Finite Group Databasebeta

← C2×#83486 #11672016

Subgroups — C2×#83486 #11672016

Order 768.

2566 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D42C211yesmaximal, characteristic
2C2×#5542C211yesmaximal, characteristic, complemented
3C2×#5218C212yesmaximal, complemented
4C2×#5215C212yesmaximal, complemented
5#83486C2116yesmaximal, direct product factor
6C2×#5075C211yesmaximal, characteristic, complemented
7C2×#5070C211yesmaximal, characteristic, complemented
8C2×#5042C212yesmaximal, complemented
9C2×#5035C212yesmaximal, complemented
10C2×C4×#214C211yesmaximal, characteristic, complemented
11C2×#5012C211yesmaximal, characteristic, complemented
12C2×#5007C211yesmaximal, characteristic, complemented
13C2×#3324—31nomaximal, Sylow
14C23×C3×D4—21no—
15C23×C3×D4C2212yes—
16C3×D42—24no—
17C3×D42C2218yes—
18C2×C3×#127C2211yescharacteristic
19C2×C3×#121—21no—
20C2×C3×#121C2212yes—
21C2×C3×#120—22no—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×#936—21no—
25C22×#214C2211yescharacteristic
26C22×#214C2212yescomplemented
27C2×#926C2212yescomplemented
28C2×#924C2212yescomplemented
29C22×#208C2212yescomplemented
30#5542—24no—
31#5542C2218yescomplemented
32C2×#887—21no—
33C2×#879—22no—
34C2×#869C2211yescharacteristic, complemented
35C2×C4×Dic6C2211yescharacteristic, complemented
36#5218C22116yescomplemented
37#5215C22116yescomplemented
38C2×#832C2211yescharacteristic
39C2×#832C2212yes—
40C2×#832C2212yescomplemented
41C2×#832C2212yescomplemented
42C2×#832C2212yescomplemented
43C2×#830C2212yescomplemented
44#5075C2218yescomplemented
45#5070C2218yescomplemented
46#5042C22116yescomplemented
47#5035C22116yescomplemented
48C4×#214C2218yescomplemented
49#5012C2218yescomplemented
50#5007C2218yescomplemented
51C2×#810C2211yescharacteristic, complemented
52C2×#808C2211yescharacteristic, complemented
53C2×#807C2212yescomplemented
54C2×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
55C2×C4×C3⋊C8C2211yescharacteristic, complemented
56C2×#556—31no—
57C2×D42D311yescharacteristic
58#3324—316no—
59C2×#486—31no—
60C2×#482—32no—
61C2×#474—31no—
62C2×#468—32no—
63C2×#460—31no—
64C2×#450—32no—
65C2×C4×SD16—31no—
66C2×#436—31no—
67C2×#418—32no—
68C25×C3—22no—
69C22×C3×D4—28no—
70C22×C3×D4—22no—
71C22×C3×D4—28no—
72C22×C3×D4—21no—
73C22×C3×D4—22no—
74C22×C3×D4—24no—
75C22×C3×D4—21no—
76C22×C3×D4—44no—
77C22×C3×D4—48no—
78C22×C3×D4D411yescharacteristic
79C22×C3×D4C2312yes—
80C22×C3×D4C2318yes—
81C22×C3×D4C2311yescharacteristic
82C23×C3×C4—21no—
83C23×C3×C4C2312yes—
84C22×Dic6—21no—
85C22×Dic6C2311yescharacteristic
86C3×#127—22no—
87C3×#127C2314yes—
88C3×#121—24no—
89C3×#121—28no—
90C3×#121C2318yes—
91C3×#120—216no—
92C3×C4×D4—22no—
93C3×C4×D4D414yes—
94C3×C4×D4C2314yes—
95C3×C4×D4C2314yes—
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4D412yes—
101C2×C3×C42C2311yescharacteristic
102C2×#217—22no—
103C2×#217—22no—
104#936—28no—
105C2×#214—24no—
106C2×#214—44no—
107C2×#214C2314yes—
108C2×#214C2318yescomplemented
109#926—24no—
110#926C2318yescomplemented
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2312yes—
114#924—24no—
115#924C2318yescomplemented
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—22no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—22no—
120C2×#208—44no—
121C2×#208C2318yescomplemented
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2312yes—
125#887—28no—
126#879—216no—
127#869—22no—
128#869C2314yescomplemented
129C4×Dic6—22no—
130C4×Dic6C2314yescomplemented
131#832C2314yes—
132#832C2318yes—
133#832C2318yescomplemented
134#832C2318yescomplemented
135#832C2318yescomplemented
136#830C2318yescomplemented
137#810C2314yescomplemented
138#808C2314yescomplemented
139#807C2318yescomplemented
140C4⋊(C3⋊C8)C2314yescomplemented
141C4×C3⋊C8C2314yescomplemented
142C23×D4—21no—
143C23×D4C2×D312yes—
144C22×SD16—31no—
145C22×SD16—32no—
146C22×M16—32no—
147#556—38no—
148#556—64no—
149D42—24no—
150D42C2×D318yes—
151C2×C4⋊Q8—31no—
152C2×#127C2×D311yescharacteristic
153C2×#126—62no—
154C2×#124—61no—
155C2×C22⋊Q8—32no—
156C2×#121—21no—
157C2×#121C2×D312yes—
158C2×#120—22no—
159C2×C4×Q8—31no—
160C2×C4×Q8—61no—
161C2×C4×D4C2×D311yescharacteristic
162C2×C4×D4C2×D311yescharacteristic
163C2×#117—32no—
164#486—38no—
165#482—316no—
166#474—38no—
167#468—316no—
168#460—38no—
169#450—316no—
170C4×SD16—38no—
171#436—38no—
172C2×#107—32no—
173C2×#106—31no—
174C2×C4⋊C8—31no—
175#418—316no—
176C2×#103—31no—
177C2×#102—32no—
178C2×#102—32no—
179C2×#102—32no—
180C2×#102—31no—
181C2×#102—32no—
182C2×C22⋊C8—32no—
183C2×C4×C8—31no—
184C24×C3—21no—
185C24×C3—22no—
186C24×C3—28no—
187C24×C3—48no—
188C24×C3—48no—
189C24×C3—84no—
190C24×C3C2×D412yes—
191C2×C3×D4—216no—
192C2×C3×D4—28no—
193C2×C3×D4—24no—
194C2×C3×D4—21no—
195C2×C3×D4—28no—
196C2×C3×D4—22no—
197C2×C3×D4—24no—
198C2×C3×D4—24no—
199C2×C3×D4—24no—
200C2×C3×D4—22no—
201C2×C3×D4—24no—
202C2×C3×D4—48no—
203C2×C3×D4—416no—
204C2×C3×D4—48no—
205C2×C3×D4—432no—
206C2×C3×D4—48no—
207C2×C3×D4—44no—
208C2×C3×D4—82no—
209C2×C3×D4C2×D414yes—
210C2×C3×D4#4114yes—
211C2×C3×D4C2414yes—
212C2×C3×D4C2412yes—
213C22×C3×C4—24no—
214C22×C3×C4—24no—
215C22×C3×C4—21no—
216C22×C3×C4—24no—
217C22×C3×C4—44no—
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D414yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2414yes—
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—22no—
227C2×Dic6—22no—
228C2×Dic6—22no—
229C2×Dic6—21no—
230C2×Dic6—24no—
231C2×Dic6—44no—
232C2×Dic6C2412yes—
233C3×C4⋊C4—21no—
234C3×C4⋊C4C2×D414yes—
235C3×C4⋊C4C2412yes—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—28no—
238C3×C22⋊C4C2×D418yes—
239C3×C42C2×D412yes—
240C3×C42C2412yes—
241#217—28no—
242#217—28no—
243#214—28no—
244#214—48no—
245C4⋊Dic3—21no—
246C4⋊Dic3—22no—
247C4⋊Dic3—24no—
248C4⋊Dic3C2414yes—
249C4⋊Dic3C2412yes—
250Dic3⋊C4—24no—
251Dic3⋊C4—28no—
252C4×Dic3—24no—
253C4×Dic3—28no—
254#208—48no—
255C2×C3⋊C8—22no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—24no—
258C2×C3⋊C8C2412yes—
259C2×C3⋊C8C2412yes—
260C2×C3⋊C8C2414yes—
261C25—22no—
262C22×Q8—31no—
263C22×Q8—61no—
264C22×D4—28no—
265C22×D4—22no—
266C22×D4—28no—
267C22×D4—22no—
268C22×D4—21no—
269C22×D4—24no—
270C22×D4—21no—
271C22×D4—44no—
272C22×D4—48no—
273C22×D4#6711yescharacteristic
274C22×D4C22×D312yes—
275C22×D4C22×D318yes—
276C22×D4C22×D311yescharacteristic
277C23×C4—21no—
278C23×C4C22×D312yes—
279C2×SD16—34no—
280C2×SD16—38no—
281C2×SD16—64no—
282C2×SD16—124no—
283C2×M16—38no—
284C2×M16—124no—
285C22×C8—31no—
286C22×C8—32no—
287C22×C8—31no—
288C4⋊Q8—34no—
289C4⋊Q8—62no—
290#127—22no—
291#127C22×D314yes—
292#126—616no—
293#124—68no—
294C22⋊Q8—38no—
295C22⋊Q8—64no—
296#121—24no—
297#121—28no—
298#121C22×D318yes—
299#120—216no—
300C4×Q8—34no—
301C4×Q8—68no—
302C4×Q8—62no—
303C4×D4—22no—
304C4×D4#6714yes—
305C4×D4C22×D314yes—
306C4×D4C22×D314yes—
307#117—38no—
308#117—64no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—62no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4#6711yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—62no—
319C2×C22⋊C4—62no—
320C2×C22⋊C4#6712yes—
321C2×C42—62no—
322C2×C42—61no—
323C2×C42C22×D311yescharacteristic
324#107—38no—
325#106—34no—
326C4⋊C8—34no—
327#103—34no—
328#102—34no—
329#102—38no—
330#102—38no—
331#102—38no—
332#102—38no—
333C22⋊C8—38no—
334C4×C8—34no—
335C23×C3—24no—
336C23×C3—24no—
337C23×C3—24no—
338C23×C3—24no—
339C23×C3—22no—
340C23×C3—24no—
341C23×C3—416no—
342C23×C3—416no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3—44no—
346C23×C3—44no—
347C23×C3—44no—
348C23×C3—88no—
349C23×C3—832no—
350C23×C3C22×D414yes—
351C23×C3C22×D411yescharacteristic
352C3×D4—24no—
353C3×D4—28no—
354C3×D4—24no—
355C3×D4—22no—
356C3×D4—24no—
357C3×D4—416no—
358C3×D4—416no—
359C3×D4—44no—
360C3×D4—48no—
361C3×D4—84no—
362C3×D4C2×#4114yes—
363C2×C3×C4—24no—
364C2×C3×C4—22no—
365C2×C3×C4—28no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—22no—
369C2×C3×C4—42no—
370C2×C3×C4—48no—
371C2×C3×C4AGL(1,ℤ/8)14yes—
372C2×C3×C4C22×D414yes—
373C2×C3×C4C22×D412yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D412yes—
376C2×C3×C4C2×#4112yes—
377C2×C3×C4H(5,𝔽2)12yes—
378C2×C3×C4C2511yescharacteristic
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—24no—
382C2×Dic3—42no—
383C2×Dic3—44no—
384C2×Dic3—42no—
385Dic6—22no—
386Dic6—22no—
387Dic6—42no—
388Dic6—48no—
389C3⋊C8—22no—
390C3⋊C8—22no—
391C3⋊C8—24no—
392C24—21no—
393C24—22no—
394C24—28no—
395C24—48no—
396C24—48no—
397C24—84no—
398C24C2×#6712yes—
399C2×Q8—32no—
400C2×Q8—61no—
401C2×Q8—64no—
402C2×Q8—62no—
403C2×Q8—62no—
404C2×Q8—124no—
405C2×D4—216no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—48no—
417C2×D4—416no—
418C2×D4—48no—
419C2×D4—432no—
420C2×D4—48no—
421C2×D4—44no—
422C2×D4—82no—
423C2×D4#23714yes—
424C2×D4C2×#6714yes—
425C2×D4C23×D314yes—
426C2×D4C23×D312yes—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—44no—
432C22×C4—61no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4C2×#6714yes—
436C22×C4C2×#6712yes—
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C23×D314yes—
440C22×C4C23×D311yescharacteristic
441SD16—68no—
442SD16—128no—
443M16—128no—
444C2×C8—32no—
445C2×C8—32no—
446C2×C8—34no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—64no—
450C4⋊C4—21no—
451C4⋊C4—32no—
452C4⋊C4—34no—
453C4⋊C4—61no—
454C4⋊C4—62no—
455C4⋊C4—64no—
456C4⋊C4—68no—
457C4⋊C4—64no—
458C4⋊C4C2×#6714yes—
459C4⋊C4C23×D312yes—
460C22⋊C4—24no—
461C22⋊C4—28no—
462C22⋊C4—68no—
463C22⋊C4—68no—
464C22⋊C4C2×#6718yes—
465C42—68no—
466C42—64no—
467C42C2×#6712yes—
468C42C23×D312yes—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—42no—
474C22×C3—48no—
475C22×C3—48no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3—44no—
479C22×C3—44no—
480C22×C3—832no—
481C22×C3—88no—
482C22×C3#54612yes—
483C22×C3C2×AGL(1,ℤ/8)12yes—
484C22×C3#57712yes—
485C22×C3C23×D411yescharacteristic
486C3×C4—24no—
487C3×C4—22no—
488C3×C4—42no—
489C3×C4#54612yes—
490C3×C4C2×AGL(1,ℤ/8)14yes—
491C3×C4C22×#4111yescharacteristic
492C3×C4C2×H(5,𝔽2)11yescharacteristic
493Dic3—42no—
494Dic3—44no—
495Dic3—42no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—416no—
503C23—416no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—88no—
510C23—832no—
511C23C22×#6714yes—
512C23C22×#6711yescharacteristic
513Q8—62no—
514Q8—62no—
515Q8—128no—
516Q8—122no—
517D4—24no—
518D4—22no—
519D4—24no—
520D4—28no—
521D4—24no—
522D4—416no—
523D4—416no—
524D4—48no—
525D4—44no—
526D4—84no—
527D4C2×#23714yes—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—42no—
535C2×C4—48no—
536C2×C4—62no—
537C2×C4—64no—
538C2×C4—62no—
539C2×C4—124no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4#93214yes—
543C2×C4C2×#23712yes—
544C2×C4#100412yes—
545C2×C4C22×#6714yes—
546C2×C4C22×#6712yes—
547C2×C4C22×#6712yes—
548C2×C4C22×#6711yescharacteristic
549C2×C4C24×D311yescharacteristic
550C8—62no—
551C8—62no—
552C8—64no—
553C2×C3—44no—
554C2×C3—44no—
555C2×C3—84no—
556C2×C3#332414yescomplemented
557C2×C3C2×#54611yescharacteristic
558C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
559C2×C3C2×#57711yescharacteristic
560C22—24no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—48no—
565C22—48no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—832no—
572C22—88no—
573C22C2×#93212yes—
574C22#580612yes—
575C22#583612yes—
576C22C23×#6711yescharacteristic
577C4—24no—
578C4—22no—
579C4—42no—
580C4—124no—
581C4—122no—
582C4—122no—
583C4C2×#93214yes—
584C4#580612yes—
585C4C22×#23711yescharacteristic
586C4C2×#100411yescharacteristic
587C3C2×#332411yescharacteristic, complemented, Sylow
588C2—44no—
589C2—44no—
590C2—84no—
591C2#8348614yesdirect product factor
592C2C22×#93211yescharacteristic
593C2C2×#580611yescharacteristic
594C2C2×#583611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
Fitting, prime radicalC2×C3×D42C2
socleC23×C3C22×D4
hypercenterC2×D42D3
hypocenterC3C2×#3324
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
2. C2×#5542C2semidirect
3. C2×#5218C2semidirect
4. C2×#5215C2semidirect
5. #83486C2direct (N × H)
6. C2×#5075C2semidirect
7. C2×#5070C2semidirect
8. C2×#5042C2semidirect
9. C2×#5035C2semidirect
10. C2×C4×#214C2semidirect
11. C2×#5012C2semidirect
12. C2×#5007C2semidirect
26. C22×#214C22semidirect
27. C2×#926C22semidirect
28. C2×#924C22semidirect
29. C22×#208C22semidirect
31. #5542C22semidirect
34. C2×#869C22semidirect
35. C2×C4×Dic6C22semidirect
36. #5218C22semidirect
37. #5215C22semidirect
40. C2×#832C22semidirect
41. C2×#832C22semidirect
42. C2×#832C22semidirect
43. C2×#830C22semidirect
44. #5075C22semidirect
45. #5070C22semidirect
46. #5042C22semidirect
47. #5035C22semidirect
48. C4×#214C22semidirect
49. #5012C22semidirect
50. #5007C22semidirect
51. C2×#810C22semidirect
52. C2×#808C22semidirect
53. C2×#807C22semidirect
54. C2×C4⋊(C3⋊C8)C22semidirect
55. C2×C4×C3⋊C8C22semidirect
108. C2×#214C23semidirect
110. #926C23semidirect
115. #924C23semidirect
121. C2×#208C23semidirect
128. #869C23semidirect
130. C4×Dic6C23semidirect
133. #832C23semidirect
134. #832C23semidirect
135. #832C23semidirect
136. #830C23semidirect
137. #810C23semidirect
138. #808C23semidirect
139. #807C23semidirect
140. C4⋊(C3⋊C8)C23semidirect
141. C4×C3⋊C8C23semidirect
556. C2×C3#3324semidirect
587. C3C2×#3324semidirect
591. C2#83486direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D422
C2×#55422
C2×#52182
C2×#52152
#834862
C2×#50752
C2×#50702
C2×#50422
C2×#50352
C2×C4×#2142
C2×#50122
C2×#50072
C2×#33243