Finite Group Databasebeta

← C2×#79683 #11672018

Subgroups — C2×#79683 #11672018

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×#5584C211yesmaximal, characteristic, complemented
2C2×D4×D12C211yesmaximal, characteristic
3C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
4C2×#5169C212yesmaximal, complemented
5C2×#5027C211yesmaximal, characteristic, complemented
6C2×#5007C211yesmaximal, characteristic, complemented
7#79683C2116yesmaximal, direct product factor
8C2×#4866C212yesmaximal, complemented
9C2×#4812C212yesmaximal, complemented
10C2×#4770C212yesmaximal, complemented
11C2×#4694C211yesmaximal, characteristic, complemented
12C2×#4688C211yesmaximal, characteristic, complemented
13C2×#3324—31nomaximal, Sylow
14C22×D3×D4C2212yes—
15C23×D12—21no—
16C22×C3×SD16C2211yescharacteristic
17C2×C3×C4×Q8C2211yescharacteristic, complemented
18C2×C3×C4×D4C2211yescharacteristic
19C2×#946C2212yescomplemented
20C22×#215C2212yescomplemented
21C2×#930—21no—
22C22×ΓL(1,𝔽25)C2212yescomplemented
23#5584—24no—
24#5584C2218yescomplemented
25D4×D12—24no—
26D4×D12C2218yes—
27C2×#898—22no—
28C2×#895C2212yes—
29C2×#892C2212yescomplemented
30C2×#882—22no—
31C2×#875—21no—
32C2×#874C2211yescharacteristic
33C2×C4×D12C2211yescharacteristic
34C2×#869C2211yescharacteristic, complemented
35C2×C4×Dic6—21no—
36C3×C4×SD16C2218yescomplemented
37C2×C3×#106C2211yescharacteristic, complemented
38C2×C3×#103C2211yescharacteristic, complemented
39C2×C3×#102C2211yescharacteristic
40C2×C3×C4×C8C2211yescharacteristic, complemented
41C2×#832C2212yes—
42#5169C22116yescomplemented
43C2×#821C2212yescomplemented
44C2×#821C2212yescomplemented
45C2×(C2×D3)⋊C8C2212yescomplemented
46#5027C2218yescomplemented
47#5007C2218yescomplemented
48C2×#809C2212yescomplemented
49C2×#807C2212yescomplemented
50C2×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
51#4866C22116yescomplemented
52#4812C22116yescomplemented
53#4770C22116yescomplemented
54#4694C2218yescomplemented
55#4688C2218yescomplemented
56C2×#556—31no—
57C2×D42—31no—
58#3324—316no—
59C2×#486—31no—
60C2×#482—32no—
61C2×#474—31no—
62C2×#468—32no—
63C2×#460—31no—
64C2×#450—32no—
65C2×C4×SD16D311yescharacteristic, complemented
66C2×#436—31no—
67C2×#418—32no—
68C24×D3—22no—
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C22×#67—22no—
73C2×D3×D4—28no—
74C2×D3×D4—24no—
75C2×D3×D4—44no—
76C2×D3×D4C2318yes—
77C22×D12—22no—
78C22×D12—21no—
79C22×D12—21no—
80C22×D12—28no—
81C22×D12—48no—
82C22×D12D411yescharacteristic
83C22×D12C2312yes—
84C22×C4×D3C2312yes—
85C22×Dic6—21no—
86C2×C3×SD16—24no—
87C2×C3×SD16C2314yes—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4×Q8—22no—
91C3×C4×Q8C2314yescomplemented
92C3×C4×D4—22no—
93C3×C4×D4C2314yes—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100#946—24no—
101#946C2318yescomplemented
102C2×#215—44no—
103C2×#215C2318yescomplemented
104#930—28no—
105C2×(C2×D3)⋊C4—22no—
106C2×(C2×D3)⋊C4—22no—
107C2×(C2×D3)⋊C4—22no—
108C2×(C2×D3)⋊C4D412yes—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3C2312yes—
112C2×Dic3⋊C4—22no—
113C2×C4×Dic3—22no—
114C22×C3⋊C8C2312yes—
115C2×ΓL(1,𝔽25)—44no—
116C2×ΓL(1,𝔽25)C2318yescomplemented
117#898—216no—
118#895—24no—
119#895C2318yes—
120#892—24no—
121#892C2318yescomplemented
122#882—216no—
123#875—28no—
124#874—22no—
125#874C2314yes—
126C4×D12D414yes—
127C4×D12C2314yes—
128#869—22no—
129#869C2314yescomplemented
130C4×Dic6—28no—
131C3×#106C2314yescomplemented
132C3×#103C2314yescomplemented
133C3×#102C2314yes—
134C3×C4×C8C2314yescomplemented
135#832C2318yes—
136#821C2318yescomplemented
137#821C2318yescomplemented
138(C2×D3)⋊C8C2318yescomplemented
139#809C2318yescomplemented
140#807C2318yescomplemented
141C4⋊(C3⋊C8)C2314yescomplemented
142C23×D4—32no—
143C23×D4—61no—
144C22×SD16—32no—
145C22×SD16C2×D311yescharacteristic
146C22×M16—32no—
147#556—38no—
148#556—64no—
149D42—38no—
150D42—64no—
151C2×C4⋊Q8—31no—
152C2×#127—31no—
153C2×#126—62no—
154C2×#124—61no—
155C2×C22⋊Q8—32no—
156C2×#121—32no—
157C2×#121—61no—
158C2×#120—62no—
159C2×C4×Q8—61no—
160C2×C4×Q8C2×D311yescharacteristic, complemented
161C2×C4×D4—31no—
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×SD16C2×D318yescomplemented
171#436—38no—
172C2×#107—32no—
173C2×#106C2×D311yescharacteristic, complemented
174C2×C4⋊C8—31no—
175#418—316no—
176C2×#103C2×D311yescharacteristic, complemented
177C2×#102—32no—
178C2×#102—32no—
179C2×#102—32no—
180C2×#102—32no—
181C2×#102C2×D311yescharacteristic
182C2×C22⋊C8—32no—
183C2×C4×C8C2×D311yescharacteristic, complemented
184C24×C3—21no—
185C23×D3—22no—
186C23×D3—28no—
187C23×D3—48no—
188C23×D3—48no—
189C23×D3—84no—
190C23×D3C2×D412yes—
191C2×C3×Q8—21no—
192C2×C3×Q8—22no—
193C2×C3×Q8—22no—
194C2×C3×Q8C2412yes—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4#4114yes—
198C2×C3×D4C2412yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—24no—
203C22×C3×C4—44no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C2×#67—28no—
208C2×#67—48no—
209C22×Dic3—22no—
210C22×Dic3C2×D412yes—
211D3×D4—216no—
212D3×D4—48no—
213D3×D4—416no—
214C2×D12—24no—
215C2×D12—24no—
216C2×D12—24no—
217C2×D12—28no—
218C2×D12—22no—
219C2×D12—24no—
220C2×D12—24no—
221C2×D12—432no—
222C2×D12—48no—
223C2×D12—44no—
224C2×D12—82no—
225C2×D12C2×D414yes—
226C2×D12C2414yes—
227C2×C4×D3—24no—
228C2×C4×D3—24no—
229C2×C4×D3C2×D414yes—
230C2×C4×D3C2414yes—
231C2×Dic6—24no—
232C2×Dic6—44no—
233C3×SD16—28no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8C2412yes—
237C2×C3×C8C2412yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—24no—
241C3×C4⋊C4C2412yes—
242C3×C4⋊C4C2412yes—
243C3×C22⋊C4—24no—
244C3×C42—24no—
245C3×C42C2×D412yes—
246C3×C42C2412yes—
247#215—48no—
248(C2×D3)⋊C4—28no—
249(C2×D3)⋊C4—28no—
250(C2×D3)⋊C4—28no—
251(C2×D3)⋊C4C2×D418yes—
252C4⋊Dic3—22no—
253C4⋊Dic3—24no—
254C4⋊Dic3C2×D414yes—
255C4⋊Dic3C2414yes—
256Dic3⋊C4—28no—
257C4×Dic3—28no—
258C2×C3⋊C8—24no—
259C2×C3⋊C8C2414yes—
260ΓL(1,𝔽25)—48no—
261C25—62no—
262C22×Q8—61no—
263C22×Q8C22×D311yescharacteristic
264C22×D4—38no—
265C22×D4—31no—
266C22×D4—32no—
267C22×D4—68no—
268C22×D4—62no—
269C22×D4—64no—
270C22×D4—62no—
271C22×D4—68no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4—128no—
275C22×D4—124no—
276C22×D4C22×D311yescharacteristic
277C23×C4—21no—
278C23×C4—32no—
279C2×SD16—24no—
280C2×SD16—38no—
281C2×SD16—124no—
282C2×SD16C22×D314yes—
283C2×M16—38no—
284C2×M16—124no—
285C22×C8—32no—
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288C4⋊Q8—34no—
289C4⋊Q8—62no—
290#127—34no—
291#127—62no—
292#126—616no—
293#124—68no—
294C22⋊Q8—38no—
295C22⋊Q8—64no—
296#121—38no—
297#121—64no—
298#121—68no—
299#120—616no—
300C4×Q8—22no—
301C4×Q8—68no—
302C4×Q8C22×D314yescomplemented
303C4×D4—22no—
304C4×D4—34no—
305C4×D4—34no—
306C4×D4C22×D314yes—
307#117—38no—
308#117—64no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—62no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—32no—
318C2×C22⋊C4—62no—
319C2×C22⋊C4—62no—
320C2×C22⋊C4—62no—
321C2×C42—21no—
322C2×C42—62no—
323C2×C42C22×D311yescharacteristic
324#107—38no—
325#106C22×D314yescomplemented
326C4⋊C8—34no—
327#103C22×D314yescomplemented
328#102—38no—
329#102—38no—
330#102—38no—
331#102—38no—
332#102C22×D314yes—
333C22⋊C8—38no—
334C4×C8C22×D314yescomplemented
335C23×C3—24no—
336C23×C3—22no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3C22×D411yescharacteristic
340C22×D3—24no—
341C22×D3—24no—
342C22×D3—24no—
343C22×D3—24no—
344C22×D3—416no—
345C22×D3—416no—
346C22×D3—44no—
347C22×D3—44no—
348C22×D3—44no—
349C22×D3—88no—
350C22×D3—832no—
351C22×D3C22×D414yes—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—42no—
355C3×D4—24no—
356C3×D4—22no—
357C3×D4C2×#4114yes—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—48no—
366C2×C3×C4AGL(1,ℤ/8)14yes—
367C2×C3×C4C22×D412yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D412yes—
370C2×C3×C4C2×#4112yes—
371C2×C3×C4H(5,𝔽2)12yes—
372C2×C3×C4C2511yescharacteristic
373#67—416no—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—44no—
377C2×Dic3C22×D414yes—
378D12—28no—
379D12—24no—
380D12—24no—
381D12—416no—
382D12—44no—
383D12—48no—
384D12—84no—
385C4×D3—28no—
386C4×D3—24no—
387C4×D3—24no—
388Dic6—48no—
389C3×C8—22no—
390C3×C8—22no—
391C3⋊C8—24no—
392C24—21no—
393C24—32no—
394C24—68no—
395C24—62no—
396C24—128no—
397C24—128no—
398C24—244no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—64no—
403C2×Q8—124no—
404C2×Q8C23×D312yes—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—34no—
408C2×D4—34no—
409C2×D4—68no—
410C2×D4—68no—
411C2×D4—616no—
412C2×D4—64no—
413C2×D4—64no—
414C2×D4—64no—
415C2×D4—62no—
416C2×D4—64no—
417C2×D4—64no—
418C2×D4—1232no—
419C2×D4—128no—
420C2×D4—128no—
421C2×D4—1216no—
422C2×D4—128no—
423C2×D4—124no—
424C2×D4—242no—
425C2×D4#23514yes—
426C2×D4C23×D312yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—32no—
432C22×C4—34no—
433C22×C4—34no—
434C22×C4—44no—
435C22×C4—64no—
436C22×C4—64no—
437C22×C4—62no—
438C22×C4D3×D411yescharacteristic
439C22×C4D3×D411yescharacteristic
440C22×C4C23×D311yescharacteristic
441SD16—28no—
442SD16—128no—
443M16—128no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—34no—
447C2×C8—64no—
448C2×C8C23×D312yes—
449C2×C8C23×D312yes—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—34no—
454C4⋊C4—34no—
455C4⋊C4—68no—
456C4⋊C4—62no—
457C4⋊C4—64no—
458C4⋊C4C23×D312yes—
459C4⋊C4C23×D312yes—
460C22⋊C4—24no—
461C22⋊C4—38no—
462C22⋊C4—68no—
463C22⋊C4—68no—
464C22⋊C4—68no—
465C42—24no—
466C42—68no—
467C42D3×D412yes—
468C42C23×D312yes—
469C22×C3—24no—
470C22×C3—42no—
471C22×C3—48no—
472C22×C3—42no—
473C22×C3#54612yes—
474C22×C3C2×AGL(1,ℤ/8)12yes—
475C22×C3#57712yes—
476C22×C3C23×D411yescharacteristic
477C2×D3—24no—
478C2×D3—24no—
479C2×D3—24no—
480C2×D3—48no—
481C2×D3—44no—
482C2×D3—44no—
483C2×D3—44no—
484C2×D3—832no—
485C2×D3—88no—
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4#54612yes—
491C3×C4C2×AGL(1,ℤ/8)14yes—
492C3×C4C22×#4111yescharacteristic
493C3×C4C2×H(5,𝔽2)11yescharacteristic
494Dic3—24no—
495Dic3—44no—
496C23—22no—
497C23—24no—
498C23—34no—
499C23—42no—
500C23—42no—
501C23—64no—
502C23—64no—
503C23—64no—
504C23—64no—
505C23—1216no—
506C23—1216no—
507C23—124no—
508C23—124no—
509C23—124no—
510C23—248no—
511C23—2432no—
512C23C2×D3×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—42no—
516Q8—128no—
517D4—22no—
518D4—24no—
519D4—64no—
520D4—64no—
521D4—68no—
522D4—1216no—
523D4—1216no—
524D4—128no—
525D4—124no—
526D4—244no—
527D4C2×#23514yes—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—34no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—48no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—68no—
541C2×C4—64no—
542C2×C4—124no—
543C2×C4#91414yes—
544C2×C4C2×#23512yes—
545C2×C4C2×D3×D412yes—
546C2×C4C2×D3×D412yes—
547C2×C4C2×D3×D411yescharacteristic
548C2×C4#100912yes—
549C2×C4C24×D311yescharacteristic
550C8—22no—
551C8—22no—
552C8—64no—
553C2×C3—44no—
554C2×C3#332414yescomplemented
555C2×C3C2×#54611yescharacteristic
556C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
557C2×C3C2×#57711yescharacteristic
558D3—44no—
559D3—84no—
560C22—24no—
561C22—48no—
562C22—42no—
563C22—42no—
564C22—64no—
565C22—64no—
566C22—64no—
567C22—128no—
568C22—124no—
569C22—124no—
570C22—124no—
571C22—2432no—
572C22—248no—
573C22#555012yes—
574C22C2×#91412yes—
575C22#577012yes—
576C22C22×D3×D411yescharacteristic
577C4—22no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—64no—
582C4—124no—
583C4#555012yes—
584C4C2×#91414yes—
585C4C22×#23511yescharacteristic
586C4C2×#100911yescharacteristic
587C3C2×#332411yescharacteristic, complemented, Sylow
588C2—44no—
589C2—124no—
590C2—244no—
591C2#7968314yesdirect product factor
592C2C2×#555011yescharacteristic
593C2C22×#91411yescharacteristic
594C2C2×#577011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC2×D4×D12C2
hypercenterC2×C4×SD16D3
hypocenterC3C2×#3324
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#5584C2semidirect
3. C2×C3×C4×SD16C2semidirect
4. C2×#5169C2semidirect
5. C2×#5027C2semidirect
6. C2×#5007C2semidirect
7. #79683C2direct (N × H)
8. C2×#4866C2semidirect
9. C2×#4812C2semidirect
10. C2×#4770C2semidirect
11. C2×#4694C2semidirect
12. C2×#4688C2semidirect
17. C2×C3×C4×Q8C22semidirect
19. C2×#946C22semidirect
20. C22×#215C22semidirect
22. C22×ΓL(1,𝔽25)C22semidirect
24. #5584C22semidirect
29. C2×#892C22semidirect
34. C2×#869C22semidirect
36. C3×C4×SD16C22semidirect
37. C2×C3×#106C22semidirect
38. C2×C3×#103C22semidirect
40. C2×C3×C4×C8C22semidirect
42. #5169C22semidirect
43. C2×#821C22semidirect
44. C2×#821C22semidirect
45. C2×(C2×D3)⋊C8C22semidirect
46. #5027C22semidirect
47. #5007C22semidirect
48. C2×#809C22semidirect
49. C2×#807C22semidirect
50. C2×C4⋊(C3⋊C8)C22semidirect
51. #4866C22semidirect
52. #4812C22semidirect
53. #4770C22semidirect
54. #4694C22semidirect
55. #4688C22semidirect
65. C2×C4×SD16D3semidirect
91. C3×C4×Q8C23semidirect
101. #946C23semidirect
103. C2×#215C23semidirect
116. C2×ΓL(1,𝔽25)C23semidirect
121. #892C23semidirect
129. #869C23semidirect
131. C3×#106C23semidirect
132. C3×#103C23semidirect
134. C3×C4×C8C23semidirect
136. #821C23semidirect
137. #821C23semidirect
138. (C2×D3)⋊C8C23semidirect
139. #809C23semidirect
140. #807C23semidirect
141. C4⋊(C3⋊C8)C23semidirect
160. C2×C4×Q8C2×D3semidirect
170. C4×SD16C2×D3semidirect
173. C2×#106C2×D3semidirect
176. C2×#103C2×D3semidirect
183. C2×C4×C8C2×D3semidirect
302. C4×Q8C22×D3semidirect
325. #106C22×D3semidirect
327. #103C22×D3semidirect
334. C4×C8C22×D3semidirect
554. C2×C3#3324semidirect
587. C3C2×#3324semidirect
591. C2#79683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55842
C2×D4×D122
C2×C3×C4×SD162
C2×#51692
C2×#50272
C2×#50072
#796832
C2×#48662
C2×#48122
C2×#47702
C2×#46942
C2×#46882
C2×#33243