Finite Group Databasebeta

← C2×#83122 #11667261

Subgroups — C2×#83122 #11667261

Order 768.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×#5470C211yesmaximal, characteristic, complemented
3#83122C2116yesmaximal, direct product factor
4C2×#5065C212yesmaximal, complemented
5C2×#5056C212yesmaximal, complemented
6C2×#5045C212yesmaximal, complemented
7C2×#5035C212yesmaximal
8C2×#5021C211yesmaximal, characteristic, complemented
9C2×#5020C211yesmaximal, characteristic, complemented
10C2×#5008C211yesmaximal, characteristic, complemented
11C2×#5007C211yesmaximal, characteristic, complemented
12C2×#4925C211yesmaximal, characteristic, complemented
13C2×#3233—31nomaximal, Sylow
14C3×#529—24no—
15C3×#529C2218yes—
16C2×C3×#126—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117—21no—
26C22×C3×C42C2211yescharacteristic
27C2×#926C2212yes—
28C2×#924—21no—
29C2×#924—21no—
30#5470—24no—
31#5470C2218yescomplemented
32C2×#870C2212yes—
33C2×#869C2211yescharacteristic
34C2×#869C2211yescharacteristic
35C2×C4×Dic6—21no—
36C2×C4×Dic6—21no—
37C2×#835C2212yes—
38C2×#835C2212yes—
39C2×#832C2212yes—
40C2×#832C2212yes—
41C2×#830C2212yes—
42#5065C22116yescomplemented
43#5056C22116yescomplemented
44#5045C22116yescomplemented
45#5035C22116yes—
46#5021C2218yescomplemented
47#5020C2218yescomplemented
48#5008C2218yescomplemented
49#5007C2218yescomplemented
50C2×#808C2212yes—
51C2×#808C2212yes—
52C2×#807C2212yes—
53C2×#807C2212yes—
54#4925C2218yescomplemented
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57C2×C4×C3⋊C8C2212yes—
58C2×#533—31no—
59C2×#529D311yescharacteristic
60#3233—316no—
61C2×#499—32no—
62C2×#488—32no—
63C2×#485—32no—
64C2×#482—32no—
65C2×#477—31no—
66C2×#475—31no—
67C2×#476—31no—
68C2×#474—31no—
69C2×#431—31no—
70C22×C3×Q8C2311yescharacteristic
71C22×C3×D4—21no—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4D411yescharacteristic
75C23×C3×C4D411yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C22×Dic6—21no—
78C22×Dic6—21no—
79C3×#126—28no—
80C3×#125—22no—
81C3×#125C2314yes—
82C3×#124—22no—
83C3×#124C2314yes—
84C3×#123—28no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8C2314yes—
87C3×#121—22no—
88C3×#121C2314yes—
89C3×C4×Q8—22no—
90C3×C4×Q8C2314yes—
91C3×C4×D4—28no—
92C3×C4×D4—22no—
93C3×C4×D4C2314yes—
94C3×#117—28no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C42—21no—
107C2×C3×C42D411yescharacteristic
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D414yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2314yes—
113C2×#217—22no—
114#926—24no—
115#926C2318yes—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2312yes—
120#924—28no—
121#924—28no—
122C2×Dic3⋊C4—22no—
123C2×Dic3⋊C4—22no—
124C2×C4×Dic3—22no—
125C22×C3⋊C8C2312yes—
126C22×C3⋊C8C2312yes—
127#870—24no—
128#870C2318yes—
129#869—22no—
130#869—22no—
131#869C2314yes—
132#869C2314yes—
133C4×Dic6—28no—
134C4×Dic6—28no—
135#835C2318yes—
136#835C2318yes—
137#832C2318yes—
138#832C2318yes—
139#830C2318yes—
140#808C2318yes—
141#808C2318yes—
142#807C2318yes—
143#807C2318yes—
144C4⋊(C3⋊C8)C2314yes—
145C4⋊(C3⋊C8)C2314yes—
146C4×C3⋊C8C2318yes—
147#533—38no—
148#533—64no—
149C2×C4⋊Q8—31no—
150C2×C4⋊Q8—31no—
151#529—24no—
152#529C2×D318yes—
153C2×#126—21no—
154C2×#125—32no—
155C2×#125C2×D311yescharacteristic
156C2×#124C2×D311yescharacteristic
157C2×#123—21no—
158C2×C22⋊Q8—61no—
159C2×C22⋊Q8—61no—
160C2×C22⋊Q8C2×D311yescharacteristic
161C2×#121C2×D311yescharacteristic
162C2×C4×Q8—61no—
163C2×C4×Q8—61no—
164C2×C4×Q8C2×D311yescharacteristic
165C2×C4×D4—21no—
166C2×C4×D4C2×D311yescharacteristic
167C2×#117—21no—
168C2×#117—32no—
169C22×C42C2×D311yescharacteristic
170#499—316no—
171#488—316no—
172#485—316no—
173#482—316no—
174#477—38no—
175#476—38no—
176#475—38no—
177#474—38no—
178#431—38no—
179C2×#107—32no—
180C2×#107—32no—
181C2×#106—32no—
182C2×#106—32no—
183C2×C4⋊C8—31no—
184C2×C4⋊C8—31no—
185C2×#103—32no—
186C2×#103—32no—
187C2×#102—32no—
188C2×#102—32no—
189C2×C22⋊C8—32no—
190C2×C4×C8—32no—
191C24×C3—21no—
192C24×C3C2×D411yescharacteristic
193C2×C3×Q8—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8C2412yes—
197C2×C3×D4—24no—
198C2×C3×D4—21no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—44no—
202C2×C3×D4C2412yes—
203C22×C3×C4—24no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—42no—
211C22×C3×C4—44no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D414yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D414yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2412yes—
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—22no—
223C22×Dic3—22no—
224C2×Dic6—24no—
225C2×Dic6—24no—
226C2×Dic6—44no—
227C2×Dic6—44no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4C2412yes—
235C3×C4⋊C4C2412yes—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C42—24no—
242C3×C42C2×D414yes—
243C3×C42C2×D414yes—
244C3×C42C2×D412yes—
245C3×C42C2×D412yes—
246C3×C42C2412yes—
247C3×C42C2412yes—
248#217—28no—
249C4⋊Dic3—24no—
250C4⋊Dic3—24no—
251C4⋊Dic3—22no—
252C4⋊Dic3—22no—
253C4⋊Dic3C2414yes—
254C4⋊Dic3C2414yes—
255Dic3⋊C4—28no—
256Dic3⋊C4—28no—
257C4×Dic3—28no—
258C2×C3⋊C8—24no—
259C2×C3⋊C8—24no—
260C2×C3⋊C8C2414yes—
261C2×C3⋊C8C2414yes—
262C22×Q8—61no—
263C22×Q8—61no—
264C22×Q8C22×D311yescharacteristic
265C22×D4—21no—
266C22×D4C22×D311yescharacteristic
267C23×C4—21no—
268C23×C4#6711yescharacteristic
269C23×C4#6711yescharacteristic
270C23×C4C22×D311yescharacteristic
271C22×C8—32no—
272C22×C8—32no—
273C4⋊Q8—34no—
274C4⋊Q8—34no—
275C4⋊Q8—62no—
276C4⋊Q8—62no—
277#126—28no—
278#125—22no—
279#125—38no—
280#125—64no—
281#125C22×D314yes—
282#124—22no—
283#124C22×D314yes—
284#123—28no—
285C22⋊Q8—22no—
286C22⋊Q8—68no—
287C22⋊Q8—68no—
288C22⋊Q8C22×D314yes—
289#121—22no—
290#121C22×D314yes—
291C4×Q8—22no—
292C4×Q8—68no—
293C4×Q8—68no—
294C4×Q8C22×D314yes—
295C4×D4—28no—
296C4×D4—22no—
297C4×D4C22×D314yes—
298#117—28no—
299#117—38no—
300#117—64no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—62no—
319C2×C42—21no—
320C2×C42—62no—
321C2×C42#6711yescharacteristic
322C2×C42#6711yescharacteristic
323C2×C42#6714yes—
324C2×C42C22×D311yescharacteristic
325C2×C42C22×D311yescharacteristic
326C2×C42C22×D314yes—
327#107—38no—
328#107—38no—
329#106—38no—
330#106—38no—
331C4⋊C8—34no—
332C4⋊C8—34no—
333#103—38no—
334#103—38no—
335#102—38no—
336#102—38no—
337C22⋊C8—38no—
338C4×C8—38no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—22no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3—42no—
346C23×C3C22×D412yes—
347C23×C3C22×D411yescharacteristic
348C3×Q8—22no—
349C3×Q8—22no—
350C3×Q8—42no—
351C3×D4—22no—
352C3×D4—22no—
353C3×D4—48no—
354C3×D4—42no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—28no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—44no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—48no—
371C2×C3×C4#13514yes—
372C2×C3×C4C22×D412yes—
373C2×C3×C4C22×D412yes—
374C2×C3×C4C22×D412yes—
375C2×C3×C4C22×D412yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D412yes—
378C2×C3×C42−1+412yes—
379C2×C3×C42−1+412yes—
380C2×C3×C4C2511yescharacteristic
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—44no—
384C2×Dic3—44no—
385Dic6—48no—
386Dic6—48no—
387C3⋊C8—24no—
388C3⋊C8—24no—
389C24—21no—
390C24C2×#6711yescharacteristic
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—64no—
395C2×Q8—64no—
396C2×Q8—124no—
397C2×Q8—124no—
398C2×Q8C23×D312yes—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—44no—
404C2×D4C23×D312yes—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4C2×#6712yes—
417C22×C4C2×#6714yes—
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C2×#6714yes—
423C22×C4C2×#6711yescharacteristic
424C22×C4C23×D312yes—
425C22×C4C23×D311yescharacteristic
426C2×C8—34no—
427C2×C8—34no—
428C2×C8—64no—
429C2×C8—64no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—34no—
437C4⋊C4—34no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—68no—
441C4⋊C4—68no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4C23×D312yes—
445C4⋊C4C23×D312yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—68no—
452C42—24no—
453C42—68no—
454C42C2×#6714yes—
455C42C2×#6712yes—
456C42C2×#6714yes—
457C42C2×#6712yes—
458C42C23×D312yes—
459C42C23×D312yes—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—44no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—84no—
468C22×C3#53612yes—
469C22×C3C2×#13512yes—
470C22×C3#57512yes—
471C22×C3C23×D411yescharacteristic
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4#53612yes—
480C3×C4C2×#13514yes—
481C3×C4C2×2−1+411yescharacteristic
482C3×C4C2×2−1+411yescharacteristic
483Dic3—44no—
484Dic3—44no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C22×#6712yes—
493C23C22×#6711yescharacteristic
494Q8—22no—
495Q8—22no—
496Q8—42no—
497Q8—128no—
498Q8—128no—
499D4—22no—
500D4—22no—
501D4—48no—
502D4—42no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—124no—
522C2×C4—124no—
523C2×C4#95014yes—
524C2×C4#101012yes—
525C2×C4#101012yes—
526C2×C4C22×#6712yes—
527C2×C4C22×#6712yes—
528C2×C4C22×#6712yes—
529C2×C4C22×#6712yes—
530C2×C4C22×#6712yes—
531C2×C4C22×#6711yescharacteristic
532C2×C4C24×D311yescharacteristic
533C8—64no—
534C8—64no—
535C2×C3—42no—
536C2×C3—82no—
537C2×C3#323314yescomplemented
538C2×C3C2×#53611yescharacteristic
539C2×C3C22×#13511yescharacteristic
540C2×C3C2×#57511yescharacteristic
541C22—22no—
542C22—22no—
543C22—22no—
544C22—44no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—84no—
549C22C2×#95012yes—
550C22#582212yes—
551C22#582512yes—
552C22C23×#6711yescharacteristic
553C4—22no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—124no—
561C4—124no—
562C4C2×#95014yes—
563C4#582512yes—
564C4C2×#101011yescharacteristic
565C4C2×#101011yescharacteristic
566C3C2×#323311yescharacteristic, complemented, Sylow
567C2—42no—
568C2—82no—
569C2#8312214yesdirect product factor
570C2C22×#95011yescharacteristic
571C2C2×#582211yescharacteristic
572C2C2×#582511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#529C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#529D3
hypocenterC3C2×#3233
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#5470C2semidirect
3. #83122C2direct (N × H)
4. C2×#5065C2semidirect
5. C2×#5056C2semidirect
6. C2×#5045C2semidirect
8. C2×#5021C2semidirect
9. C2×#5020C2semidirect
10. C2×#5008C2semidirect
11. C2×#5007C2semidirect
12. C2×#4925C2semidirect
31. #5470C22semidirect
42. #5065C22semidirect
43. #5056C22semidirect
44. #5045C22semidirect
46. #5021C22semidirect
47. #5020C22semidirect
48. #5008C22semidirect
49. #5007C22semidirect
54. #4925C22semidirect
537. C2×C3#3233semidirect
566. C3C2×#3233semidirect
569. C2#83122direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×#54702
#831222
C2×#50652
C2×#50562
C2×#50452
C2×#50352
C2×#50212
C2×#50202
C2×#50082
C2×#50072
C2×#49252
C2×#32333