Finite Group Databasebeta

← C2×#78878 #11664474

Subgroups — C2×#78878 #11664474

Order 768.

2398 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2C22×#888C211yesmaximal, 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#78878C2116yesmaximal, direct product factor
9C2×#4794C212yesmaximal, complemented
10C2×#4792C212yesmaximal, complemented
11C2×#4765C212yesmaximal, complemented
12C2×#4760C212yesmaximal, complemented
13C2×#3141—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C23×Dic6—21no—
17C22×C3×M16C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19#5832—24no—
20#5832C2218yes—
21C2×#947C2211yescharacteristic
22C2×#944C2211yescharacteristic
23C2×#940C2211yescharacteristic
24C2×#939—21no—
25C2×#939—21no—
26C2×#936C2211yescharacteristic
27C2×C4×#67—21no—
28C2×C4×#67—21no—
29C22×C4⋊Dic3C2211yescharacteristic
30C22×Dic3⋊C4—21no—
31C22×C4×Dic3C2211yescharacteristic
32C22×#208C2211yescharacteristic
33C2×#888—24no—
34C2×#888—21no—
35C2×#888—21no—
36C2×#888C2212yes—
37C2×#888C2212yes—
38C2×#888C2218yescomplemented
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#4794C22116yescomplemented
55#4792C22116yescomplemented
56#4765C22116yescomplemented
57#4760C22116yescomplemented
58C2×#532—31no—
59C22×C4⋊Q8—31no—
60#3141—316no—
61C2×#489—32no—
62C2×#489—32no—
63C2×#487—32no—
64C2×#486—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×Dic6—22no—
93C22×Dic6—22no—
94C22×Dic6—28no—
95C22×Dic6—48no—
96C22×Dic6C2311yescharacteristic
97C2×C3×M16—24no—
98C2×C3×M16C2314yes—
99C22×C3×C8C2312yes—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2314yes—
104C2×#217—21no—
105C2×#217—21no—
106#947—22no—
107#947C2314yes—
108#944—22no—
109#944C2314yes—
110#940—22no—
111#940C2314yes—
112#939—28no—
113#939—28no—
114#936—22no—
115#936C2314yes—
116C4×#67—28no—
117C4×#67—28no—
118C2×(C2×D3)⋊C4—21no—
119C2×(C2×D3)⋊C4—21no—
120C2×C4⋊Dic3—21no—
121C2×C4⋊Dic3—22no—
122C2×C4⋊Dic3C2314yes—
123C2×C4⋊Dic3C2312yes—
124C2×Dic3⋊C4—22no—
125C2×Dic3⋊C4—22no—
126C2×Dic3⋊C4—28no—
127C2×Dic3⋊C4—21no—
128C2×Dic3⋊C4—21no—
129C2×C4×Dic3D411yescharacteristic
130C2×C4×Dic3D411yescharacteristic
131C2×C4×Dic3D414yes—
132C2×C4×Dic3C2311yescharacteristic
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2314yes—
135C2×#208—24no—
136C2×#208C2314yes—
137C22×C3⋊C8C2312yes—
138#888—24no—
139#888—24no—
140#888—28no—
141#888—28no—
142#888C2318yes—
143#888C2318yes—
144C3×#103C2318yes—
145C3×#102C2318yes—
146#835C2318yes—
147#832C2318yes—
148#821C2318yes—
149#816C2318yes—
150#815C2318yes—
151C8×Dic3C2318yes—
152#810C2318yes—
153#809C2318yes—
154C22×#41—31no—
155C22×#41C2×D311yescharacteristic
156C23×Q8—61no—
157C22×M16—31no—
158C22×M16C2×D311yescharacteristic
159#532—38no—
160#532—64no—
161C2×C4⋊Q8—32no—
162C2×C4⋊Q8—32no—
163C2×C4⋊Q8—38no—
164C2×C4⋊Q8—31no—
165C2×C4⋊Q8—61no—
166C2×C4⋊Q8—61no—
167C2×C4⋊Q8—64no—
168C2×#127—31no—
169C2×#124—31no—
170C2×#124—31no—
171C2×#121—61no—
172C2×#121—61no—
173C2×C4×D4—61no—
174C2×C4×D4—61no—
175C22×C4⋊C4—31no—
176C22×C4⋊C4—61no—
177C22×C4⋊C4C2×D311yescharacteristic
178C22×C42—31no—
179#489—316no—
180#489—316no—
181#487—316no—
182#486—316no—
183#417—38no—
184#417—38no—
185#417—38no—
186#417C2×D318yescomplemented
187C2×#103—32no—
188C2×#103—32no—
189C2×#103—32no—
190C2×#103C2×D312yes—
191C2×#102—32no—
192C2×#102—32no—
193C2×#102—32no—
194C2×#102C2×D312yes—
195C4×M16—38no—
196C2×#97—32no—
197C2×C4×C8—32no—
198C24×C3—21no—
199C24×C3C2×D411yescharacteristic
200C23×D3—21no—
201C3×#41—28no—
202C3×#41—28no—
203C3×#41—48no—
204C2×C3×Q8—21no—
205C2×C3×Q8—22no—
206C2×C3×Q8—22no—
207C2×C3×Q8C2412yes—
208C2×C3×D4—24no—
209C2×C3×D4—21no—
210C2×C3×D4—22no—
211C2×C3×D4—22no—
212C2×C3×D4—44no—
213C2×C3×D4C2412yes—
214C22×C3×C4—21no—
215C22×C3×C4—24no—
216C22×C3×C4—22no—
217C22×C3×C4—24no—
218C22×C3×C4—44no—
219C22×C3×C4—44no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D412yes—
222C22×C3×C4#4114yes—
223C22×C3×C4#4114yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2412yes—
226C2×#67—24no—
227C2×#67—24no—
228C2×#67—24no—
229C2×#67—44no—
230C2×#67—44no—
231C2×#67—44no—
232C22×Dic3—24no—
233C22×Dic3—24no—
234C22×Dic3—24no—
235C22×Dic3—22no—
236C22×Dic3—21no—
237C22×Dic3—44no—
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D414yes—
240C22×Dic3C2×D411yescharacteristic
241C22×Dic3C2×D414yes—
242C22×Dic3C2×D411yescharacteristic
243C22×Dic3C2×D411yescharacteristic
244#235—28no—
245#235—28no—
246#235—48no—
247C2×D12—22no—
248C2×D12—22no—
249C2×D12—21no—
250C2×D12C2412yes—
251C2×C4×D3—24no—
252C2×C4×D3—44no—
253C2×Dic6—22no—
254C2×Dic6—22no—
255C2×Dic6—21no—
256C2×Dic6—28no—
257C2×Dic6—28no—
258C2×Dic6—48no—
259C2×Dic6—432no—
260C2×Dic6—48no—
261C2×Dic6C2412yes—
262C3×M16—28no—
263C2×C3×C8—24no—
264C2×C3×C8C2414yes—
265C3×C4⋊C4—22no—
266C3×C4⋊C4—24no—
267C3×C4⋊C4C2414yes—
268#217—24no—
269#217—24no—
270(C2×D3)⋊C4—24no—
271(C2×D3)⋊C4—24no—
272C4⋊Dic3—22no—
273C4⋊Dic3—24no—
274C4⋊Dic3C2414yes—
275Dic3⋊C4—28no—
276Dic3⋊C4—28no—
277Dic3⋊C4—24no—
278Dic3⋊C4—24no—
279C4×Dic3C2×D414yes—
280C4×Dic3C2×D414yes—
281C4×Dic3C2×D412yes—
282C4×Dic3C2×D412yes—
283C4×Dic3C2412yes—
284C4×Dic3C2412yes—
285#208—28no—
286C2×C3⋊C8—24no—
287C2×C3⋊C8C2414yes—
288C2×#41—22no—
289C2×#41—24no—
290C2×#41—24no—
291C2×#41—34no—
292C2×#41—62no—
293C2×#41—64no—
294C2×#41—64no—
295C2×#41C22×D314yes—
296C22×Q8—31no—
297C22×Q8—68no—
298C22×Q8—62no—
299C22×Q8—62no—
300C22×Q8—128no—
301C22×Q8C22×D311yescharacteristic
302C22×D4—21no—
303C22×D4—31no—
304C22×D4—61no—
305C22×D4—61no—
306C22×D4—61no—
307C22×D4C22×D311yescharacteristic
308C23×C4—21no—
309C23×C4—21no—
310C23×C4—31no—
311C23×C4—31no—
312C23×C4—61no—
313C23×C4—61no—
314C23×C4C22×D311yescharacteristic
315C2×M16—24no—
316C2×M16—34no—
317C2×M16—64no—
318C2×M16C22×D314yes—
319C22×C8—32no—
320C22×C8C22×D312yes—
321C4⋊Q8—38no—
322C4⋊Q8—38no—
323C4⋊Q8—34no—
324C4⋊Q8—68no—
325C4⋊Q8—68no—
326C4⋊Q8—64no—
327C4⋊Q8—64no—
328C4⋊Q8—62no—
329#127—34no—
330#127—62no—
331#124—34no—
332#124—34no—
333#124—62no—
334#124—62no—
335#121—68no—
336#121—68no—
337C4×D4—68no—
338C4×D4—68no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—68no—
344C2×C4⋊C4—62no—
345C2×C4⋊C4—62no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4—61no—
348C2×C4⋊C4—62no—
349C2×C4⋊C4—61no—
350C2×C4⋊C4C22×D312yes—
351C2×C4⋊C4C22×D314yes—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C22⋊C4—61no—
355C2×C22⋊C4—61no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—34no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—34no—
362#103—38no—
363#103—38no—
364#103—38no—
365#103C22×D318yes—
366#102—38no—
367#102—38no—
368#102—38no—
369#102C22×D318yes—
370#97—38no—
371C4×C8—38no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—42no—
376C23×C3—42no—
377C23×C3—42no—
378C23×C3#12414yes—
379C23×C3C22×D412yes—
380C23×C3C22×D411yescharacteristic
381C22×D3—22no—
382C22×D3—42no—
383C22×D3—42no—
384C22×D3—42no—
385C3×Q8—22no—
386C3×Q8—22no—
387C3×Q8—42no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—48no—
391C3×D4—42no—
392C2×C3×C4—22no—
393C2×C3×C4—24no—
394C2×C3×C4—48no—
395C2×C3×C4—42no—
396C2×C3×C4—44no—
397C2×C3×C4—48no—
398C2×C3×C4#12414yes—
399C2×C3×C4#12418yes—
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D412yes—
402C2×C3×C4C2×#4112yes—
403C2×C3×C4C2×#4114yes—
404C2×C3×C4C2×#4112yes—
405C2×C3×C4C2×#4114yes—
406C2×C3×C4C2511yescharacteristic
407#67—48no—
408#67—48no—
409#67—48no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—28no—
413C2×Dic3—28no—
414C2×Dic3—24no—
415C2×Dic3—22no—
416C2×Dic3—48no—
417C2×Dic3—42no—
418C2×Dic3—44no—
419C2×Dic3AGL(1,ℤ/8)14yes—
420C2×Dic3#13714yes—
421C2×Dic3C22×D412yes—
422C2×Dic3C22×D412yes—
423C2×Dic3C22×D412yes—
424C2×Dic3C22×D412yes—
425D12—22no—
426D12—22no—
427D12—42no—
428C4×D3—48no—
429Dic6—22no—
430Dic6—22no—
431Dic6—416no—
432Dic6—42no—
433Dic6—416no—
434C3×C8—24no—
435C3⋊C8—24no—
436C24—21no—
437C24—61no—
438C24D3×D411yescharacteristic
439#41—28no—
440#41—28no—
441#41—48no—
442#41—68no—
443#41—68no—
444#41—128no—
445C2×Q8—21no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—32no—
449C2×Q8—68no—
450C2×Q8—68no—
451C2×Q8—61no—
452C2×Q8—62no—
453C2×Q8—62no—
454C2×Q8—128no—
455C2×Q8—1232no—
456C2×Q8—128no—
457C2×Q8C23×D312yes—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—32no—
463C2×D4—44no—
464C2×D4—64no—
465C2×D4—64no—
466C2×D4—64no—
467C2×D4—62no—
468C2×D4—61no—
469C2×D4—62no—
470C2×D4—124no—
471C2×D4—124no—
472C2×D4—124no—
473C2×D4C23×D312yes—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—34no—
481C22×C4—34no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—64no—
487C22×C4—64no—
488C22×C4—64no—
489C22×C4—61no—
490C22×C4—62no—
491C22×C4—64no—
492C22×C4—124no—
493C22×C4—124no—
494C22×C4#23514yes—
495C22×C4D3×D412yes—
496C22×C4D3×D411yescharacteristic
497C22×C4#23714yes—
498C22×C4C23×D312yes—
499C22×C4C23×D311yescharacteristic
500M16—28no—
501M16—68no—
502C2×C8—24no—
503C2×C8—34no—
504C2×C8—64no—
505C2×C8C23×D314yes—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—34no—
509C4⋊C4—68no—
510C4⋊C4—68no—
511C4⋊C4—64no—
512C4⋊C4—64no—
513C4⋊C4—64no—
514C4⋊C4—62no—
515C4⋊C4C23×D314yes—
516C22⋊C4—64no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C42—34no—
521C42—34no—
522C42—32no—
523C42—32no—
524C42—32no—
525C42—32no—
526C22×C3—28no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—84no—
533C22×C3C2×#12414yes—
534C22×C3C2×#12412yes—
535C22×C3C2×AGL(1,ℤ/8)12yes—
536C22×C3C2×#13712yes—
537C22×C3C23×D411yescharacteristic
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—84no—
542C3×C4—42no—
543C3×C4—44no—
544C3×C4C2×#12412yes—
545C3×C4C2×#12414yes—
546C3×C4C22×#4111yescharacteristic
547C3×C4C22×#4111yescharacteristic
548Dic3—22no—
549Dic3—22no—
550Dic3—42no—
551Dic3—44no—
552Dic3C2×AGL(1,ℤ/8)14yes—
553Dic3C2×#13714yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—62no—
561C23—122no—
562C23—122no—
563C23—122no—
564C23#88514yes—
565C23C2×D3×D412yes—
566C23C2×D3×D411yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—42no—
570Q8—62no—
571Q8—62no—
572Q8—1216no—
573Q8—1216no—
574Q8—122no—
575D4—22no—
576D4—22no—
577D4—48no—
578D4—42no—
579D4—62no—
580D4—62no—
581D4—128no—
582D4—128no—
583D4—128no—
584D4—122no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—32no—
588C2×C4—32no—
589C2×C4—34no—
590C2×C4—34no—
591C2×C4—32no—
592C2×C4—32no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—68no—
598C2×C4—68no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—64no—
603C2×C4—128no—
604C2×C4—122no—
605C2×C4—124no—
606C2×C4—128no—
607C2×C4#88514yes—
608C2×C4#88518yes—
609C2×C4C2×#23512yes—
610C2×C4C2×#23514yes—
611C2×C4C2×D3×D412yes—
612C2×C4C2×D3×D411yescharacteristic
613C2×C4C2×#23712yes—
614C2×C4C2×#23714yes—
615C2×C4C24×D311yescharacteristic
616C8—24no—
617C8—64no—
618C2×C3—24no—
619C2×C3—82no—
620C2×C3#314114yescomplemented
621C2×C3C22×#12411yescharacteristic
622C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
623C2×C3C22×#13711yescharacteristic
624D3—82no—
625C22—28no—
626C22—22no—
627C22—22no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—84no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22—244no—
636C22C2×#88514yes—
637C22C2×#88512yes—
638C22D3×AGL(1,ℤ/8)12yes—
639C22D3×#13712yes—
640C22C22×D3×D411yescharacteristic
641C4—34no—
642C4—34no—
643C4—44no—
644C4—42no—
645C4—62no—
646C4—62no—
647C4—124no—
648C4—122no—
649C4C2×#88514yes—
650C4C2×#88512yes—
651C4C22×#23511yescharacteristic
652C4C22×#23711yescharacteristic
653C3C2×#314111yescharacteristic, complemented, Sylow
654C2—24no—
655C2—82no—
656C2—242no—
657C2#7887814yesdirect product factor
658C2C22×#88511yescharacteristic
659C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
660C2C2×D3×#13711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#417C2
socleC23×C3C22×D4
prime radicalC2×#5832C2
hypercenterC2×#417D3
hypocenterC3C2×#3141
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×#888C2semidirect
3. C2×C3×#417C2semidirect
4. C2×#5232C2semidirect
5. C2×#5133C2semidirect
6. C2×Dic3×M16C2semidirect
7. C2×#4966C2semidirect
8. #78878C2direct (N × H)
9. C2×#4794C2semidirect
10. C2×#4792C2semidirect
11. C2×#4765C2semidirect
12. C2×#4760C2semidirect
38. C2×#888C22semidirect
39. C3×#417C22semidirect
42. #5232C22semidirect
45. #5133C22semidirect
46. Dic3×M16C22semidirect
52. #4966C22semidirect
54. #4794C22semidirect
55. #4792C22semidirect
56. #4765C22semidirect
57. #4760C22semidirect
68. C2×#417D3semidirect
186. #417C2×D3semidirect
620. C2×C3#3141semidirect
653. C3C2×#3141semidirect
657. C2#78878direct (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×#8882
C2×C3×#4172
C2×#52322
C2×#51332
C2×Dic3×M162
C2×#49662
#788782
C2×#47942
C2×#47922
C2×#47652
C2×#47602
C2×#31413