Finite Group Databasebeta

← C2×#78879 #11664482

Subgroups — C2×#78879 #11664482

Order 768.

2182 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2C22×#890C211yesmaximal, characteristic, complemented
3C2×C3×#417C211yesmaximal, characteristic, complemented
4C2×#5232C211yesmaximal, characteristic, complemented
5C2×#5133C211yesmaximal, characteristic, complemented
6C2×Dic3×M16C211yesmaximal, characteristic, complemented
7C2×#4966C211yesmaximal, characteristic, complemented
8#78879C2116yesmaximal, direct product factor
9C2×#4797C212yesmaximal, complemented
10C2×#4800C212yesmaximal, complemented
11C2×#4764C212yesmaximal, complemented
12C2×#4768C212yesmaximal, complemented
13C2×#3142—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18#5832—24no—
19#5832C2218yes—
20C2×#947C2211yescharacteristic
21C2×#944C2211yescharacteristic
22C2×#940C2211yescharacteristic
23C2×#939—21no—
24C2×#939—21no—
25C2×#936C2211yescharacteristic
26C2×C4×#67—21no—
27C2×C4×#67—21no—
28C22×C4⋊Dic3—21no—
29C22×C4⋊Dic3C2211yescharacteristic
30C22×Dic3⋊C4—21no—
31C22×C4×Dic3C2211yescharacteristic
32C22×#208C2211yescharacteristic
33C2×#890—24no—
34C2×#890—21no—
35C2×#890—21no—
36C2×#890C2212yes—
37C2×#890C2212yes—
38C2×#890C2218yescomplemented
39C3×#417C2218yescomplemented
40C2×C3×#103C2212yes—
41C2×C3×#102C2212yes—
42#5232C2218yescomplemented
43C2×#835C2212yes—
44C2×#832C2212yes—
45#5133C2218yescomplemented
46Dic3×M16C2218yescomplemented
47C2×#821C2212yes—
48C2×#816C2212yes—
49C2×#815C2212yes—
50C2×C8×Dic3C2212yes—
51C2×#810C2212yes—
52#4966C2218yescomplemented
53C2×#809C2212yes—
54#4800C22116yescomplemented
55#4797C22116yescomplemented
56#4768C22116yescomplemented
57#4764C22116yescomplemented
58C2×#532—31no—
59C22×#125—31no—
60#3142—316no—
61C2×#491—32no—
62C2×#490—32no—
63C2×#488—32no—
64C2×#488—32no—
65C2×#417—31no—
66C2×#417—31no—
67C2×#417—31no—
68C2×#417D311yescharacteristic, complemented
69C2×C4×M16—31no—
70C2×C3×#41—22no—
71C2×C3×#41—24no—
72C2×C3×#41—24no—
73C2×C3×#41C2314yes—
74C22×C3×Q8C2311yescharacteristic
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×#67—21no—
82C22×#67—21no—
83C23×Dic3—21no—
84C23×Dic3D411yescharacteristic
85C23×Dic3D411yescharacteristic
86C2×#235—22no—
87C2×#235—24no—
88C2×#235—24no—
89C2×#235C2314yes—
90C22×D12C2311yescharacteristic
91C22×C4×D3—21no—
92C22×Dic6C2311yescharacteristic
93C2×C3×M16—24no—
94C2×C3×M16C2314yes—
95C22×C3×C8C2312yes—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2314yes—
100C2×#217—21no—
101C2×#217—21no—
102#947—22no—
103#947C2314yes—
104#944—22no—
105#944C2314yes—
106#940—22no—
107#940C2314yes—
108#939—28no—
109#939—28no—
110#936—22no—
111#936C2314yes—
112C4×#67—28no—
113C4×#67—28no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—22no—
118C2×C4⋊Dic3—22no—
119C2×C4⋊Dic3—22no—
120C2×C4⋊Dic3—28no—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2312yes—
123C2×Dic3⋊C4—22no—
124C2×Dic3⋊C4—22no—
125C2×Dic3⋊C4—28no—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—21no—
128C2×C4×Dic3D411yescharacteristic
129C2×C4×Dic3D411yescharacteristic
130C2×C4×Dic3D414yes—
131C2×C4×Dic3C2311yescharacteristic
132C2×C4×Dic3C2311yescharacteristic
133C2×C4×Dic3C2314yes—
134C2×#208—24no—
135C2×#208C2314yes—
136C22×C3⋊C8C2312yes—
137#890—24no—
138#890—24no—
139#890—28no—
140#890—28no—
141#890C2318yes—
142#890C2318yes—
143C3×#103C2318yes—
144C3×#102C2318yes—
145#835C2318yes—
146#832C2318yes—
147#821C2318yes—
148#816C2318yes—
149#815C2318yes—
150C8×Dic3C2318yes—
151#810C2318yes—
152#809C2318yes—
153C22×#41—31no—
154C22×#41C2×D311yescharacteristic
155C22×M16—31no—
156C22×M16C2×D311yescharacteristic
157#532—38no—
158#532—64no—
159C2×C4⋊Q8—31no—
160C2×#127—31no—
161C2×#125—32no—
162C2×#125—32no—
163C2×#125—38no—
164C2×#125—61no—
165C2×#125—61no—
166C2×#125—64no—
167C2×#124—31no—
168C2×#124—31no—
169C2×#121—61no—
170C2×#121—61no—
171C2×C4×D4—61no—
172C2×C4×D4—61no—
173C22×C4⋊C4—31no—
174C22×C4⋊C4—61no—
175C22×C4⋊C4—61no—
176C22×C4⋊C4C2×D311yescharacteristic
177C22×C42—31no—
178#491—316no—
179#490—316no—
180#488—316no—
181#488—316no—
182#417—38no—
183#417—38no—
184#417—38no—
185#417C2×D318yescomplemented
186C2×#103—32no—
187C2×#103—32no—
188C2×#103—32no—
189C2×#103C2×D312yes—
190C2×#102—32no—
191C2×#102—32no—
192C2×#102—32no—
193C2×#102C2×D312yes—
194C4×M16—38no—
195C2×#97—32no—
196C2×C4×C8—32no—
197C24×C3—21no—
198C24×C3C2×D411yescharacteristic
199C23×D3—21no—
200C3×#41—28no—
201C3×#41—28no—
202C3×#41—48no—
203C2×C3×Q8—21no—
204C2×C3×Q8—22no—
205C2×C3×Q8—22no—
206C2×C3×Q8C2412yes—
207C2×C3×D4—24no—
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—22no—
211C2×C3×D4—44no—
212C2×C3×D4C2412yes—
213C22×C3×C4—24no—
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—24no—
217C22×C3×C4—44no—
218C22×C3×C4—44no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4#4114yes—
222C22×C3×C4#4114yes—
223C22×C3×C4C2411yescharacteristic
224C22×C3×C4C2412yes—
225C2×#67—24no—
226C2×#67—24no—
227C2×#67—24no—
228C2×#67—44no—
229C2×#67—44no—
230C2×#67—44no—
231C22×Dic3—24no—
232C22×Dic3—22no—
233C22×Dic3—21no—
234C22×Dic3—42no—
235C22×Dic3—42no—
236C22×Dic3—44no—
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D411yescharacteristic
240C22×Dic3C2×D411yescharacteristic
241C22×Dic3C2×D414yes—
242C22×Dic3C2×D414yes—
243#235—28no—
244#235—28no—
245#235—48no—
246C2×D12—22no—
247C2×D12—22no—
248C2×D12—21no—
249C2×D12C2412yes—
250C2×C4×D3—24no—
251C2×C4×D3—44no—
252C2×Dic6—22no—
253C2×Dic6—22no—
254C2×Dic6—21no—
255C2×Dic6C2412yes—
256C3×M16—28no—
257C2×C3×C8—24no—
258C2×C3×C8C2414yes—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4C2414yes—
262#217—24no—
263#217—24no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266C4⋊Dic3—22no—
267C4⋊Dic3—24no—
268C4⋊Dic3—28no—
269C4⋊Dic3—28no—
270C4⋊Dic3C2414yes—
271Dic3⋊C4—28no—
272Dic3⋊C4—28no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275C4×Dic3C2×D414yes—
276C4×Dic3C2×D414yes—
277C4×Dic3C2×D412yes—
278C4×Dic3C2×D412yes—
279C4×Dic3C2412yes—
280C4×Dic3C2412yes—
281#208—28no—
282C2×C3⋊C8—24no—
283C2×C3⋊C8C2414yes—
284C2×#41—22no—
285C2×#41—24no—
286C2×#41—24no—
287C2×#41—34no—
288C2×#41—62no—
289C2×#41—64no—
290C2×#41—64no—
291C2×#41C22×D314yes—
292C22×Q8—31no—
293C22×Q8C22×D311yescharacteristic
294C22×D4—21no—
295C22×D4—31no—
296C22×D4—61no—
297C22×D4—61no—
298C22×D4—61no—
299C22×D4C22×D311yescharacteristic
300C23×C4—21no—
301C23×C4—21no—
302C23×C4—31no—
303C23×C4—31no—
304C23×C4—61no—
305C23×C4—61no—
306C23×C4C22×D311yescharacteristic
307C2×M16—24no—
308C2×M16—34no—
309C2×M16—64no—
310C2×M16C22×D314yes—
311C22×C8—32no—
312C22×C8C22×D312yes—
313C4⋊Q8—34no—
314C4⋊Q8—62no—
315#127—34no—
316#127—62no—
317#125—38no—
318#125—38no—
319#125—68no—
320#125—68no—
321#125—64no—
322#125—64no—
323#124—34no—
324#124—34no—
325#124—62no—
326#124—62no—
327#121—68no—
328#121—68no—
329C4×D4—68no—
330C4×D4—68no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4—68no—
336C2×C4⋊C4—62no—
337C2×C4⋊C4—62no—
338C2×C4⋊C4—68no—
339C2×C4⋊C4—62no—
340C2×C4⋊C4—62no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—62no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4C22×D312yes—
346C2×C4⋊C4C22×D314yes—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C42—31no—
352C2×C42—31no—
353C2×C42—34no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—34no—
357#103—38no—
358#103—38no—
359#103—38no—
360#103C22×D318yes—
361#102—38no—
362#102—38no—
363#102—38no—
364#102C22×D318yes—
365#97—38no—
366C4×C8—38no—
367C23×C3—22no—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—42no—
371C23×C3—42no—
372C23×C3—42no—
373C23×C3#12414yes—
374C23×C3C22×D412yes—
375C23×C3C22×D411yescharacteristic
376C22×D3—22no—
377C22×D3—42no—
378C22×D3—42no—
379C22×D3—42no—
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—42no—
383C3×D4—22no—
384C3×D4—22no—
385C3×D4—48no—
386C3×D4—42no—
387C2×C3×C4—24no—
388C2×C3×C4—22no—
389C2×C3×C4—48no—
390C2×C3×C4—42no—
391C2×C3×C4—44no—
392C2×C3×C4—48no—
393C2×C3×C4#12414yes—
394C2×C3×C4#12418yes—
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C22×D412yes—
397C2×C3×C4C2×#4112yes—
398C2×C3×C4C2×#4114yes—
399C2×C3×C4C2×#4112yes—
400C2×C3×C4C2×#4114yes—
401C2×C3×C4C2511yescharacteristic
402#67—48no—
403#67—48no—
404#67—48no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—24no—
410C2×Dic3—22no—
411C2×Dic3—44no—
412C2×Dic3—44no—
413C2×Dic3—48no—
414C2×Dic3—42no—
415C2×Dic3—44no—
416C2×Dic3C22×D412yes—
417C2×Dic3C22×D412yes—
418C2×Dic3C22×D412yes—
419C2×Dic3C22×D412yes—
420D12—22no—
421D12—22no—
422D12—42no—
423C4×D3—48no—
424Dic6—22no—
425Dic6—22no—
426Dic6—42no—
427C3×C8—24no—
428C3⋊C8—24no—
429C24—21no—
430C24—61no—
431C24D3×D411yescharacteristic
432#41—28no—
433#41—28no—
434#41—48no—
435#41—68no—
436#41—68no—
437#41—128no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—32no—
442C2×Q8—61no—
443C2×Q8—62no—
444C2×Q8—62no—
445C2×Q8C23×D312yes—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—32no—
451C2×D4—44no—
452C2×D4—64no—
453C2×D4—64no—
454C2×D4—64no—
455C2×D4—62no—
456C2×D4—61no—
457C2×D4—62no—
458C2×D4—124no—
459C2×D4—124no—
460C2×D4—124no—
461C2×D4C23×D312yes—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—34no—
471C22×C4—34no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—64no—
475C22×C4—61no—
476C22×C4—62no—
477C22×C4—64no—
478C22×C4—122no—
479C22×C4—122no—
480C22×C4—124no—
481C22×C4—124no—
482C22×C4#23514yes—
483C22×C4D3×D412yes—
484C22×C4D3×D411yescharacteristic
485C22×C4#23714yes—
486C22×C4C23×D312yes—
487C22×C4C23×D311yescharacteristic
488M16—28no—
489M16—68no—
490C2×C8—24no—
491C2×C8—34no—
492C2×C8—64no—
493C2×C8C23×D314yes—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—34no—
497C4⋊C4—68no—
498C4⋊C4—68no—
499C4⋊C4—68no—
500C4⋊C4—68no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—62no—
505C4⋊C4C23×D314yes—
506C22⋊C4—64no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—64no—
510C42—34no—
511C42—34no—
512C42—32no—
513C42—32no—
514C42—32no—
515C42—32no—
516C22×C3—28no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—84no—
523C22×C3C2×#12414yes—
524C22×C3C2×#12412yes—
525C22×C3#57512yes—
526C22×C3#57512yes—
527C22×C3C23×D411yescharacteristic
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—42no—
531C2×D3—84no—
532C3×C4—42no—
533C3×C4—44no—
534C3×C4C2×#12412yes—
535C3×C4C2×#12414yes—
536C3×C4C22×#4111yescharacteristic
537C3×C4C22×#4111yescharacteristic
538Dic3—22no—
539Dic3—22no—
540Dic3—22no—
541Dic3—22no—
542Dic3—42no—
543Dic3—44no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—62no—
551C23—122no—
552C23—122no—
553C23—122no—
554C23#88514yes—
555C23C2×D3×D412yes—
556C23C2×D3×D411yescharacteristic
557Q8—22no—
558Q8—22no—
559Q8—42no—
560Q8—62no—
561Q8—62no—
562Q8—122no—
563D4—22no—
564D4—22no—
565D4—48no—
566D4—42no—
567D4—62no—
568D4—62no—
569D4—128no—
570D4—128no—
571D4—128no—
572D4—122no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—32no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—124no—
590C2×C4—124no—
591C2×C4—128no—
592C2×C4—122no—
593C2×C4—124no—
594C2×C4—128no—
595C2×C4#88514yes—
596C2×C4#88518yes—
597C2×C4C2×#23512yes—
598C2×C4C2×#23514yes—
599C2×C4C2×D3×D412yes—
600C2×C4C2×D3×D411yescharacteristic
601C2×C4C2×#23712yes—
602C2×C4C2×#23714yes—
603C2×C4C24×D311yescharacteristic
604C8—24no—
605C8—64no—
606C2×C3—24no—
607C2×C3—82no—
608C2×C3#314214yescomplemented
609C2×C3C22×#12411yescharacteristic
610C2×C3C2×#57511yescharacteristic
611C2×C3C2×#57511yescharacteristic
612D3—82no—
613C22—28no—
614C22—22no—
615C22—22no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—84no—
620C22—122no—
621C22—122no—
622C22—122no—
623C22—244no—
624C22C2×#88514yes—
625C22C2×#88512yes—
626C22#577312yes—
627C22#577712yes—
628C22C22×D3×D411yescharacteristic
629C4—44no—
630C4—42no—
631C4—62no—
632C4—62no—
633C4—62no—
634C4—62no—
635C4—124no—
636C4—122no—
637C4C2×#88514yes—
638C4C2×#88512yes—
639C4C22×#23511yescharacteristic
640C4C22×#23711yescharacteristic
641C3C2×#314211yescharacteristic, complemented, Sylow
642C2—24no—
643C2—82no—
644C2—242no—
645C2#7887914yesdirect product factor
646C2C22×#88511yescharacteristic
647C2C2×#577311yescharacteristic
648C2C2×#577711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#417C2
socleC23×C3C22×D4
prime radicalC2×#5832C2
hypercenterC2×#417D3
hypocenterC3C2×#3142
LCSG ▷ C2×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
2. C22×#890C2semidirect
3. C2×C3×#417C2semidirect
4. C2×#5232C2semidirect
5. C2×#5133C2semidirect
6. C2×Dic3×M16C2semidirect
7. C2×#4966C2semidirect
8. #78879C2direct (N × H)
9. C2×#4797C2semidirect
10. C2×#4800C2semidirect
11. C2×#4764C2semidirect
12. C2×#4768C2semidirect
38. C2×#890C22semidirect
39. C3×#417C22semidirect
42. #5232C22semidirect
45. #5133C22semidirect
46. Dic3×M16C22semidirect
52. #4966C22semidirect
54. #4800C22semidirect
55. #4797C22semidirect
56. #4768C22semidirect
57. #4764C22semidirect
68. C2×#417D3semidirect
185. #417C2×D3semidirect
608. C2×C3#3142semidirect
641. C3C2×#3142semidirect
645. C2#78879direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#58322
C22×#8902
C2×C3×#4172
C2×#52322
C2×#51332
C2×Dic3×M162
C2×#49662
#788792
C2×#47972
C2×#48002
C2×#47642
C2×#47682
C2×#31423