Finite Group Databasebeta

← C2×#81679 #11664410

Subgroups — C2×#81679 #11664410

Order 768.

2182 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic, complemented
2C2×#5668C211yesmaximal, characteristic
3C2×C3×#489C211yesmaximal, characteristic, complemented
4C2×#5089C211yesmaximal, characteristic, complemented
5C2×#5055C211yesmaximal, characteristic, complemented
6#81679C2116yesmaximal, direct product factor
7C2×#4804C212yesmaximal, complemented
8C2×#4797C212yesmaximal, complemented
9C2×#4772C212yesmaximal, complemented
10C2×#4764C212yesmaximal, complemented
11C2×#4714C211yesmaximal, characteristic, complemented
12C2×D3×#97C211yesmaximal, characteristic, complemented
13C2×#3135—31nomaximal, Sylow
14C22×#239C2211yescharacteristic
15C22×#237C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#124C2211yescharacteristic
18C2×#946—21no—
19C2×Q8×Dic3—21no—
20C2×#944C2211yescharacteristic
21C2×#940C2211yescharacteristic
22C22×D3×C8C2212yes—
23#5728—24no—
24#5728C2218yescomplemented
25#5668—24no—
26#5668C2218yes—
27C2×#897—21no—
28C2×#892C2212yes—
29C2×#890C2212yes—
30C2×#887—21no—
31C2×#884—21no—
32C2×#884—21no—
33C2×#67⋊C4—21no—
34C2×#67⋊C4—21no—
35C2×#875C2211yescharacteristic
36C2×C42×D3C2211yescharacteristic
37C2×#869C2211yescharacteristic
38C3×#489C2218yescomplemented
39C2×C3×#103C2212yes—
40C2×C3×#102C2212yes—
41C2×C3×#97C2211yescharacteristic
42C2×#835C2212yes—
43C2×#832C2212yes—
44C2×#821C2212yes—
45C2×#816C2212yes—
46C2×#815C2212yes—
47#5089C2218yescomplemented
48#5055C2218yescomplemented
49C2×#810C2212yes—
50C2×#809C2212yes—
51C2×#803C2211yescharacteristic
52#4804C22116yescomplemented
53#4797C22116yescomplemented
54#4772C22116yescomplemented
55#4764C22116yescomplemented
56#4714C2218yescomplemented
57D3×#97C2218yescomplemented
58C2×#533—31no—
59C2×#532—31no—
60#3135—316no—
61C2×#491—32no—
62C2×#490—32no—
63C2×#489—31no—
64C2×#489—31no—
65C2×#489—31no—
66C2×#489D311yescharacteristic, complemented
67C2×#416—32no—
68C2×#416—32no—
69C22×#97—31no—
70C22×C3×Q8—21no—
71C22×C3×Q8C2311yescharacteristic
72C22×C3×D4C2311yescharacteristic
73C22×#67—21no—
74C22×#67—21no—
75C22×#67—21no—
76C23×Dic3—21no—
77C2×#239—22no—
78C2×#239—28no—
79C2×#239C2314yes—
80C2×#237—22no—
81C2×#237—28no—
82C2×#237C2314yes—
83C22×D12—21no—
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×C4×D3D411yescharacteristic
87C22×C4×D3D411yescharacteristic
88C22×C4×D3C2311yescharacteristic
89C22×Dic6—21no—
90C22×Dic6C2311yescharacteristic
91C22×C3×C8C2312yes—
92C3×C4⋊Q8—22no—
93C3×C4⋊Q8C2314yes—
94C3×#124—22no—
95C3×#124C2314yes—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C42C2311yescharacteristic
101#946—28no—
102Q8×Dic3—28no—
103#944—22no—
104#944C2314yes—
105#940—22no—
106#940C2314yes—
107C2×(C2×D3)⋊C4—22no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—22no—
112C2×C4⋊Dic3C2312yes—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—22no—
116C2×C4×Dic3—22no—
117C2×C4×Dic3D411yescharacteristic
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2312yes—
121C2×D3×C8—28no—
122C2×D3×C8C2318yes—
123#897—28no—
124#892—24no—
125#892C2318yes—
126#890—24no—
127#890C2318yes—
128#887—28no—
129#884—28no—
130#884—28no—
131#67⋊C4—28no—
132#67⋊C4—28no—
133#875—22no—
134#875C2314yes—
135C42×D3D414yes—
136C42×D3C2314yes—
137#869—22no—
138#869C2314yes—
139C3×#103C2318yes—
140C3×#102C2318yes—
141C3×#97C2314yes—
142#835C2318yes—
143#832C2318yes—
144#821C2318yes—
145#816C2318yes—
146#815C2318yes—
147#810C2318yes—
148#809C2318yes—
149#803C2314yes—
150C22×#41—31no—
151C22×#41—31no—
152C23×C8—32no—
153#533—38no—
154#533—64no—
155#532—38no—
156#532—64no—
157C2×C4⋊Q8—31no—
158C2×C4⋊Q8—31no—
159C2×C4⋊Q8C2×D311yescharacteristic
160C2×#127—31no—
161C2×#125—32no—
162C2×#124—31no—
163C2×#124C2×D311yescharacteristic
164C2×C22⋊Q8—61no—
165C2×C22⋊Q8—61no—
166C2×#121—61no—
167C2×#121—61no—
168C2×C4×Q8—61no—
169C2×C4×Q8—61no—
170C2×C4×D4—61no—
171C2×C4×D4—61no—
172C2×#117—32no—
173C22×C42—31no—
174#491—316no—
175#490—316no—
176#489—38no—
177#489—38no—
178#489—38no—
179#489C2×D318yescomplemented
180#416—316no—
181#416—316no—
182C2×#103—32no—
183C2×#103—32no—
184C2×#103—32no—
185C2×#103C2×D312yes—
186C2×#102—32no—
187C2×#102—32no—
188C2×#102—32no—
189C2×#102C2×D312yes—
190C2×#97—32no—
191C2×#97—31no—
192C2×#97—38no—
193C2×#97C2×D311yescharacteristic
194C24×C3—21no—
195C23×D3—21no—
196C23×D3C2×D411yescharacteristic
197C2×C3×Q8—21no—
198C2×C3×Q8—24no—
199C2×C3×Q8—24no—
200C2×C3×Q8—44no—
201C2×C3×Q8C2412yes—
202C2×C3×D4—21no—
203C2×C3×D4—24no—
204C2×C3×D4C2412yes—
205C22×C3×C4—22no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C2×#67—24no—
211C2×#67—24no—
212C2×#67—24no—
213C2×#67—44no—
214C2×#67—44no—
215C2×#67—44no—
216C22×Dic3—22no—
217C22×Dic3—21no—
218C22×Dic3—24no—
219C22×Dic3—44no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223#239—216no—
224#239—48no—
225#237—216no—
226#237—48no—
227C2×D12—24no—
228C2×D12—24no—
229C2×D12—21no—
230C2×D12—44no—
231C2×D12C2412yes—
232C2×C4×D3—24no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—44no—
236C2×C4×D3C2×D414yes—
237C2×C4×D3C2×D414yes—
238C2×C4×D3C2×D412yes—
239C2×C4×D3#4118yes—
240C2×C4×D3C2412yes—
241C2×Dic6—24no—
242C2×Dic6—24no—
243C2×Dic6—21no—
244C2×Dic6—44no—
245C2×Dic6C2412yes—
246C2×C3×C8—24no—
247C2×C3×C8C2414yes—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4C2414yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C42C2×D412yes—
254C3×C42C2412yes—
255(C2×D3)⋊C4—28no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258C4⋊Dic3—22no—
259C4⋊Dic3—24no—
260C4⋊Dic3—28no—
261C4⋊Dic3C2414yes—
262Dic3⋊C4—24no—
263Dic3⋊C4—24no—
264Dic3⋊C4—28no—
265C4×Dic3—28no—
266C4×Dic3C2×D412yes—
267C4×Dic3C2×D414yes—
268C4×Dic3C2×D414yes—
269C4×Dic3C2412yes—
270C2×C3⋊C8—24no—
271C2×C3⋊C8C2414yes—
272D3×C8—216no—
273C2×#41—34no—
274C2×#41—34no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—68no—
278C2×#41—68no—
279C22×Q8—21no—
280C22×Q8—31no—
281C22×Q8—61no—
282C22×Q8C22×D311yescharacteristic
283C22×D4—31no—
284C22×D4—61no—
285C22×D4—61no—
286C22×D4—61no—
287C22×D4—61no—
288C22×D4C22×D311yescharacteristic
289C23×C4—31no—
290C23×C4—31no—
291C23×C4—31no—
292C23×C4—61no—
293C23×C4—61no—
294C22×C8—32no—
295C22×C8—38no—
296C22×C8—68no—
297C22×C8C22×D312yes—
298C4⋊Q8—22no—
299C4⋊Q8—34no—
300C4⋊Q8—34no—
301C4⋊Q8—62no—
302C4⋊Q8—62no—
303C4⋊Q8C22×D314yes—
304#127—34no—
305#127—62no—
306#125—38no—
307#125—64no—
308#124—22no—
309#124—34no—
310#124—62no—
311#124C22×D314yes—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314#121—68no—
315#121—68no—
316C4×Q8—68no—
317C4×Q8—68no—
318C4×D4—68no—
319C4×D4—68no—
320#117—38no—
321#117—64no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—62no—
327C2×C4⋊C4—62no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4C22×D312yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—62no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—34no—
339C2×C42—34no—
340C2×C42—62no—
341C2×C42C22×D311yescharacteristic
342#103—38no—
343#103—38no—
344#103—38no—
345#103C22×D318yes—
346#102—38no—
347#102—38no—
348#102—38no—
349#102C22×D318yes—
350#97—38no—
351#97—34no—
352#97C22×D314yes—
353C23×C3—22no—
354C23×C3—42no—
355C23×C3—42no—
356C23×C3—42no—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3—42no—
362C22×D3—42no—
363C22×D3—42no—
364C22×D3#12414yes—
365C22×D3C22×D412yes—
366C3×Q8—24no—
367C3×Q8—48no—
368C3×Q8—42no—
369C3×D4—24no—
370C3×D4—42no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—42no—
376C2×C3×C4—44no—
377C2×C3×C4C22×D412yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4114yes—
381C2×C3×C4C2511yescharacteristic
382#67—48no—
383#67—48no—
384#67—48no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—24no—
388C2×Dic3—22no—
389C2×Dic3—42no—
390C2×Dic3—44no—
391C2×Dic3—48no—
392C2×Dic3#12414yes—
393C2×Dic3C22×D412yes—
394C2×Dic3C22×D412yes—
395C2×Dic3C22×D412yes—
396D12—24no—
397D12—48no—
398D12—42no—
399C4×D3—44no—
400C4×D3—44no—
401C4×D3—48no—
402C4×D3#12418yes—
403C4×D3C2×#4118yes—
404Dic6—24no—
405Dic6—48no—
406Dic6—42no—
407C3×C8—24no—
408C3⋊C8—24no—
409C24—21no—
410C24—31no—
411C24—61no—
412#41—616no—
413#41—616no—
414#41—128no—
415#41—128no—
416C2×Q8—24no—
417C2×Q8—21no—
418C2×Q8—24no—
419C2×Q8—32no—
420C2×Q8—44no—
421C2×Q8—64no—
422C2×Q8—61no—
423C2×Q8—64no—
424C2×Q8—124no—
425C2×Q8C23×D312yes—
426C2×D4—21no—
427C2×D4—24no—
428C2×D4—32no—
429C2×D4—64no—
430C2×D4—64no—
431C2×D4—64no—
432C2×D4—64no—
433C2×D4—61no—
434C2×D4—64no—
435C2×D4—124no—
436C2×D4—124no—
437C2×D4—124no—
438C2×D4—124no—
439C2×D4C23×D312yes—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—32no—
444C22×C4—38no—
445C22×C4—32no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—34no—
449C22×C4—34no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—124no—
457C22×C4—124no—
458C22×C4D3×D411yescharacteristic
459C22×C4D3×D411yescharacteristic
460C22×C4C23×D311yescharacteristic
461C2×C8—24no—
462C2×C8—34no—
463C2×C8—616no—
464C2×C8—64no—
465C2×C8C23×D314yes—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—34no—
469C4⋊C4—64no—
470C4⋊C4—64no—
471C4⋊C4—68no—
472C4⋊C4—68no—
473C4⋊C4—62no—
474C4⋊C4—64no—
475C4⋊C4C23×D314yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—68no—
479C22⋊C4—64no—
480C22⋊C4—64no—
481C42—34no—
482C42—34no—
483C42—32no—
484C42—32no—
485C42—68no—
486C42D3×D412yes—
487C42C23×D312yes—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—84no—
492C22×C3C2×#12412yes—
493C22×C3#57512yes—
494C22×C3#57512yes—
495C22×C3C23×D411yescharacteristic
496C2×D3—28no—
497C2×D3—22no—
498C2×D3—22no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—84no—
503C2×D3C2×#12414yes—
504C3×C4—22no—
505C3×C4—22no—
506C3×C4—42no—
507C3×C4—44no—
508C3×C4C2×#12412yes—
509C3×C4C22×#4112yes—
510Dic3—22no—
511Dic3—22no—
512Dic3—42no—
513Dic3—44no—
514Dic3C2×#12414yes—
515C23—22no—
516C23—34no—
517C23—32no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—62no—
522C23—62no—
523C23—62no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23C2×D3×D411yescharacteristic
528Q8—24no—
529Q8—48no—
530Q8—42no—
531Q8—64no—
532Q8—128no—
533Q8—122no—
534D4—24no—
535D4—42no—
536D4—64no—
537D4—128no—
538D4—128no—
539D4—128no—
540D4—128no—
541D4—122no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—34no—
547C2×C4—32no—
548C2×C4—38no—
549C2×C4—38no—
550C2×C4—32no—
551C2×C4—32no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—64no—
557C2×C4—62no—
558C2×C4—124no—
559C2×C4—124no—
560C2×C4—128no—
561C2×C4—128no—
562C2×C4—122no—
563C2×C4—124no—
564C2×C4C2×D3×D412yes—
565C2×C4C2×D3×D412yes—
566C2×C4C2×D3×D411yescharacteristic
567C2×C4D3×#4114yes—
568C2×C4C24×D311yescharacteristic
569C8—24no—
570C8—64no—
571C2×C3—82no—
572C2×C3#313514yescomplemented
573C2×C3C22×#12411yescharacteristic
574C2×C3C2×#57511yescharacteristic
575C2×C3C2×#57511yescharacteristic
576D3—24no—
577D3—82no—
578C22—34no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—68no—
583C22—62no—
584C22—62no—
585C22—84no—
586C22—122no—
587C22—122no—
588C22—122no—
589C22—244no—
590C22D3×#12412yes—
591C22#577312yes—
592C22#577712yes—
593C22C22×D3×D411yescharacteristic
594C4—22no—
595C4—22no—
596C4—34no—
597C4—42no—
598C4—44no—
599C4—62no—
600C4—62no—
601C4—122no—
602C4—124no—
603C4D3×#12412yes—
604C4C2×D3×#4112yes—
605C3C2×#313511yescharacteristic, complemented, Sylow
606C2—64no—
607C2—82no—
608C2—242no—
609C2#8167914yesdirect product factor
610C2C2×D3×#12411yescharacteristic
611C2C2×#577311yescharacteristic
612C2C2×#577711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC2×#5668C2
hypercenterC2×#489D3
hypocenterC3C2×#3135
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
3. C2×C3×#489C2semidirect
4. C2×#5089C2semidirect
5. C2×#5055C2semidirect
6. #81679C2direct (N × H)
7. C2×#4804C2semidirect
8. C2×#4797C2semidirect
9. C2×#4772C2semidirect
10. C2×#4764C2semidirect
11. C2×#4714C2semidirect
12. C2×D3×#97C2semidirect
24. #5728C22semidirect
38. C3×#489C22semidirect
47. #5089C22semidirect
48. #5055C22semidirect
52. #4804C22semidirect
53. #4797C22semidirect
54. #4772C22semidirect
55. #4764C22semidirect
56. #4714C22semidirect
57. D3×#97C22semidirect
66. C2×#489D3semidirect
179. #489C2×D3semidirect
572. C2×C3#3135semidirect
605. C3C2×#3135semidirect
609. C2#81679direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#56682
C2×C3×#4892
C2×#50892
C2×#50552
#816792
C2×#48042
C2×#47972
C2×#47722
C2×#47642
C2×#47142
C2×D3×#972
C2×#31353