Finite Group Databasebeta

← C2×#83470 #11664428

Subgroups — C2×#83470 #11664428

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×#544C211yesmaximal, characteristic, complemented
2C2×#5525C211yesmaximal, characteristic
3#83470C2116yesmaximal, direct product factor
4C2×#5089C212yesmaximal, complemented
5C2×#5088C212yesmaximal, complemented
6C2×#5067C212yesmaximal, complemented
7C2×#5068C212yesmaximal, complemented
8C2×#5001C211yesmaximal, characteristic, complemented
9C2×#5002C211yesmaximal, characteristic, complemented
10C2×#4997C211yesmaximal, characteristic, complemented
11C2×#4966C211yesmaximal, characteristic, complemented
12C22×#809C211yesmaximal, characteristic, complemented
13C2×#3136—31nomaximal, Sylow
14C22×#235C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C3×#544—24no—
17C3×#544C2218yescomplemented
18C2×C3×C4⋊Q8—21no—
19C2×C3×C4⋊Q8C2212yescomplemented
20C2×C3×#125—21no—
21C2×C3×#125C2212yescomplemented
22C2×C3×C22⋊Q8—22no—
23C2×C3×#117C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×#208C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28#5525—24no—
29#5525C2218yes—
30C2×#895—21no—
31C2×#895—21no—
32C2×#882—22no—
33C2×#875C2212yes—
34C2×#874C2212yes—
35#5089C22116yescomplemented
36#5088C22116yescomplemented
37#5068C22116yescomplemented
38#5067C22116yescomplemented
39#5002C2218yescomplemented
40#5001C2218yescomplemented
41#4997C2218yescomplemented
42C2×#810C2211yescharacteristic, complemented
43C2×#810C2211yescharacteristic, complemented
44C2×#810C2212yescomplemented
45#4966C2218yescomplemented
46C2×#809C2212yes—
47C2×#809C2212yes—
48C2×#809C2212yes—
49C2×#809C2212yes—
50C2×#809C2218yescomplemented
51C2×#809C2211yescharacteristic, complemented
52C2×#809C2211yescharacteristic, complemented
53C2×#809C2212yescomplemented
54C2×#803C2212yescomplemented
55C2×C4×C3⋊C8C2212yescomplemented
56C2×#544D311yescharacteristic, complemented
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×#405—31no—
68C24×D3—21no—
69C22×C3×Q8—22no—
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C22×#67—21no—
73C2×#235—22no—
74C2×#235—24no—
75C2×#235—24no—
76C2×#235C2314yes—
77C22×D12—24no—
78C22×D12—24no—
79C22×D12—21no—
80C22×D12—22no—
81C22×D12—22no—
82C22×D12—22no—
83C22×D12C2312yes—
84C22×D12C2314yes—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×Dic6C2311yescharacteristic
88C3×C4⋊Q8—28no—
89C3×C4⋊Q8—24no—
90C3×C4⋊Q8C2318yescomplemented
91C3×#125—28no—
92C3×#125—24no—
93C3×#125C2318yescomplemented
94C3×C22⋊Q8—216no—
95C3×#117—22no—
96C3×#117D414yes—
97C3×#117C2314yes—
98C3×#117C2314yes—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4D412yes—
112C2×C3×C42—21no—
113C2×C3×C42C2312yes—
114C2×(C2×D3)⋊C4—22no—
115C2×(C2×D3)⋊C4—22no—
116C2×#208—24no—
117C2×#208C2314yes—
118C22×C3⋊C8—24no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2312yes—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2314yes—
123#895—28no—
124#895—28no—
125#882—216no—
126#875—24no—
127#875C2318yes—
128#874—24no—
129#874C2318yes—
130#810C2318yescomplemented
131#810C2314yescomplemented
132#810C2314yescomplemented
133#809C2318yes—
134#809C2318yes—
135#809C2318yes—
136#809C2318yes—
137#809C2318yescomplemented
138#809C2314yescomplemented
139#809C2314yescomplemented
140#803C2318yescomplemented
141C4×C3⋊C8C2318yescomplemented
142C22×#41—31no—
143C23×D4—31no—
144C22×M16—31no—
145C23×C8—31no—
146#544—24no—
147#544C2×D318yescomplemented
148#535—38no—
149#535—64no—
150C2×C4⋊Q8—21no—
151C2×C4⋊Q8C2×D312yescomplemented
152C2×#127—32no—
153C2×#125—21no—
154C2×#125C2×D312yescomplemented
155C2×#124—32no—
156C2×C22⋊Q8—22no—
157C2×#121—61no—
158C2×#121—61no—
159C2×#120—62no—
160C2×#117C2×D311yescharacteristic
161C2×#117C2×D311yescharacteristic
162C22×C4⋊C4C2×D311yescharacteristic
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×C3C2×D411yescharacteristic
185C23×D3—22no—
186C23×D3—21no—
187C23×D3—24no—
188C23×D3—44no—
189C23×D3—82no—
190C23×D3—82no—
191C2×C3×Q8—28no—
192C2×C3×Q8—84no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—24no—
197C22×C3×C4—82no—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D412yes—
202C22×C3×C4#4114yes—
203C22×C3×C4#4114yes—
204C22×C3×C4C2411yescharacteristic
205C22×C3×C4C2412yes—
206C2×#67—24no—
207C2×#67—44no—
208C22×Dic3—21no—
209#235—28no—
210#235—28no—
211#235—48no—
212C2×D12—24no—
213C2×D12—24no—
214C2×D12—28no—
215C2×D12—28no—
216C2×D12—22no—
217C2×D12—22no—
218C2×D12—21no—
219C2×D12—22no—
220C2×D12—24no—
221C2×D12—28no—
222C2×D12—28no—
223C2×D12—44no—
224C2×D12—48no—
225C2×D12—84no—
226C2×D12—84no—
227C2×D12C2414yes—
228C2×D12C2412yes—
229C2×C4×D3—24no—
230C2×C4×D3—44no—
231C2×Dic6—22no—
232C2×Dic6—22no—
233C2×Dic6—21no—
234C2×Dic6C2412yes—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—28no—
238C3×C4⋊C4—28no—
239C3×C4⋊C4—28no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2×D414yes—
243C3×C4⋊C4C2×D414yes—
244C3×C4⋊C4C2414yes—
245C3×C4⋊C4C2412yes—
246C3×C4⋊C4C2412yes—
247C3×C22⋊C4—24no—
248C3×C22⋊C4C2×D418yes—
249C3×C42—24no—
250C3×C42C2×D414yes—
251C3×C42C2414yes—
252(C2×D3)⋊C4—28no—
253(C2×D3)⋊C4—28no—
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—61no—
263C2×#41—34no—
264C2×#41—62no—
265C2×#41—64no—
266C2×#41—64no—
267C22×Q8—22no—
268C22×Q8—31no—
269C22×D4—32no—
270C22×D4—31no—
271C22×D4—34no—
272C22×D4—61no—
273C22×D4—64no—
274C22×D4—64no—
275C22×D4—61no—
276C22×D4—62no—
277C22×D4—62no—
278C22×D4—62no—
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—28no—
290C4⋊Q8—24no—
291C4⋊Q8C22×D318yescomplemented
292#127—38no—
293#127—64no—
294#125—28no—
295#125—24no—
296#125C22×D318yescomplemented
297#124—38no—
298#124—64no—
299C22⋊Q8—216no—
300#121—68no—
301#121—68no—
302#120—616no—
303#117—22no—
304#117#6714yes—
305#117C22×D314yes—
306#117C22×D314yes—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4#6711yescharacteristic
313C2×C4⋊C4#6711yescharacteristic
314C2×C4⋊C4C22×D312yes—
315C2×C4⋊C4C22×D314yes—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—62no—
320C2×C22⋊C4—62no—
321C2×C22⋊C4#6712yes—
322C2×C42—21no—
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#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—
353C3×Q8—88no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—44no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—84no—
363C2×C3×C4#12414yes—
364C2×C3×C4#12418yes—
365C2×C3×C4C22×D414yes—
366C2×C3×C4C22×D414yes—
367C2×C3×C4C22×D412yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4112yes—
370C2×C3×C4C2×#4112yes—
371C2×C3×C4C2×#4114yes—
372C2×C3×C4C2×#4114yes—
373C2×C3×C4C2511yescharacteristic
374#67—48no—
375C2×Dic3—22no—
376C2×Dic3—42no—
377D12—24no—
378D12—24no—
379D12—22no—
380D12—22no—
381D12—44no—
382D12—48no—
383D12—42no—
384D12—88no—
385D12—88no—
386C4×D3—48no—
387Dic6—22no—
388Dic6—22no—
389Dic6—42no—
390C3⋊C8—22no—
391C3⋊C8—22no—
392C3⋊C8—24no—
393C24—62no—
394C24—61no—
395C24—64no—
396C24—124no—
397C24—242no—
398C24—242no—
399C24C2×#6711yescharacteristic
400#41—68no—
401#41—68no—
402#41—128no—
403C2×Q8—28no—
404C2×Q8—32no—
405C2×Q8—61no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×Q8—84no—
409C2×D4—34no—
410C2×D4—32no—
411C2×D4—64no—
412C2×D4—64no—
413C2×D4—64no—
414C2×D4—68no—
415C2×D4—68no—
416C2×D4—62no—
417C2×D4—62no—
418C2×D4—64no—
419C2×D4—62no—
420C2×D4—61no—
421C2×D4—68no—
422C2×D4—68no—
423C2×D4—124no—
424C2×D4—124no—
425C2×D4—128no—
426C2×D4—244no—
427C2×D4—244no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—61no—
433C22×C4—64no—
434C22×C4—82no—
435C22×C4—124no—
436C22×C4#23914yes—
437C22×C4#23914yes—
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—24no—
453C4⋊C4—22no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4C2×#6714yes—
460C4⋊C4C2×#6714yes—
461C4⋊C4C23×D314yes—
462C4⋊C4C23×D312yes—
463C4⋊C4C23×D312yes—
464C22⋊C4—24no—
465C22⋊C4—68no—
466C22⋊C4—68no—
467C22⋊C4C2×#6718yes—
468C42—24no—
469C42C2×#6714yes—
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×C4—44no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4C2×#12412yes—
494C3×C4C2×#12414yes—
495C3×C4C22×#4111yescharacteristic
496C3×C4C22×#4111yescharacteristic
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#94714yes—
513C23C22×#6712yes—
514C23C22×#6711yescharacteristic
515Q8—62no—
516Q8—62no—
517Q8—88no—
518Q8—122no—
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—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—62no—
538C2×C4—84no—
539C2×C4—122no—
540C2×C4—128no—
541C2×C4#94714yes—
542C2×C4#94718yes—
543C2×C4C2×#23912yes—
544C2×C4C2×#23912yes—
545C2×C4C2×#23914yes—
546C2×C4C2×#23914yes—
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#313614yescomplemented
557C2×C3C22×#12411yescharacteristic
558C2×C3C2×#57611yescharacteristic
559C2×C3C2×#57711yescharacteristic
560D3—82no—
561D3—84no—
562C22—28no—
563C22—22no—
564C22—22no—
565C22—122no—
566C22—124no—
567C22—122no—
568C22—124no—
569C22—122no—
570C22—124no—
571C22—2416no—
572C22—248no—
573C22—244no—
574C22C2×#94714yes—
575C22C2×#94712yes—
576C22#583512yes—
577C22#583712yes—
578C22C23×#6711yescharacteristic
579C4—24no—
580C4—24no—
581C4—44no—
582C4—42no—
583C4—42no—
584C4—122no—
585C4C2×#94714yes—
586C4C2×#94712yes—
587C4C22×#23911yescharacteristic
588C4C22×#23911yescharacteristic
589C3C2×#313611yescharacteristic, complemented, Sylow
590C2—24no—
591C2—244no—
592C2—242no—
593C2#8347014yesdirect product factor
594C2C22×#94711yescharacteristic
595C2C2×#583511yescharacteristic
596C2C2×#583711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#544C2
socleC23×C3C22×D4
prime radicalC2×#5525C2
hypercenterC2×#544D3
hypocenterC3C2×#3136
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×C3×#544C2semidirect
3. #83470C2direct (N × H)
4. C2×#5089C2semidirect
5. C2×#5088C2semidirect
6. C2×#5067C2semidirect
7. C2×#5068C2semidirect
8. C2×#5001C2semidirect
9. C2×#5002C2semidirect
10. C2×#4997C2semidirect
11. C2×#4966C2semidirect
12. C22×#809C2semidirect
17. C3×#544C22semidirect
19. C2×C3×C4⋊Q8C22semidirect
21. C2×C3×#125C22semidirect
35. #5089C22semidirect
36. #5088C22semidirect
37. #5068C22semidirect
38. #5067C22semidirect
39. #5002C22semidirect
40. #5001C22semidirect
41. #4997C22semidirect
42. C2×#810C22semidirect
43. C2×#810C22semidirect
44. C2×#810C22semidirect
45. #4966C22semidirect
50. C2×#809C22semidirect
51. C2×#809C22semidirect
52. C2×#809C22semidirect
53. C2×#809C22semidirect
54. C2×#803C22semidirect
55. C2×C4×C3⋊C8C22semidirect
56. C2×#544D3semidirect
90. C3×C4⋊Q8C23semidirect
93. C3×#125C23semidirect
130. #810C23semidirect
131. #810C23semidirect
132. #810C23semidirect
137. #809C23semidirect
138. #809C23semidirect
139. #809C23semidirect
140. #803C23semidirect
141. C4×C3⋊C8C23semidirect
147. #544C2×D3semidirect
151. C2×C4⋊Q8C2×D3semidirect
154. C2×#125C2×D3semidirect
291. C4⋊Q8C22×D3semidirect
296. #125C22×D3semidirect
556. C2×C3#3136semidirect
589. C3C2×#3136semidirect
593. C2#83470direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×#55252
#834702
C2×#50892
C2×#50882
C2×#50672
C2×#50682
C2×#50012
C2×#50022
C2×#49972
C2×#49662
C22×#8092
C2×#31363