescharacteristic
470#41—28no—
471#41—28no—
472#41—44no—
473#41—44no—
474#41—68no—
475#41—68no—
476#41—124no—
477#41—124no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—32no—
484C2×Q8—61no—
485C2×Q8—61no—
486C2×Q8—61no—
487C2×Q8—61no—
488C2×Q8—61no—
489C2×Q8C2×D1212yes—
490C2×D4—24no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—32no—
497C2×D4—34no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4—68no—
501C2×D4—64no—
502C2×D4—68no—
503C2×D4—64no—
504C2×D4—68no—
505C2×D4—62no—
506C2×D4—61no—
507C2×D4—61no—
508C2×D4—62no—
509C2×D4—61no—
510C2×D4—61no—
511C2×D4—62no—
512C2×D4—62no—
513C2×D4—61no—
514C2×D4—62no—
515C2×D4—124no—
516C2×D4—124no—
517C2×D4—122no—
518C2×D4—122no—
519C2×D4—122no—
520C2×D4—124no—
521C2×D4—124no—
522C2×D4—122no—
523C2×D4C2×D1212yes—
524C22×C4—21no—
525C22×C4—24no—
526C22×C4—22no—
527C22×C4—32no—
528C22×C4—32no—
529C22×C4—42no—
530C22×C4—42no—
531C22×C4—44no—
532C22×C4—42no—
533C22×C4—64no—
534C22×C4—64no—
535C22×C4—61no—
536C22×C4—64no—
537C22×C4—122no—
538C22×C4—122no—
539C22×C4—122no—
540C22×C4C2×D1212yes—
541C22×C4#23512yes—
542C22×C4D3×D411yescharacteristic
543C22×C4D3×D412yes—
544C22×C4D3×D412yes—
545C22×C4D3×D412yes—
546C22×C4C2×#6712yes—
547C22×C4C23×D311yescharacteristic
548SD16—616no—
549SD16—616no—
550D8—616no—
551D8—616no—
552M16—28no—
553M16—68no—
554C2×C8—22no—
555C2×C8—22no—
556C2×C8—34no—
557C2×C8—62no—
558C2×C8—62no—
559C2×C8C2×#6714yes—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—42no—
563C4⋊C4—68no—
564C22⋊C4—68no—
565C22⋊C4—68no—
566C22⋊C4—128no—
567C22⋊C4—128no—
568C42—64no—
569C42—61no—
570C42—61no—
571C42—61no—
572C42—61no—
573C22×C3—22no—
574C22×C3—22no—
575C22×C3—24no—
576C22×C3—24no—
577C22×C3—42no—
578C22×C3—44no—
579C22×C3—81no—
580C22×C3—81no—
581C22×C3—82no—
582C22×C3#39214yes—
583C22×C3C2×#12011yescharacteristic
584C22×C3C2×#12111yescharacteristic
585C22×C3C2×#12711yescharacteristic
586C22×C3C2×AGL(1,ℤ/8)12yes—
587C22×C3#57512yes—
588C2×D3—42no—
589C2×D3—44no—
590C2×D3—44no—
591C2×D3—48no—
592C2×D3—816no—
593C2×D3—81no—
594C2×D3—82no—
595C2×D3—81no—
596C2×D3—82no—
597C2×D3—84no—
598C2×D3—82no—
599C3×C4—42no—
600C3×C4—82no—
601C3×C4#39214yes—
602C3×C4C2×#12011yescharacteristic
603C3×C4C2×#12011yescharacteristic
604C3×C4C2×#12111yescharacteristic
605C3×C4C2×#12111yescharacteristic
606Dic3—24no—
607Dic3—42no—
608Dic3—42no—
609C23—22no—
610C23—22no—
611C23—22no—
612C23—41no—
613C23—42no—
614C23—42no—
615C23—41no—
616C23—62no—
617C23—64no—
618C23—1216no—
619C23—128no—
620C23—122no—
621C23—121no—
622C23—124no—
623C23—122no—
624C23—122no—
625C23—124no—
626C23—121no—
627C23—122no—
628C23—244no—
629C23—248no—
630C23—244no—
631C23#88212yes—
632C23#88412yes—
633C23#94012yes—
634C23C2×D3×D411yescharacteristic
635Q8—22no—
636Q8—22no—
637Q8—41no—
638Q8—41no—
639Q8—62no—
640Q8—62no—
641Q8—121no—
642Q8—121no—
643D4—22no—
644D4—22no—
645D4—48no—
646D4—41no—
647D4—41no—
648D4—64no—
649D4—62no—
650D4—62no—
651D4—64no—
652D4—1216no—
653D4—128no—
654D4—128no—
655D4—121no—
656D4—122no—
657D4—122no—
658D4—121no—
659C2×C4—22no—
660C2×C4—34no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—48no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—62no—
667C2×C4—62no—
668C2×C4—62no—
669C2×C4—62no—
670C2×C4—62no—
671C2×C4—62no—
672C2×C4—84no—
673C2×C4—128no—
674C2×C4—121no—
675C2×C4—121no—
676C2×C4—128no—
677C2×C4#88212yes—
678C2×C4#88214yes—
679C2×C4#88412yes—
680C2×C4#88414yes—
681C2×C4#93014yes—
682C2×C4#93714yes—
683C2×C4#94012yes—
684C2×C4C22×D1211yescharacteristic
685C2×C4C2×#23511yescharacteristic
686C2×C4C2×D3×D411yescharacteristic
687C2×C4C2×D3×D411yescharacteristic
688C2×C4C2×D3×D411yescharacteristic
689C2×C4C22×#6711yescharacteristic
690C8—24no—
691C8—64no—
692C2×C3—24no—
693C2×C3—82no—
694C2×C3C2×#39211yescharacteristic
695C2×C3#300612yes—
696C2×C3#303612yes—
697C2×C3#315111yescharacteristic
698C2×C3#315311yescharacteristic
699D3—82no—
700D3—84no—
701C22—22no—
702C22—24no—
703C22—24no—
704C22—22no—
705C22—44no—
706C22—42no—
707C22—81no—
708C22—82no—
709C22—81no—
710C22—124no—
711C22—128no—
712C22—122no—
713C22—124no—
714C22—2416no—
715C22—242no—
716C22—241no—
717C22—244no—
718C22—242no—
719C22—242no—
720C22—241no—
721C22#496014yes—
722C22C2×#88211yescharacteristic
723C22C2×#88411yescharacteristic
724C22D3×AGL(1,ℤ/8)12yes—
725C22#577712yes—
726C22C2×#94011yescharacteristic
727C4—42no—
728C4—64no—
729C4—82no—
730C4—122no—
731C4—122no—
732C4#496014yes—
733C4C2×#88211yescharacteristic
734C4C2×#88411yescharacteristic
735C4C2×#93011yescharacteristic
736C4C2×#93711yescharacteristic
737C3#2342811yescharacteristic, complemented, Sylow
738C2—24no—
739C2—82no—
740C2—244no—
741C2—242no—
742C2#7888412yes—
743C2#7891012yes—
744C2C2×#496011yescharacteristic
745C2#8393711yescharacteristic
746C2#8397311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#417C2
socleC23×C3C22×D4
hypercenterC2×#417D3
hypocenterC3#23428
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×#5832C2semidirect
2. C22×#949C2semidirect
3. C22×#908C2semidirect
4. C2×C3×#417C2semidirect
5. C2×#4988C2semidirect
6. C2×#4966C2semidirect
7. #71979C2semidirect
8. #70282C2semidirect
9. #70277C2semidirect
10. #70214C2semidirect
11. #70209C2semidirect
23. C2×#947C22semidirect
24. C2×#944C22semidirect
26. C2×#940C22semidirect
28. C2×#936C22semidirect
40. C2×C3×#103C22semidirect
41. C2×C3×#102C22semidirect
45. C2×#810C22semidirect
47. C2×#809C22semidirect
48. #4550C22semidirect
49. #4550C22semidirect
54. C2×#417D3semidirect
89. C22×D12D4semidirect
96. C2×C3×M16D4semidirect
132. C2×#208D4semidirect
174. C2×#103C2×D3semidirect
176. C2×#102C2×D3semidirect
346. C2×M16#67semidirect
444. C2×Dic3AGL(1,ℤ/8)semidirect
737. C3#23428semidirect

Maximal subgroups

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

subgroupindex
C2×#58322
C22×#9492
C22×#9082
C2×C3×#4172
C2×#49882
C2×#49662
#719792
#702822
#702772
#702142
#702092
#234283