Finite Group Databasebeta

← C2×#83380 #11664423

Subgroups — C2×#83380 #11664423

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×C3×#535C211yesmaximal, characteristic
2C2×#5519C211yesmaximal, characteristic, complemented
3C2×#5232C211yesmaximal, characteristic, complemented
4C2×#5233C211yesmaximal, characteristic, complemented
5C2×#5215C211yesmaximal, characteristic, complemented
6C22×#832C211yesmaximal, characteristic, complemented
7#83380C2116yesmaximal, direct product factor
8C2×#5071C212yesmaximal, complemented
9C2×#5070C212yesmaximal, complemented
10C2×#5055C212yesmaximal, complemented
11C2×#5056C212yesmaximal, complemented
12C2×#4997C211yesmaximal, characteristic, complemented
13C2×#3136—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C23×C3×D4C2211yescharacteristic
16C3×#535—24no—
17C3×#535C2218yes—
18C2×C3×#127C2212yes—
19C2×C3×#124C2212yes—
20C2×C3×#121—21no—
21C2×C3×#121—21no—
22C2×C3×#120—22no—
23C2×C3×#117C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C22×#208C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28#5519—24no—
29#5519C2218yescomplemented
30C2×#890—21no—
31C2×#888—21no—
32C2×#878—22no—
33C2×#870C2212yescomplemented
34C2×#869C2212yescomplemented
35#5233C2218yescomplemented
36#5232C2218yescomplemented
37C2×#835C2211yescharacteristic, complemented
38C2×#835C2211yescharacteristic, complemented
39C2×#835C2212yescomplemented
40#5215C2218yescomplemented
41C2×#832C2212yes—
42C2×#832C2212yes—
43C2×#832C2212yes—
44C2×#832C2212yes—
45C2×#832C2218yescomplemented
46C2×#832C2211yescharacteristic, complemented
47C2×#832C2211yescharacteristic, complemented
48C2×#832C2212yescomplemented
49#5071C22116yescomplemented
50#5070C22116yescomplemented
51#5056C22116yescomplemented
52#5055C22116yescomplemented
53#4997C2218yescomplemented
54C2×#803C2212yescomplemented
55C2×C4×C3⋊C8C2212yescomplemented
56C2×#544—31no—
57C2×#535D311yescharacteristic
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×#405—31no—
68C25×C3—21no—
69C2×C3×#41—22no—
70C2×C3×#41—24no—
71C2×C3×#41—24no—
72C2×C3×#41C2314yes—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×D4—21no—
75C22×C3×D4—21no—
76C22×C3×D4—22no—
77C22×C3×D4—22no—
78C22×C3×D4—22no—
79C22×C3×D4—24no—
80C22×C3×D4—24no—
81C22×C3×D4C2312yes—
82C22×C3×D4C2314yes—
83C22×C3×D4C2311yescharacteristic
84C23×C3×C4—21no—
85C23×C3×C4C2311yescharacteristic
86C23×Dic3—21no—
87C22×Dic6—22no—
88C3×#127—24no—
89C3×#127C2318yes—
90C3×#124—24no—
91C3×#124C2318yes—
92C3×#121—28no—
93C3×#121—28no—
94C3×#120—216no—
95C3×#117D414yes—
96C3×#117C2314yes—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4D412yes—
102C2×C3×C42C2312yes—
103C2×#217—21no—
104#926—22no—
105#926C2314yes—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2312yes—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2314yes—
113C2×Dic3⋊C4—22no—
114C2×Dic3⋊C4—22no—
115C2×C4×Dic3—21no—
116C2×#208—24no—
117C2×#208C2314yes—
118C22×C3⋊C8—24no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2312yes—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2314yes—
123#890—28no—
124#888—28no—
125#878—216no—
126#870—24no—
127#870C2318yescomplemented
128#869—24no—
129#869C2318yescomplemented
130#835C2318yescomplemented
131#835C2314yescomplemented
132#835C2314yescomplemented
133#832C2318yes—
134#832C2318yes—
135#832C2318yes—
136#832C2318yes—
137#832C2318yescomplemented
138#832C2314yescomplemented
139#832C2314yescomplemented
140#803C2318yescomplemented
141C4×C3⋊C8C2318yescomplemented
142C22×#41C2×D311yescharacteristic
143C23×D4C2×D311yescharacteristic
144C22×M16—31no—
145C23×C8—31no—
146#544—38no—
147#544—64no—
148#535—24no—
149#535C2×D318yes—
150C2×C4⋊Q8—32no—
151C2×C4⋊Q8—61no—
152C2×#127C2×D312yes—
153C2×#125—32no—
154C2×#125—61no—
155C2×#124C2×D312yes—
156C2×C22⋊Q8—62no—
157C2×#121—21no—
158C2×#121—21no—
159C2×#120—22no—
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#405—38no—
182C2×#97—32no—
183C2×C4×C8—32no—
184C24×C3—22no—
185C24×C3—21no—
186C24×C3—24no—
187C24×C3—44no—
188C24×C3—82no—
189C24×C3—82no—
190C24×C3C2×D411yescharacteristic
191C3×#41—28no—
192C3×#41—28no—
193C3×#41—48no—
194C2×C3×Q8—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8C2412yes—
198C2×C3×D4—24no—
199C2×C3×D4—24no—
200C2×C3×D4—22no—
201C2×C3×D4—21no—
202C2×C3×D4—28no—
203C2×C3×D4—28no—
204C2×C3×D4—24no—
205C2×C3×D4—24no—
206C2×C3×D4—28no—
207C2×C3×D4—28no—
208C2×C3×D4—22no—
209C2×C3×D4—22no—
210C2×C3×D4—44no—
211C2×C3×D4—44no—
212C2×C3×D4—48no—
213C2×C3×D4—84no—
214C2×C3×D4—84no—
215C2×C3×D4C2414yes—
216C2×C3×D4C2412yes—
217C22×C3×C4—21no—
218C22×C3×C4—24no—
219C22×C3×C4—44no—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D412yes—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D412yes—
224C22×C3×C4#4114yes—
225C22×C3×C4#4114yes—
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2412yes—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—22no—
231C22×Dic3—24no—
232C22×Dic3—82no—
233C2×Dic6—28no—
234C2×Dic6—84no—
235C3×C4⋊C4C2×D414yes—
236C3×C4⋊C4C2×D414yes—
237C3×C22⋊C4—28no—
238C3×C22⋊C4—28no—
239C3×C22⋊C4C2×D418yes—
240C3×C42C2×D414yes—
241C3×C42C2414yes—
242#217—24no—
243C4⋊Dic3—21no—
244C4⋊Dic3—22no—
245C4⋊Dic3—28no—
246C4⋊Dic3—21no—
247C4⋊Dic3—24no—
248C4⋊Dic3C2412yes—
249C4⋊Dic3C2414yes—
250C4⋊Dic3C2412yes—
251Dic3⋊C4—28no—
252Dic3⋊C4—28no—
253C4×Dic3—24no—
254#208—28no—
255C2×C3⋊C8—22no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—28no—
258C2×C3⋊C8—24no—
259C2×C3⋊C8C2412yes—
260C2×C3⋊C8C2414yes—
261C2×C3⋊C8C2412yes—
262C25—21no—
263C2×#41—22no—
264C2×#41—24no—
265C2×#41—24no—
266C2×#41C22×D314yes—
267C22×Q8—62no—
268C22×Q8C22×D311yescharacteristic
269C22×D4—21no—
270C22×D4—24no—
271C22×D4—24no—
272C22×D4—21no—
273C22×D4—22no—
274C22×D4—22no—
275C22×D4—22no—
276C22×D4C22×D312yes—
277C22×D4C22×D311yescharacteristic
278C22×D4C22×D314yes—
279C23×C4—21no—
280C23×C4—61no—
281C23×C4C22×D311yescharacteristic
282C2×M16—34no—
283C2×M16—64no—
284C22×C8—31no—
285C22×C8—31no—
286C22×C8—34no—
287C22×C8—32no—
288C22×C8—64no—
289C4⋊Q8—38no—
290C4⋊Q8—68no—
291C4⋊Q8—64no—
292#127—24no—
293#127C22×D318yes—
294#125—38no—
295#125—68no—
296#125—64no—
297#124—24no—
298#124C22×D318yes—
299C22⋊Q8—616no—
300#121—28no—
301#121—28no—
302#120—216no—
303#117—34no—
304#117—62no—
305#117#6714yes—
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⋊C4#6711yescharacteristic
317C2×C4⋊C4#6711yescharacteristic
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4#6712yes—
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#97—38no—
335C4×C8—38no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—24no—
339C23×C3—22no—
340C23×C3—42no—
341C23×C3—48no—
342C23×C3—44no—
343C23×C3—42no—
344C23×C3—44no—
345C23×C3—42no—
346C23×C3—44no—
347C23×C3—84no—
348C23×C3—84no—
349C23×C3—816no—
350C23×C3#12414yes—
351C23×C3C22×D412yes—
352C23×C3C22×D411yescharacteristic
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—42no—
356C3×D4—24no—
357C3×D4—24no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—48no—
361C3×D4—48no—
362C3×D4—44no—
363C3×D4—42no—
364C3×D4—88no—
365C3×D4—88no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—22no—
369C2×C3×C4—42no—
370C2×C3×C4—48no—
371C2×C3×C4#12414yes—
372C2×C3×C4#12418yes—
373C2×C3×C4C22×D414yes—
374C2×C3×C4C22×D414yes—
375C2×C3×C4C22×D412yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4112yes—
378C2×C3×C4C2×#4112yes—
379C2×C3×C4C2×#4114yes—
380C2×C3×C4C2×#4114yes—
381C2×C3×C4C2511yescharacteristic
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—24no—
385C2×Dic3—42no—
386C2×Dic3—44no—
387C2×Dic3—42no—
388C2×Dic3—84no—
389Dic6—88no—
390C3⋊C8—22no—
391C3⋊C8—22no—
392C3⋊C8—24no—
393C24—22no—
394C24—21no—
395C24—24no—
396C24—44no—
397C24—82no—
398C24—82no—
399C24C2×#6711yescharacteristic
400#41—28no—
401#41—28no—
402#41—48no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—68no—
407C2×Q8—244no—
408C2×Q8C23×D312yes—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—48no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4C23×D314yes—
427C2×D4C23×D312yes—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—62no—
434C22×C4—64no—
435C22×C4—242no—
436C22×C4#23714yes—
437C22×C4#23714yes—
438C22×C4C2×#6712yes—
439C22×C4C2×#6712yes—
440C22×C4C2×#6712yes—
441C22×C4C2×#6711yescharacteristic
442C22×C4C23×D312yes—
443C22×C4C23×D311yescharacteristic
444M16—68no—
445C2×C8—32no—
446C2×C8—34no—
447C2×C8—32no—
448C2×C8—62no—
449C2×C8—62no—
450C2×C8—68no—
451C2×C8—64no—
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×#6714yes—
463C4⋊C4C2×#6714yes—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—64no—
467C22⋊C4C2×#6718yes—
468C42—64no—
469C42C2×#6714yes—
470C42C23×D314yes—
471C22×C3—28no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—42no—
475C22×C3—44no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3—42no—
479C22×C3—44no—
480C22×C3—816no—
481C22×C3—84no—
482C22×C3—88no—
483C22×C3C2×#12414yes—
484C22×C3C2×#12412yes—
485C22×C3#57612yes—
486C22×C3#57712yes—
487C22×C3C23×D411yescharacteristic
488C3×C4—24no—
489C3×C4—24no—
490C3×C4—42no—
491C3×C4C2×#12414yes—
492C3×C4C2×#12412yes—
493C3×C4C22×#4111yescharacteristic
494C3×C4C22×#4111yescharacteristic
495Dic3—42no—
496Dic3—44no—
497Dic3—42no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—42no—
503C23—48no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—84no—
510C23—84no—
511C23—816no—
512C23#93614yes—
513C23C22×#6712yes—
514C23C22×#6711yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—42no—
518Q8—248no—
519D4—24no—
520D4—24no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—48no—
525D4—44no—
526D4—42no—
527D4—88no—
528D4—88no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—64no—
537C2×C4—124no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—244no—
541C2×C4#93614yes—
542C2×C4#93618yes—
543C2×C4C2×#23712yes—
544C2×C4C2×#23712yes—
545C2×C4C2×#23714yes—
546C2×C4C2×#23714yes—
547C2×C4C22×#6714yes—
548C2×C4C22×#6714yes—
549C2×C4C22×#6712yes—
550C2×C4C22×#6711yescharacteristic
551C2×C4C24×D311yescharacteristic
552C8—62no—
553C8—62no—
554C8—64no—
555C2×C3—24no—
556C2×C3—82no—
557C2×C3—84no—
558C2×C3#313614yescomplemented
559C2×C3C22×#12411yescharacteristic
560C2×C3C2×#57611yescharacteristic
561C2×C3C2×#57711yescharacteristic
562C22—28no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—44no—
567C22—42no—
568C22—44no—
569C22—42no—
570C22—44no—
571C22—816no—
572C22—88no—
573C22—84no—
574C22C2×#93614yes—
575C22C2×#93612yes—
576C22#583512yes—
577C22#583612yes—
578C22C23×#6711yescharacteristic
579C4—24no—
580C4—24no—
581C4—42no—
582C4—124no—
583C4—122no—
584C4—122no—
585C4C2×#93614yes—
586C4C2×#93612yes—
587C4C22×#23711yescharacteristic
588C4C22×#23711yescharacteristic
589C3C2×#313611yescharacteristic, complemented, Sylow
590C2—24no—
591C2—84no—
592C2—82no—
593C2#8338014yesdirect product factor
594C2C22×#93611yescharacteristic
595C2C2×#583511yescharacteristic
596C2C2×#583611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
Fitting, prime radicalC2×C3×#535C2
socleC23×C3C22×D4
hypercenterC2×#535D3
hypocenterC3C2×#3136
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. C2×#5519C2semidirect
3. C2×#5232C2semidirect
4. C2×#5233C2semidirect
5. C2×#5215C2semidirect
6. C22×#832C2semidirect
7. #83380C2direct (N × H)
8. C2×#5071C2semidirect
9. C2×#5070C2semidirect
10. C2×#5055C2semidirect
11. C2×#5056C2semidirect
12. C2×#4997C2semidirect
29. #5519C22semidirect
33. C2×#870C22semidirect
34. C2×#869C22semidirect
35. #5233C22semidirect
36. #5232C22semidirect
37. C2×#835C22semidirect
38. C2×#835C22semidirect
39. C2×#835C22semidirect
40. #5215C22semidirect
45. C2×#832C22semidirect
46. C2×#832C22semidirect
47. C2×#832C22semidirect
48. C2×#832C22semidirect
49. #5071C22semidirect
50. #5070C22semidirect
51. #5056C22semidirect
52. #5055C22semidirect
53. #4997C22semidirect
54. C2×#803C22semidirect
55. C2×C4×C3⋊C8C22semidirect
127. #870C23semidirect
129. #869C23semidirect
130. #835C23semidirect
131. #835C23semidirect
132. #835C23semidirect
137. #832C23semidirect
138. #832C23semidirect
139. #832C23semidirect
140. #803C23semidirect
141. C4×C3⋊C8C23semidirect
558. C2×C3#3136semidirect
589. C3C2×#3136semidirect
593. C2#83380direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C2×#55192
C2×#52322
C2×#52332
C2×#52152
C22×#8322
#833802
C2×#50712
C2×#50702
C2×#50552
C2×#50562
C2×#49972
C2×#31363