Finite Group Databasebeta

← C2×#78563 #11664425

Subgroups — C2×#78563 #11664425

Order 768.

2262 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5525C211yesmaximal, characteristic
2C2×#5519C211yesmaximal, characteristic, complemented
3C2×C3×#405C211yesmaximal, characteristic, complemented
4C2×#5133C211yesmaximal, characteristic, complemented
5C2×#5131C211yesmaximal, characteristic, complemented
6C2×#5113C211yesmaximal, characteristic, complemented
7C22×#821C211yesmaximal, characteristic, complemented
8#78563C2116yesmaximal, direct product factor
9C2×#4713C212yesmaximal, complemented
10C2×#4714C212yesmaximal, complemented
11C2×#4697C212yesmaximal, complemented
12C2×#4694C212yesmaximal, complemented
13C2×#3136—31nomaximal, Sylow
14C22×#235C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C23×C3×C8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21#5525—24no—
22#5525C2218yes—
23#5519—24no—
24#5519C2218yescomplemented
25C2×#895—21no—
26C2×#895—21no—
27C2×#890—21no—
28C2×#888—21no—
29C2×#882—22no—
30C2×#878—22no—
31C2×#875C2212yes—
32C2×#874C2212yes—
33C2×#870C2212yescomplemented
34C2×#869C2212yescomplemented
35C3×#405C2218yescomplemented
36C2×C3×#97C2212yescomplemented
37C2×C3×C4×C8C2212yescomplemented
38#5133C2218yescomplemented
39#5131C2218yescomplemented
40#5113C2218yescomplemented
41C2×#821C2212yes—
42C2×#821C2212yes—
43C2×#821C2212yes—
44C2×#821C2212yes—
45C2×#821C2218yescomplemented
46C2×#821C2211yescharacteristic, complemented
47C2×#821C2211yescharacteristic, complemented
48C2×#821C2212yescomplemented
49C2×#816C2211yescharacteristic, complemented
50C2×#816C2211yescharacteristic, complemented
51C2×#816C2212yescomplemented
52#4714C22116yescomplemented
53#4713C22116yescomplemented
54#4697C22116yescomplemented
55#4694C22116yescomplemented
56C2×#544—31no—
57C2×#535—31no—
58#3136—316no—
59C2×#490—32no—
60C2×#489—32no—
61C2×#488—32no—
62C2×#486—32no—
63C2×#417—31no—
64C2×#418—31no—
65C2×#416—31no—
66C22×#102—31no—
67C2×#405D311yescharacteristic, complemented
68C24×D3—21no—
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
71C23×Dic3—21no—
72C2×#235—22no—
73C2×#235—24no—
74C2×#235—24no—
75C2×#235C2314yes—
76C22×D12—24no—
77C22×D12—24no—
78C22×D12—21no—
79C22×D12—22no—
80C22×D12—22no—
81C22×D12—22no—
82C22×D12C2314yes—
83C22×D12C2312yes—
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×Dic6—22no—
87C22×Dic6C2311yescharacteristic
88C2×C3×M16—24no—
89C2×C3×M16C2314yes—
90C22×C3×C8—24no—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2314yes—
94C22×C3×C8C2312yes—
95C3×#117D414yes—
96C3×#117C2314yes—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D412yes—
100C2×C3×C42C2312yes—
101C2×#217—21no—
102C2×(C2×D3)⋊C4—22no—
103C2×(C2×D3)⋊C4—22no—
104#926—22no—
105#926C2314yes—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3C2314yes—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—22no—
114C2×Dic3⋊C4—22no—
115C2×C4×Dic3—21no—
116#895—28no—
117#895—28no—
118#890—28no—
119#888—28no—
120#882—216no—
121#878—216no—
122#875—24no—
123#875C2318yes—
124#874—24no—
125#874C2318yes—
126#870—24no—
127#870C2318yescomplemented
128#869—24no—
129#869C2318yescomplemented
130C3×#97C2318yescomplemented
131C3×C4×C8C2318yescomplemented
132#821C2318yes—
133#821C2318yes—
134#821C2318yes—
135#821C2318yes—
136#821C2318yescomplemented
137#821C2314yescomplemented
138#821C2314yescomplemented
139#816C2318yescomplemented
140#816C2314yescomplemented
141#816C2314yescomplemented
142C22×#41—31no—
143C23×D4—31no—
144C22×M16C2×D311yescharacteristic
145C23×C8C2×D311yescharacteristic
146#544—38no—
147#544—64no—
148#535—38no—
149#535—64no—
150C2×C4⋊Q8—32no—
151C2×C4⋊Q8—61no—
152C2×#127—32no—
153C2×#125—32no—
154C2×#125—61no—
155C2×#124—32no—
156C2×C22⋊Q8—62no—
157C2×#121—61no—
158C2×#121—61no—
159C2×#120—62no—
160C2×#117—31no—
161C2×#117C2×D311yescharacteristic
162C22×C4⋊C4—31no—
163#490—316no—
164#489—316no—
165#488—316no—
166#486—316no—
167#418—38no—
168#417—38no—
169#416—38no—
170C2×#103—31no—
171C2×#103—32no—
172C2×#103—31no—
173C2×#102—31no—
174C2×#102—32no—
175C2×#102—32no—
176C2×#102—32no—
177C2×#102—32no—
178C2×#102—32no—
179C2×#102—31no—
180C2×#102—38no—
181#405C2×D318yescomplemented
182C2×#97C2×D312yescomplemented
183C2×C4×C8C2×D312yescomplemented
184C24×C3C2×D411yescharacteristic
185C23×D3—22no—
186C23×D3—21no—
187C23×D3—24no—
188C23×D3—44no—
189C23×D3—82no—
190C23×D3—82no—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D412yes—
195C22×C3×C4#4114yes—
196C22×C3×C4#4114yes—
197C22×C3×C4C2411yescharacteristic
198C22×C3×C4C2412yes—
199C2×#67—24no—
200C2×#67—44no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—22no—
204C22×Dic3—21no—
205C22×Dic3—24no—
206C22×Dic3—82no—
207#235—28no—
208#235—28no—
209#235—48no—
210C2×D12—24no—
211C2×D12—24no—
212C2×D12—28no—
213C2×D12—28no—
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—21no—
217C2×D12—22no—
218C2×D12—24no—
219C2×D12—28no—
220C2×D12—28no—
221C2×D12—44no—
222C2×D12—48no—
223C2×D12—84no—
224C2×D12—84no—
225C2×D12C2414yes—
226C2×D12C2412yes—
227C2×C4×D3—24no—
228C2×C4×D3—44no—
229C2×Dic6—22no—
230C2×Dic6—22no—
231C2×Dic6—21no—
232C2×Dic6—28no—
233C2×Dic6—84no—
234C2×Dic6C2412yes—
235C3×M16—28no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C2×C3×C8—28no—
239C2×C3×C8—24no—
240C2×C3×C8C2412yes—
241C2×C3×C8C2414yes—
242C2×C3×C8C2412yes—
243C3×C4⋊C4C2×D414yes—
244C3×C4⋊C4C2×D414yes—
245C3×C22⋊C4C2×D418yes—
246C3×C42C2×D414yes—
247C3×C42C2414yes—
248#217—24no—
249(C2×D3)⋊C4—28no—
250(C2×D3)⋊C4—28no—
251C4⋊Dic3—21no—
252C4⋊Dic3—22no—
253C4⋊Dic3—28no—
254C4⋊Dic3—21no—
255C4⋊Dic3—24no—
256C4⋊Dic3C2412yes—
257C4⋊Dic3C2414yes—
258C4⋊Dic3C2412yes—
259Dic3⋊C4—28no—
260Dic3⋊C4—28no—
261C4×Dic3—24no—
262C25—61no—
263C2×#41—34no—
264C2×#41—62no—
265C2×#41—64no—
266C2×#41—64no—
267C22×Q8—31no—
268C22×Q8—62no—
269C22×D4—32no—
270C22×D4—31no—
271C22×D4—34no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4—62no—
275C22×D4—62no—
276C22×D4—62no—
277C22×D4—64no—
278C22×D4—64no—
279C23×C4—61no—
280C23×C4—61no—
281C23×C4C22×D311yescharacteristic
282C2×M16—24no—
283C2×M16C22×D314yes—
284C22×C8—24no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D314yes—
288C22×C8C22×D312yes—
289C4⋊Q8—38no—
290C4⋊Q8—68no—
291C4⋊Q8—64no—
292#127—38no—
293#127—64no—
294#125—38no—
295#125—68no—
296#125—64no—
297#124—38no—
298#124—64no—
299C22⋊Q8—616no—
300#121—68no—
301#121—68no—
302#120—616no—
303#117—34no—
304#117—62no—
305#117D1214yes—
306#117C22×D314yes—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—62no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—62no—
315C2×C4⋊C4—62no—
316C2×C4⋊C4D1211yescharacteristic
317C2×C4⋊C4D1211yescharacteristic
318C2×C22⋊C4—61no—
319C2×C22⋊C4—62no—
320C2×C22⋊C4—62no—
321C2×C22⋊C4D1212yes—
322C2×C42—61no—
323C2×C42C22×D312yes—
324#103—34no—
325#103—34no—
326#103—38no—
327#102—38no—
328#102—38no—
329#102—38no—
330#102—38no—
331#102—34no—
332#102—34no—
333#102—38no—
334#97C22×D318yescomplemented
335C4×C8C22×D318yescomplemented
336C23×C3—22no—
337C23×C3—22no—
338C23×C3#12414yes—
339C23×C3C22×D412yes—
340C23×C3C22×D411yescharacteristic
341C22×D3—24no—
342C22×D3—22no—
343C22×D3—42no—
344C22×D3—48no—
345C22×D3—44no—
346C22×D3—42no—
347C22×D3—44no—
348C22×D3—42no—
349C22×D3—44no—
350C22×D3—84no—
351C22×D3—84no—
352C22×D3—816no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4#12414yes—
356C2×C3×C4#12418yes—
357C2×C3×C4C22×D414yes—
358C2×C3×C4C22×D414yes—
359C2×C3×C4C22×D412yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C2×#4112yes—
362C2×C3×C4C2×#4112yes—
363C2×C3×C4C2×#4114yes—
364C2×C3×C4C2×#4114yes—
365C2×C3×C4C2511yescharacteristic
366#67—48no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—24no—
370C2×Dic3—22no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—44no—
374C2×Dic3—42no—
375C2×Dic3—84no—
376D12—24no—
377D12—24no—
378D12—22no—
379D12—22no—
380D12—44no—
381D12—48no—
382D12—42no—
383D12—88no—
384D12—88no—
385C4×D3—48no—
386Dic6—22no—
387Dic6—22no—
388Dic6—42no—
389Dic6—88no—
390C3×C8—22no—
391C3×C8—22no—
392C3×C8—24no—
393C24—62no—
394C24—61no—
395C24—64no—
396C24—124no—
397C24—242no—
398C24—242no—
399C24C2×D1211yescharacteristic
400#41—68no—
401#41—68no—
402#41—128no—
403C2×Q8—32no—
404C2×Q8—61no—
405C2×Q8—68no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×Q8—244no—
409C2×D4—34no—
410C2×D4—32no—
411C2×D4—64no—
412C2×D4—64no—
413C2×D4—62no—
414C2×D4—61no—
415C2×D4—68no—
416C2×D4—68no—
417C2×D4—64no—
418C2×D4—64no—
419C2×D4—68no—
420C2×D4—68no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—124no—
424C2×D4—124no—
425C2×D4—128no—
426C2×D4—244no—
427C2×D4—244no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—64no—
433C22×C4—64no—
434C22×C4—124no—
435C22×C4—242no—
436C22×C4C2×D1212yes—
437C22×C4C2×D1212yes—
438C22×C4C2×D1212yes—
439C22×C4C2×D1211yescharacteristic
440C22×C4#23514yes—
441C22×C4#23514yes—
442C22×C4C23×D312yes—
443C22×C4C23×D311yescharacteristic
444M16—28no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—28no—
448C2×C8—24no—
449C2×C8C23×D312yes—
450C2×C8C23×D314yes—
451C2×C8C23×D312yes—
452C4⋊C4—34no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—64no—
456C4⋊C4—62no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—68no—
460C4⋊C4—68no—
461C4⋊C4—68no—
462C4⋊C4C2×D1214yes—
463C4⋊C4C2×D1214yes—
464C22⋊C4—64no—
465C22⋊C4—68no—
466C22⋊C4—68no—
467C22⋊C4C2×D1218yes—
468C42—64no—
469C42C2×D1214yes—
470C42C23×D314yes—
471C22×C3—28no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3C2×#12414yes—
475C22×C3C2×#12412yes—
476C22×C3#57612yes—
477C22×C3#57712yes—
478C22×C3C23×D411yescharacteristic
479C2×D3—42no—
480C2×D3—44no—
481C2×D3—42no—
482C2×D3—44no—
483C2×D3—42no—
484C2×D3—44no—
485C2×D3—816no—
486C2×D3—84no—
487C2×D3—88no—
488C3×C4—24no—
489C3×C4—24no—
490C3×C4C2×#12412yes—
491C3×C4C2×#12414yes—
492C3×C4C22×#4111yescharacteristic
493C3×C4C22×#4111yescharacteristic
494Dic3—42no—
495Dic3—42no—
496Dic3—44no—
497Dic3—42no—
498C23—22no—
499C23—22no—
500C23—64no—
501C23—62no—
502C23—122no—
503C23—128no—
504C23—124no—
505C23—122no—
506C23—124no—
507C23—122no—
508C23—124no—
509C23—244no—
510C23—244no—
511C23—2416no—
512C23#87514yes—
513C23C22×D1212yes—
514C23C22×D1211yescharacteristic
515Q8—62no—
516Q8—62no—
517Q8—122no—
518Q8—248no—
519D4—64no—
520D4—64no—
521D4—62no—
522D4—62no—
523D4—128no—
524D4—128no—
525D4—124no—
526D4—122no—
527D4—248no—
528D4—248no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—64no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—124no—
538C2×C4—122no—
539C2×C4—128no—
540C2×C4—244no—
541C2×C4#87514yes—
542C2×C4#87518yes—
543C2×C4C22×D1214yes—
544C2×C4C22×D1214yes—
545C2×C4C22×D1212yes—
546C2×C4C22×D1211yescharacteristic
547C2×C4C2×#23512yes—
548C2×C4C2×#23512yes—
549C2×C4C2×#23514yes—
550C2×C4C2×#23514yes—
551C2×C4C24×D311yescharacteristic
552C8—22no—
553C8—22no—
554C8—24no—
555C2×C3—24no—
556C2×C3#313614yescomplemented
557C2×C3C22×#12411yescharacteristic
558C2×C3C2×#57611yescharacteristic
559C2×C3C2×#57711yescharacteristic
560D3—82no—
561D3—84no—
562C22—22no—
563C22—22no—
564C22—28no—
565C22—122no—
566C22—124no—
567C22—122no—
568C22—124no—
569C22—122no—
570C22—124no—
571C22—2416no—
572C22—248no—
573C22—244no—
574C22C2×#87514yes—
575C22C2×#87512yes—
576C22#575112yes—
577C22#575212yes—
578C22C23×D1211yescharacteristic
579C4—24no—
580C4—24no—
581C4—124no—
582C4—122no—
583C4—122no—
584C4—122no—
585C4C2×#87514yes—
586C4C2×#87512yes—
587C4C22×#23511yescharacteristic
588C4C22×#23511yescharacteristic
589C3C2×#313611yescharacteristic, complemented, Sylow
590C2—24no—
591C2—244no—
592C2—242no—
593C2#7856314yesdirect product factor
594C2C22×#87511yescharacteristic
595C2C2×#575211yescharacteristic
596C2C2×#575111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#405C2
socleC23×C3C22×D4
prime radicalC2×#5525C2
hypercenterC2×#405D3
hypocenterC3C2×#3136
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. C2×#5519C2semidirect
3. C2×C3×#405C2semidirect
4. C2×#5133C2semidirect
5. C2×#5131C2semidirect
6. C2×#5113C2semidirect
7. C22×#821C2semidirect
8. #78563C2direct (N × H)
9. C2×#4713C2semidirect
10. C2×#4714C2semidirect
11. C2×#4697C2semidirect
12. C2×#4694C2semidirect
24. #5519C22semidirect
33. C2×#870C22semidirect
34. C2×#869C22semidirect
35. C3×#405C22semidirect
36. C2×C3×#97C22semidirect
37. C2×C3×C4×C8C22semidirect
38. #5133C22semidirect
39. #5131C22semidirect
40. #5113C22semidirect
45. C2×#821C22semidirect
46. C2×#821C22semidirect
47. C2×#821C22semidirect
48. C2×#821C22semidirect
49. C2×#816C22semidirect
50. C2×#816C22semidirect
51. C2×#816C22semidirect
52. #4714C22semidirect
53. #4713C22semidirect
54. #4697C22semidirect
55. #4694C22semidirect
67. C2×#405D3semidirect
127. #870C23semidirect
129. #869C23semidirect
130. C3×#97C23semidirect
131. C3×C4×C8C23semidirect
136. #821C23semidirect
137. #821C23semidirect
138. #821C23semidirect
139. #816C23semidirect
140. #816C23semidirect
141. #816C23semidirect
181. #405C2×D3semidirect
182. C2×#97C2×D3semidirect
183. C2×C4×C8C2×D3semidirect
334. #97C22×D3semidirect
335. C4×C8C22×D3semidirect
556. C2×C3#3136semidirect
589. C3C2×#3136semidirect
593. C2#78563direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55252
C2×#55192
C2×C3×#4052
C2×#51332
C2×#51312
C2×#51132
C22×#8212
#785632
C2×#47132
C2×#47142
C2×#46972
C2×#46942
C2×#31363