Finite Group Databasebeta

← C2×#81314 #11667226

Subgroups — C2×#81314 #11667226

Order 768.

2286 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5596C211yesmaximal, characteristic
2C22×#888C211yesmaximal, characteristic, complemented
3C2×C3×#481C211yesmaximal, characteristic, complemented
4C2×#5034C211yesmaximal, characteristic, complemented
5C2×#4968C211yesmaximal, characteristic, complemented
6#81314C2116yesmaximal, direct product factor
7C2×#4870C212yesmaximal, complemented
8C2×#4855C212yesmaximal, complemented
9C2×#4760C212yesmaximal, complemented
10C2×#4761C212yesmaximal, complemented
11C2×#4735C211yesmaximal, characteristic, complemented
12C2×#4718C211yesmaximal, characteristic, complemented
13C2×#3227—31nomaximal, Sylow
14C22×#237—21no—
15C23×Dic6—21no—
16C2×C3×#121C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#939—21no—
20C2×D4×Dic3C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×Dic3⋊C4—21no—
24C22×C4×Dic3C2211yescharacteristic
25#5596—24no—
26#5596C2218yes—
27C2×#893C2211yescharacteristic
28C2×#888—24no—
29C2×#888—21no—
30C2×#888—21no—
31C2×#888C2211yescharacteristic
32C2×#888C2212yes—
33C2×#888C2212yes—
34C2×#888C2218yescomplemented
35C2×#885—21no—
36C2×#67⋊C4—21no—
37C3×#481C2218yescomplemented
38C2×C3×#106C2212yes—
39C2×C3×#102C2212yes—
40C2×C3×C22⋊C8C2211yescharacteristic
41C2×#832C2212yes—
42C2×#830C2211yescharacteristic
43C2×#821C2212yes—
44C2×#817C2212yes—
45C2×Dic3⋊C8C2212yes—
46C2×C8×Dic3C2212yes—
47#5034C2218yescomplemented
48#4968C2218yescomplemented
49C2×#809C2212yes—
50C2×#808C2212yes—
51C2×#808C2212yes—
52#4870C22116yescomplemented
53#4855C22116yescomplemented
54#4761C22116yescomplemented
55#4760C22116yescomplemented
56#4735C2218yescomplemented
57#4718C2218yescomplemented
58C2×#532—31no—
59C22×C4⋊Q8—31no—
60#3227—316no—
61C2×#498—32no—
62C2×#486—32no—
63C2×#481—31no—
64C2×#481—31no—
65C2×#481—31no—
66C2×#481D311yescharacteristic, complemented
67C2×#476—32no—
68C2×#476—32no—
69C2×#431—31no—
70C22×C3×D4—21no—
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C22×#67—21no—
75C22×#67—21no—
76C22×#67—21no—
77C23×Dic3—21no—
78C23×Dic3—21no—
79C23×Dic3D411yescharacteristic
80C23×Dic3D411yescharacteristic
81C2×#237—28no—
82C2×#237—48no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×Dic6—22no—
86C22×Dic6—22no—
87C22×Dic6—28no—
88C22×Dic6—21no—
89C22×Dic6—48no—
90C22×C3×C8C2312yes—
91C3×#121—22no—
92C3×#121C2314yes—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2314yes—
98C2×C3×C22⋊C4—21no—
99C2×#217—21no—
100#940—22no—
101#940C2314yes—
102#939—28no—
103D4×Dic3—22no—
104D4×Dic3C2314yes—
105#930—22no—
106#930C2314yes—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—22no—
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2314yes—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—22no—
116C2×Dic3⋊C4—22no—
117C2×Dic3⋊C4—28no—
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D414yes—
121C2×C4×Dic3C2314yes—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2312yes—
125#893—22no—
126#893C2314yes—
127#888—24no—
128#888—24no—
129#888—28no—
130#888—28no—
131#888—22no—
132#888C2318yes—
133#888C2318yes—
134#888C2314yes—
135#885—28no—
136#67⋊C4—28no—
137C3×#106C2318yes—
138C3×#102C2318yes—
139C3×C22⋊C8C2314yes—
140#832C2318yes—
141#830C2314yes—
142#821C2318yes—
143#817C2318yes—
144Dic3⋊C8C2318yes—
145C8×Dic3C2318yes—
146#809C2318yes—
147#808C2318yes—
148#808C2318yes—
149C22×#41—61no—
150C23×Q8—61no—
151#532—38no—
152#532—64no—
153C2×C4⋊Q8—32no—
154C2×C4⋊Q8—31no—
155C2×C4⋊Q8—32no—
156C2×C4⋊Q8—38no—
157C2×C4⋊Q8—61no—
158C2×C4⋊Q8—61no—
159C2×C4⋊Q8—64no—
160C2×#127—31no—
161C2×#124—61no—
162C2×#121—31no—
163C2×#121—61no—
164C2×#121C2×D311yescharacteristic
165C2×C4×D4—31no—
166C2×C4×D4—31no—
167C2×C4×D4—61no—
168C22×C4⋊C4—31no—
169C22×C4⋊C4—61no—
170C22×C4⋊C4C2×D311yescharacteristic
171C22×C42—31no—
172#498—316no—
173#486—316no—
174#481—38no—
175#481—38no—
176#481—38no—
177#481C2×D318yescomplemented
178#476—316no—
179#476—316no—
180#431—38no—
181C2×#106—32no—
182C2×#106—32no—
183C2×#106—32no—
184C2×#106C2×D312yes—
185C2×C4⋊C8—32no—
186C2×#102—32no—
187C2×#102—32no—
188C2×#102—32no—
189C2×#102C2×D312yes—
190C2×C22⋊C8—31no—
191C2×C22⋊C8C2×D311yescharacteristic
192C2×C4×C8—32no—
193C24×C3—21no—
194C24×C3C2×D411yescharacteristic
195C23×D3—21no—
196C2×C3×D4—24no—
197C2×C3×D4—21no—
198C2×C3×D4—24no—
199C2×C3×D4—44no—
200C2×C3×D4C2412yes—
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—44no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2412yes—
209C22×C3×C4C2411yescharacteristic
210C2×#67—24no—
211C2×#67—24no—
212C2×#67—24no—
213C2×#67—44no—
214C2×#67—44no—
215C2×#67—44no—
216C22×Dic3—24no—
217C22×Dic3—24no—
218C22×Dic3—22no—
219C22×Dic3—21no—
220C22×Dic3—24no—
221C22×Dic3—44no—
222C22×Dic3—44no—
223C22×Dic3SD1618yes—
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D414yes—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D414yes—
229C22×Dic3C2×D411yescharacteristic
230#237—432no—
231C2×D12—24no—
232C2×D12—21no—
233C2×D12C2412yes—
234C2×C4×D3—24no—
235C2×C4×D3—44no—
236C2×Dic6—24no—
237C2×Dic6—28no—
238C2×Dic6—28no—
239C2×Dic6—48no—
240C2×Dic6—432no—
241C2×Dic6—44no—
242C2×Dic6—48no—
243C2×C3×C8—24no—
244C2×C3×C8C2414yes—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4C2412yes—
249C3×C4⋊C4C2414yes—
250C3×C22⋊C4—24no—
251#217—24no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—24no—
254C4⋊Dic3—24no—
255C4⋊Dic3—22no—
256C4⋊Dic3—21no—
257C4⋊Dic3C2412yes—
258C4⋊Dic3C2414yes—
259Dic3⋊C4—24no—
260Dic3⋊C4—28no—
261Dic3⋊C4—28no—
262C4×Dic3C2×D414yes—
263C4×Dic3C2×D414yes—
264C4×Dic3C2×D412yes—
265C4×Dic3C2×D412yes—
266C4×Dic3C2412yes—
267C4×Dic3C2412yes—
268C2×C3⋊C8—24no—
269C2×C3⋊C8C2414yes—
270C2×#41—68no—
271C2×#41—128no—
272C22×Q8—61no—
273C22×Q8—68no—
274C22×Q8—62no—
275C22×Q8—62no—
276C22×Q8—128no—
277C22×D4—21no—
278C22×D4—31no—
279C22×D4—61no—
280C22×D4—61no—
281C22×D4—61no—
282C22×D4C22×D311yescharacteristic
283C23×C4—21no—
284C23×C4—31no—
285C23×C4—31no—
286C23×C4—61no—
287C23×C4—61no—
288C23×C4—61no—
289C23×C4C22×D311yescharacteristic
290C22×C8—32no—
291C22×C8C22×D312yes—
292C4⋊Q8—38no—
293C4⋊Q8—38no—
294C4⋊Q8—34no—
295C4⋊Q8—64no—
296C4⋊Q8—68no—
297C4⋊Q8—68no—
298C4⋊Q8—64no—
299C4⋊Q8—62no—
300#127—34no—
301#127—62no—
302#124—68no—
303#121—22no—
304#121—34no—
305#121—68no—
306#121—62no—
307#121C22×D314yes—
308C4×D4—34no—
309C4×D4—34no—
310C4×D4—68no—
311C4×D4—62no—
312C4×D4—62no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—34no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—68no—
319C2×C4⋊C4—62no—
320C2×C4⋊C4—62no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—62no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D312yes—
325C2×C4⋊C4C22×D314yes—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—34no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—34no—
337#106—38no—
338#106—38no—
339#106—38no—
340#106C22×D318yes—
341C4⋊C8—38no—
342#102—38no—
343#102—38no—
344#102—38no—
345#102C22×D318yes—
346C22⋊C8—34no—
347C22⋊C8C22×D314yes—
348C4×C8—38no—
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—42no—
353C23×C3—42no—
354C23×C3—42no—
355C23×C3C2×SD1614yes—
356C23×C3C22×D412yes—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C22×D3—42no—
360C22×D3—42no—
361C22×D3—42no—
362C3×D4—24no—
363C3×D4—48no—
364C3×D4—42no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—48no—
371C2×C3×C4—42no—
372C2×C3×C4—44no—
373C2×C3×C4C22×D412yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C42−1+414yes—
376C2×C3×C4C2511yescharacteristic
377#67—48no—
378#67—48no—
379#67—48no—
380C2×Dic3—22no—
381C2×Dic3—28no—
382C2×Dic3—24no—
383C2×Dic3—22no—
384C2×Dic3—48no—
385C2×Dic3—42no—
386C2×Dic3—44no—
387C2×Dic3—48no—
388C2×Dic3C2×SD1614yes—
389C2×Dic3C2×SD1614yes—
390C2×Dic3C2×SD16116yes—
391C2×Dic3AGL(1,ℤ/8)14yes—
392C2×Dic3C22×D412yes—
393C2×Dic3C22×D412yes—
394C2×Dic3C22×D412yes—
395C2×Dic3C22×D412yes—
396D12—24no—
397D12—42no—
398C4×D3—48no—
399Dic6—416no—
400Dic6—48no—
401Dic6—416no—
402C3×C8—24no—
403C3⋊C8—24no—
404C24—21no—
405C24—61no—
406C24D3×D411yescharacteristic
407#41—1232no—
408C2×Q8—64no—
409C2×Q8—68no—
410C2×Q8—68no—
411C2×Q8—128no—
412C2×Q8—124no—
413C2×Q8—1232no—
414C2×Q8—128no—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—24no—
418C2×D4—32no—
419C2×D4—44no—
420C2×D4—64no—
421C2×D4—64no—
422C2×D4—64no—
423C2×D4—61no—
424C2×D4—64no—
425C2×D4—124no—
426C2×D4—124no—
427C2×D4—124no—
428C2×D4C23×D312yes—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—38no—
434C22×C4—34no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—34no—
439C22×C4—31no—
440C22×C4—44no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—124no—
448C22×C4—124no—
449C22×C4—124no—
450C22×C4D3×D412yes—
451C22×C4D3×D411yescharacteristic
452C22×C4C23×D312yes—
453C22×C4C23×D311yescharacteristic
454C2×C8—24no—
455C2×C8—34no—
456C2×C8—64no—
457C2×C8C23×D314yes—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—34no—
462C4⋊C4—32no—
463C4⋊C4—68no—
464C4⋊C4—68no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—62no—
468C4⋊C4—61no—
469C4⋊C4C23×D314yes—
470C4⋊C4C23×D312yes—
471C22⋊C4—24no—
472C22⋊C4—64no—
473C22⋊C4—64no—
474C22⋊C4—64no—
475C42—34no—
476C42—32no—
477C42—34no—
478C42—32no—
479C42—32no—
480C42—32no—
481C22×C3—22no—
482C22×C3—28no—
483C22×C3—22no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—84no—
488C22×C3#53612yes—
489C22×C3C22×SD1614yes—
490C22×C3C22×SD1612yes—
491C22×C3C2×AGL(1,ℤ/8)12yes—
492C22×C3C23×D411yescharacteristic
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—84no—
497C3×C4—22no—
498C3×C4—42no—
499C3×C4—44no—
500C3×C4#53612yes—
501C3×C4C2×2−1+412yes—
502Dic3—22no—
503Dic3—42no—
504Dic3—44no—
505Dic3C22×SD1614yes—
506Dic3C22×SD1614yes—
507Dic3C2×AGL(1,ℤ/8)14yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—62no—
515C23—122no—
516C23—122no—
517C23—122no—
518C23D3×SD1614yes—
519C23C2×D3×D412yes—
520C23C2×D3×D411yescharacteristic
521Q8—1216no—
522Q8—128no—
523Q8—1216no—
524D4—24no—
525D4—48no—
526D4—42no—
527D4—64no—
528D4—128no—
529D4—128no—
530D4—128no—
531D4—122no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—32no—
538C2×C4—32no—
539C2×C4—32no—
540C2×C4—34no—
541C2×C4—34no—
542C2×C4—316no—
543C2×C4—34no—
544C2×C4—32no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—68no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—64no—
552C2×C4—128no—
553C2×C4—128no—
554C2×C4—128no—
555C2×C4—122no—
556C2×C4—124no—
557C2×C4C2×D3×D412yes—
558C2×C4C2×D3×D411yescharacteristic
559C2×C4#101014yes—
560C2×C4C24×D311yescharacteristic
561C8—24no—
562C8—64no—
563C2×C3—24no—
564C2×C3—82no—
565C2×C3#322714yescomplemented
566C2×C3C2×#53611yescharacteristic
567C2×C3C23×SD1611yescharacteristic
568C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
569D3—82no—
570C22—22no—
571C22—28no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—84no—
577C22—122no—
578C22—122no—
579C22—122no—
580C22—244no—
581C22#564812yes—
582C22C2×D3×SD1614yes—
583C22C2×D3×SD1612yes—
584C22D3×AGL(1,ℤ/8)12yes—
585C22C22×D3×D411yescharacteristic
586C4—22no—
587C4—34no—
588C4—34no—
589C4—34no—
590C4—42no—
591C4—44no—
592C4—62no—
593C4—122no—
594C4—124no—
595C4#564812yes—
596C4C2×#101012yes—
597C3C2×#322711yescharacteristic, complemented, Sylow
598C2—24no—
599C2—82no—
600C2—242no—
601C2#8131414yesdirect product factor
602C2C2×#564811yescharacteristic
603C2C22×D3×SD1611yescharacteristic
604C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#481C2
socleC23×C3C22×D4
prime radicalC2×#5596C2
hypercenterC2×#481D3
hypocenterC3C2×#3227
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. C22×#888C2semidirect
3. C2×C3×#481C2semidirect
4. C2×#5034C2semidirect
5. C2×#4968C2semidirect
6. #81314C2direct (N × H)
7. C2×#4870C2semidirect
8. C2×#4855C2semidirect
9. C2×#4760C2semidirect
10. C2×#4761C2semidirect
11. C2×#4735C2semidirect
12. C2×#4718C2semidirect
34. C2×#888C22semidirect
37. C3×#481C22semidirect
47. #5034C22semidirect
48. #4968C22semidirect
52. #4870C22semidirect
53. #4855C22semidirect
54. #4761C22semidirect
55. #4760C22semidirect
56. #4735C22semidirect
57. #4718C22semidirect
66. C2×#481D3semidirect
177. #481C2×D3semidirect
565. C2×C3#3227semidirect
597. C3C2×#3227semidirect
601. C2#81314direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55962
C22×#8882
C2×C3×#4812
C2×#50342
C2×#49682
#813142
C2×#48702
C2×#48552
C2×#47602
C2×#47612
C2×#47352
C2×#47182
C2×#32273