Finite Group Databasebeta

← #10690331

Subgroups — #10690331

Order 768.

966 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2329C211yesmaximal, characteristic, complemented
2C2×#5046C211yesmaximal, characteristic
3C2×#4538C211yesmaximal, characteristic, complemented
4#69521C211yesmaximal, characteristic
5#69426C211yesmaximal, characteristic
6#67624C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9042—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×#215—21no—
12C22×(C2×D3)⋊C4—21no—
13C22×(C2×D3)⋊C4—21no—
14C3×#391—21no—
15C3×#389—21no—
16C3×#387—21no—
17C3×#387—21no—
18C3×#385—21no—
19C3×#380C2211yescharacteristic
20C3×C4×C22⋊C4—21no—
21C2×C3×#95C2211yescharacteristic
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5046—22no—
26#5046C414yescomplemented
27C2×#809—21no—
28C2×#809—21no—
29C2×#809—21no—
30#4538—22no—
31C2×#450—31no—
32#2329D311yescharacteristic, complemented
33#1786—31no—
34C2×#328—31no—
35#1438—31no—
36#1294—31no—
37#1280—31no—
38C24×D3—21no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—42no—
50C22×D12—42no—
51C22×D12C2×C412yes—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8C2×C412yes—
55C3×C22⋊Q8D412yes—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C42—21no—
65C2×C3×C42—21no—
66C2×#215—24no—
67C2×#215—42no—
68C2×#215—42no—
69C2×(C2×D3)⋊C4—42no—
70C2×(C2×D3)⋊C4—42no—
71C2×(C2×D3)⋊C4—42no—
72C2×(C2×D3)⋊C4—42no—
73C2×(C2×D3)⋊C4—42no—
74C2×(C2×D3)⋊C4—42no—
75C22×C3⋊C8—21no—
76C22×C3⋊C8—21no—
77C3×#95—21no—
78C3×#95—21no—
79C3×#95—21no—
80C3×#95—21no—
81C3×#95—21no—
82C3×#95—21no—
83C3×#95—21no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86#830—21no—
87#830—21no—
88#830—21no—
89#830C2×C412yes—
90#809—24no—
91#809—42no—
92#809—42no—
93C23×D4—31no—
94C22×SD16—61no—
95C2×C22⋊Q8C2×D311yescharacteristic
96C22×C22⋊C4—61no—
97C22×C22⋊C4—61no—
98#450—34no—
99#450—62no—
100C2×#102—61no—
101C2×#102—61no—
102C2×#102—61no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8—31no—
105#391—21no—
106#389—21no—
107#387—21no—
108#387—21no—
109#385—21no—
110#380C2×D311yescharacteristic
111C4×C22⋊C4—21no—
112C2×#95—31no—
113C2×#95C2×D311yescharacteristic
114#328—62no—
115C24×C3C2×D411yescharacteristic
116C23×D3—22no—
117C23×D3—22no—
118C23×D3—22no—
119C23×D3—41no—
120C23×D3—41no—
121C23×D3—41no—
122C23×D3—41no—
123C23×D3—41no—
124C23×D3—41no—
125C23×D3—41no—
126C23×D3—41no—
127C23×D3—41no—
128C23×D3—41no—
129C23×D3—41no—
130C23×D3—41no—
131C2×C3×Q8—21no—
132C2×C3×Q8—41no—
133C2×C3×Q8—41no—
134C2×C3×Q8C22⋊C412yescomplemented
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4SD1612yescomplemented
151C22×C3×C4C22×C411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C22×C3×C4#4111yescharacteristic
157C22×Dic3—22no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—44no—
171C2×D12—44no—
172C2×D12—44no—
173C2×D12—41no—
174C2×D12—41no—
175C2×D12—41no—
176C2×D12—41no—
177C2×D12—44no—
178C2×D12—41no—
179C2×D12—41no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—41no—
184C3×C4⋊C4—41no—
185C3×C4⋊C4—81no—
186C3×C4⋊C4—81no—
187C3×C4⋊C4C22⋊C412yescomplemented
188C3×C22⋊C4—22no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—41no—
191C3×C22⋊C4—41no—
192C3×C42—81no—
193C3×C42—81no—
194#215—48no—
195(C2×D3)⋊C4—48no—
196(C2×D3)⋊C4—48no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—41no—
199C2×C3⋊C8—41no—
200C2×C3⋊C8—41no—
201C2×C3⋊C8—41no—
202C2×C3⋊C8—41no—
203C25—61no—
204C22×Q8#6711yescharacteristic
205C22×D4—32no—
206C22×D4—61no—
207C22×D4—61no—
208C22×D4—61no—
209C22×D4—61no—
210C22×D4—61no—
211C22×D4—122no—
212C22×D4—122no—
213C23×C4—61no—
214C23×C4#6711yescharacteristic
215C23×C4#6711yescharacteristic
216C23×C4C22×D311yescharacteristic
217C2×SD16—64no—
218C2×SD16—122no—
219C2×SD16—122no—
220C22×C8—61no—
221C22×C8—61no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8C4×D312yes—
225C22⋊Q8D1212yes—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4#6711yescharacteristic
230C2×C4⋊C4#6711yescharacteristic
231C2×C4⋊C4#6711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—122no—
235C2×C22⋊C4—122no—
236C2×C22⋊C4—122no—
237C2×C22⋊C4—122no—
238C2×C22⋊C4—122no—
239C2×C22⋊C4—122no—
240C2×C42—21no—
241C2×C42—21no—
242#102—64no—
243#102—122no—
244#102—122no—
245C22⋊C8—32no—
246C22⋊C8—61no—
247C22⋊C8—61no—
248C22⋊C8—61no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—61no—
257#95—61no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3C4×D411yescharacteristic
263C23×C3#12011yescharacteristic
264C23×C3#12111yescharacteristic
265C23×C3C2×SD1611yescharacteristic
266C23×C3#13511yescharacteristic
267C23×C3AGL(1,ℤ/8)11yescharacteristic
268C23×C3#13711yescharacteristic
269C22×D3—24no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—81no—
289C22×D3—81no—
290C22×D3—81no—
291C22×D3—81no—
292C22×D3—81no—
293C22×D3—81no—
294C22×D3—81no—
295C22×D3—81no—
296C22×D3—81no—
297C22×D3—81no—
298C22×D3—81no—
299C22×D3—81no—
300C22×D3—81no—
301C22×D3—81no—
302C22×D3—81no—
303C22×D3—81no—
304C3×Q8—42no—
305C3×Q8—81no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—81no—
335C2×C3×C4—81no—
336C2×C3×C4C2×C22⋊C411yescharacteristic
337C2×C3×C4C4×D411yescharacteristic
338C2×C3×C4#12111yescharacteristic
339C2×C3×C4#12311yescharacteristic
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—81no—
344C2×Dic3—81no—
345C2×Dic3—81no—
346C2×Dic3—81no—
347D12—24no—
348D12—24no—
349D12—42no—
350D12—42no—
351D12—42no—
352D12—42no—
353D12—42no—
354D12—42no—
355C3⋊C8—42no—
356C3⋊C8—81no—
357C24—62no—
358C24—62no—
359C24—62no—
360C24—121no—
361C24—121no—
362C24—121no—
363C24—121no—
364C24—121no—
365C24—121no—
366C24—121no—
367C24—121no—
368C24—121no—
369C24—121no—
370C24—121no—
371C24—121no—
372C24C2×#6711yescharacteristic
373C2×Q8—21no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8(C2×D3)⋊C412yescomplemented
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—121no—
384C2×D4—121no—
385C2×D4—121no—
386C2×D4—121no—
387C2×D4—124no—
388C2×D4—124no—
389C2×D4—124no—
390C2×D4—124no—
391C2×D4—121no—
392C2×D4—121no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—62no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4—121no—
412C22×C4—121no—
413C22×C4—121no—
414C22×C4—121no—
415C22×C4#21512yescomplemented
416C22×C4C2×C4×D311yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4#23511yescharacteristic
419C22×C4#23511yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422SD16—128no—
423C2×C8—62no—
424C2×C8—121no—
425C2×C8—121no—
426C2×C8—121no—
427C2×C8—121no—
428C2×C8—121no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4(C2×D3)⋊C412yescomplemented
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—128no—
442C22⋊C4—128no—
443C42—81no—
444C42—81no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3#38611yescharacteristic
458C22×C3C4×SD1611yescharacteristic
459C22×C3#44011yescharacteristic
460C22×C3#44811yescharacteristic
461C22×C3#44911yescharacteristic
462C22×C3#45311yescharacteristic
463C22×C3#45411yescharacteristic
464C22×C3B(4,𝔽2)11yescharacteristic
465C22×C3B(4,𝔽2)11yescharacteristic
466C22×C3#46011yescharacteristic
467C22×C3#46411yescharacteristic
468C2×D3—44no—
469C2×D3—44no—
470C2×D3—44no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3—82no—
474C2×D3—82no—
475C2×D3—82no—
476C2×D3—82no—
477C2×D3—82no—
478C2×D3—82no—
479C2×D3—82no—
480C2×D3—82no—
481C2×D3—82no—
482C2×D3—82no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4—81no—
493Dic3—82no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—64no—
499C23—122no—
500C23—122no—
501C23—122no—
502C23—122no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23—122no—
511C23—122no—
512C23—122no—
513C23—122no—
514C23—122no—
515C23—122no—
516C23—122no—
517C23—241no—
518C23—241no—
519C23—241no—
520C23—241no—
521C23—241no—
522C23—241no—
523C23—241no—
524C23—241no—
525C23—241no—
526C23—241no—
527C23—241no—
528C23—241no—
529C23—241no—
530C23—241no—
531C23—241no—
532C23—241no—
533C23C4×#6711yescharacteristic
534C23#93011yescharacteristic
535C23C2×#21511yescharacteristic
536C23#94211yescharacteristic
537C23#94911yescharacteristic
538C23#95011yescharacteristic
539C23#95311yescharacteristic
540Q8—42no—
541Q8—81no—
542D4—64no—
543D4—64no—
544D4—122no—
545D4—122no—
546D4—122no—
547D4—122no—
548D4—122no—
549D4—122no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—241no—
584C2×C4—241no—
585C2×C4—241no—
586C2×C4—241no—
587C2×C4C2×(C2×D3)⋊C411yescharacteristic
588C2×C4C4×#6711yescharacteristic
589C2×C4#92911yescharacteristic
590C2×C4#93011yescharacteristic
591C8—122no—
592C8—241no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3#159611yescharacteristic
598C2×C3#159711yescharacteristic
599C2×C3#159911yescharacteristic
600C2×C3#160011yescharacteristic
601C2×C3#188011yescharacteristic
602C2×C3#188311yescharacteristic
603C2×C3#188911yescharacteristic
604D3—84no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—124no—
618C22—124no—
619C22—124no—
620C22—242no—
621C22—242no—
622C22—242no—
623C22—242no—
624C22—242no—
625C22—242no—
626C22—242no—
627C22—242no—
628C22—242no—
629C22—242no—
630C22—242no—
631C22—242no—
632C22C4×#21511yescharacteristic
633C22#502611yescharacteristic
634C22#502711yescharacteristic
635C22#502811yescharacteristic
636C22#521311yescharacteristic
637C22#522711yescharacteristic
638C22#523811yescharacteristic
639C22#524111yescharacteristic
640C22#524411yescharacteristic
641C22#524411yescharacteristic
642C22#524511yescharacteristic
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—242no—
654C3#904211yescharacteristic, complemented, Sylow
655C2—21no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—244no—
660C2#7163311yescharacteristic
661C2#7165411yescharacteristic
662C2#7166711yescharacteristic
663C2#7188411yescharacteristic
664C2#7188811yescharacteristic
665C2#7189211yescharacteristic
666C2#7190211yescharacteristic
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2329C2
socleC23×C3#120
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenter#2329D3
hypocenterC3#9042
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2329

Complements

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

NH ≅ G/Ntype
1. C3×#2329C2semidirect
3. C2×#4538C2semidirect
6. #67624C2semidirect
7. #67613C2semidirect
26. #5046C4semidirect
32. #2329D3semidirect
134. C2×C3×Q8C22⋊C4semidirect
150. C22×C3×C4SD16semidirect
187. C3×C4⋊C4C22⋊C4semidirect
376. C2×Q8(C2×D3)⋊C4semidirect
415. C22×C4#215semidirect
436. C4⋊C4(C2×D3)⋊C4semidirect
654. C3#9042semidirect

Maximal subgroups

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

subgroupindex
C3×#23292
C2×#50462
C2×#45382
#695212
#694262
#676242
#676132
#90423