Finite Group Databasebeta

← C2×#81462 #11667222

Subgroups — C2×#81462 #11667222

Order 768.

1822 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5624C211yesmaximal, characteristic
2C22×#890C211yesmaximal, characteristic, complemented
3C2×C3×#484C211yesmaximal, characteristic, complemented
4C2×#5043C211yesmaximal, characteristic, complemented
5C2×#4968C211yesmaximal, characteristic
6#81462C2116yesmaximal, direct product factor
7C2×#4891C212yesmaximal, complemented
8C2×#4878C212yesmaximal, complemented
9C2×#4800C212yesmaximal, complemented
10C2×#4796C212yesmaximal, complemented
11C2×#4736C211yesmaximal, characteristic
12C2×#4718C211yesmaximal, characteristic, complemented
13C2×#3232—31nomaximal, Sylow
14C2×C3×C22⋊Q8C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#947C2211yescharacteristic
17C2×Q8×Dic3C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C22×C4⋊Dic3—21no—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4—21no—
22C22×C4×Dic3C2211yescharacteristic
23#5624—24no—
24#5624C2218yes—
25C2×#898—21no—
26C2×#893C2211yescharacteristic
27C2×#892—21no—
28C2×#890—24no—
29C2×#890—21no—
30C2×#890—21no—
31C2×#890C2211yescharacteristic
32C2×#890C2212yes—
33C2×#890C2212yes—
34C2×#890C2218yescomplemented
35C2×#886—21no—
36C2×#67⋊C4—21no—
37C3×#484C2218yescomplemented
38C2×C3×#107C2212yes—
39C2×C3×#103C2212yes—
40C2×C3×C22⋊C8C2211yescharacteristic
41C2×#835C2212yes—
42C2×#830C2211yescharacteristic
43C2×#821C2212yes—
44C2×#818C2212yes—
45C2×Dic3⋊C8C2212yes—
46C2×C8×Dic3C2212yes—
47#5043C2218yescomplemented
48#4968C2218yes—
49C2×#809C2212yes—
50C2×#808C2212yes—
51C2×#808C2212yes—
52#4891C22116yescomplemented
53#4878C22116yescomplemented
54#4800C22116yescomplemented
55#4796C22116yescomplemented
56#4736C2218yes—
57#4718C2218yescomplemented
58C2×#529—31no—
59C22×#125—31no—
60#3232—316no—
61C2×#499—32no—
62C2×#488—32no—
63C2×#484D311yescharacteristic, complemented
64C2×#481—31no—
65C2×#483—31no—
66C2×#480—31no—
67C2×#479—32no—
68C2×#478—32no—
69C2×#431—31no—
70C22×C3×Q8C2311yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4C2311yescharacteristic
73C22×#67—21no—
74C23×Dic3—21no—
75C23×Dic3D411yescharacteristic
76C23×Dic3D411yescharacteristic
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C3×C8C2312yes—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8C2314yes—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C22⋊C4—21no—
89#947—22no—
90#947C2314yes—
91Q8×Dic3—22no—
92Q8×Dic3C2314yes—
93#930—22no—
94#930C2314yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—28no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2314yes—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—22no—
111C2×Dic3⋊C4—22no—
112C2×Dic3⋊C4—28no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D414yes—
117C2×C4×Dic3C2314yes—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2312yes—
121#898—28no—
122#893—22no—
123#893C2314yes—
124#892—28no—
125#890—24no—
126#890—24no—
127#890—28no—
128#890—28no—
129#890—22no—
130#890C2314yes—
131#890C2318yes—
132#890C2318yes—
133#886—28no—
134#67⋊C4—28no—
135C3×#107C2318yes—
136C3×#103C2318yes—
137C3×C22⋊C8C2314yes—
138#835C2318yes—
139#830C2314yes—
140#821C2318yes—
141#818C2318yes—
142Dic3⋊C8C2318yes—
143C8×Dic3C2318yes—
144#809C2318yes—
145#808C2318yes—
146#808C2318yes—
147#529—38no—
148#529—64no—
149C2×#126—61no—
150C2×#125—32no—
151C2×#125—32no—
152C2×#125—31no—
153C2×#125—38no—
154C2×#125—61no—
155C2×#125—61no—
156C2×#125—64no—
157C2×#124—31no—
158C2×#123—61no—
159C2×C22⋊Q8C2×D311yescharacteristic
160C2×#121—31no—
161C2×C4×Q8—31no—
162C2×C4×D4—31no—
163C2×C4×D4—61no—
164C2×#117—61no—
165C22×C4⋊C4—31no—
166C22×C4⋊C4—61no—
167C22×C4⋊C4—61no—
168C22×C4⋊C4C2×D311yescharacteristic
169C22×C42—31no—
170#499—316no—
171#488—316no—
172#484C2×D318yescomplemented
173#483—38no—
174#481—38no—
175#480—38no—
176#479—316no—
177#478—316no—
178#431—38no—
179C2×#107—32no—
180C2×#107C2×D312yes—
181C2×#106—32no—
182C2×#106—32no—
183C2×C4⋊C8—32no—
184C2×#103—32no—
185C2×#103C2×D312yes—
186C2×#102—32no—
187C2×#102—32no—
188C2×C22⋊C8—31no—
189C2×C22⋊C8C2×D311yescharacteristic
190C2×C4×C8—32no—
191C24×C3C2×D411yescharacteristic
192C23×D3—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—24no—
195C2×C3×Q8C2412yes—
196C22×C3×C4—24no—
197C22×C3×C4—21no—
198C22×C3×C4—24no—
199C22×C3×C4—22no—
200C22×C3×C4—21no—
201C22×C3×C4—44no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2412yes—
205C22×C3×C4C2411yescharacteristic
206C2×#67—24no—
207C2×#67—44no—
208C22×Dic3—24no—
209C22×Dic3—21no—
210C22×Dic3—24no—
211C22×Dic3—22no—
212C22×Dic3—42no—
213C22×Dic3—44no—
214C22×Dic3C2×D414yes—
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D414yes—
220C2×D12—24no—
221C2×D12—21no—
222C2×D12C2412yes—
223C2×C4×D3—24no—
224C2×C4×D3—44no—
225C2×C3×C8—24no—
226C2×C3×C8C2414yes—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4C2412yes—
232C3×C4⋊C4C2414yes—
233C3×C22⋊C4—24no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—24no—
238C4⋊Dic3—24no—
239C4⋊Dic3—22no—
240C4⋊Dic3—21no—
241C4⋊Dic3—24no—
242C4⋊Dic3—24no—
243C4⋊Dic3—28no—
244C4⋊Dic3—28no—
245C4⋊Dic3C2412yes—
246C4⋊Dic3C2414yes—
247Dic3⋊C4—24no—
248Dic3⋊C4—28no—
249Dic3⋊C4—28no—
250C4×Dic3—24no—
251C4×Dic3C2×D412yes—
252C4×Dic3C2×D414yes—
253C4×Dic3C2×D412yes—
254C4×Dic3C2×D414yes—
255C4×Dic3C2412yes—
256C4×Dic3C2412yes—
257C2×C3⋊C8—24no—
258C2×C3⋊C8C2414yes—
259C22×Q8C22×D311yescharacteristic
260C22×D4—31no—
261C22×D4—61no—
262C23×C4—21no—
263C23×C4—31no—
264C23×C4—31no—
265C23×C4—61no—
266C23×C4—61no—
267C23×C4C22×D311yescharacteristic
268C22×C8—32no—
269C22×C8C22×D312yes—
270#126—68no—
271#125—38no—
272#125—38no—
273#125—34no—
274#125—68no—
275#125—64no—
276#125—64no—
277#125—68no—
278#125—62no—
279#124—34no—
280#124—62no—
281#123—68no—
282C22⋊Q8—22no—
283C22⋊Q8C22×D314yes—
284#121—34no—
285#121—62no—
286C4×Q8—34no—
287C4×Q8—62no—
288C4×D4—34no—
289C4×D4—62no—
290C4×D4—68no—
291#117—68no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—32no—
296C2×C4⋊C4—34no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—68no—
299C2×C4⋊C4—62no—
300C2×C4⋊C4—62no—
301C2×C4⋊C4—68no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4C22×D312yes—
310C2×C4⋊C4C22×D314yes—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—34no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42—34no—
323C2×C42—61no—
324#107—38no—
325#107C22×D318yes—
326#106—38no—
327#106—38no—
328C4⋊C8—38no—
329#103—38no—
330#103C22×D318yes—
331#102—38no—
332#102—38no—
333C22⋊C8—34no—
334C22⋊C8C22×D314yes—
335C4×C8—38no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3#13514yes—
339C23×C3C22×D412yes—
340C23×C3C22×D411yescharacteristic
341C22×D3—22no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C3×Q8—24no—
346C3×Q8—42no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—48no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4C22×D412yes—
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C42−1+414yes—
360C2×C3×C4C2511yescharacteristic
361#67—48no—
362C2×Dic3—28no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—44no—
370C2×Dic3—48no—
371C2×Dic3—44no—
372C2×Dic3—42no—
373C2×Dic3C22×D412yes—
374C2×Dic3C22×D412yes—
375C2×Dic3C22×D412yes—
376C2×Dic3C22×D412yes—
377D12—24no—
378D12—42no—
379C4×D3—48no—
380C3×C8—24no—
381C3⋊C8—24no—
382C24—61no—
383C24D3×D411yescharacteristic
384C2×Q8—21no—
385C2×Q8—24no—
386C2×Q8C23×D312yes—
387C2×D4—32no—
388C2×D4—64no—
389C2×D4—61no—
390C2×D4—64no—
391C2×D4—124no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—34no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—34no—
403C22×C4—44no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—124no—
410C22×C4—122no—
411C22×C4—124no—
412C22×C4D3×D412yes—
413C22×C4D3×D411yescharacteristic
414C22×C4C23×D312yes—
415C22×C4C23×D311yescharacteristic
416C2×C8—24no—
417C2×C8—34no—
418C2×C8—64no—
419C2×C8C23×D314yes—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—21no—
424C4⋊C4—34no—
425C4⋊C4—32no—
426C4⋊C4—68no—
427C4⋊C4—68no—
428C4⋊C4—68no—
429C4⋊C4—68no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—64no—
434C4⋊C4—62no—
435C4⋊C4—61no—
436C4⋊C4C23×D314yes—
437C4⋊C4C23×D312yes—
438C22⋊C4—24no—
439C22⋊C4—64no—
440C22⋊C4—64no—
441C22⋊C4—64no—
442C22⋊C4—64no—
443C42—34no—
444C42—32no—
445C42—32no—
446C42—34no—
447C42—32no—
448C42—32no—
449C42—64no—
450C22×C3—22no—
451C22×C3—28no—
452C22×C3—22no—
453C22×C3#53612yes—
454C22×C3C2×#13514yes—
455C22×C3C2×#13512yes—
456C22×C3#57512yes—
457C22×C3C23×D411yescharacteristic
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—84no—
462C3×C4—22no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—44no—
466C3×C4#53612yes—
467C3×C4C2×2−1+412yes—
468Dic3—22no—
469Dic3—22no—
470Dic3—22no—
471Dic3—22no—
472Dic3—44no—
473Dic3—42no—
474C23—22no—
475C23—22no—
476C23—62no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23#91914yes—
481C23C2×D3×D412yes—
482C23C2×D3×D411yescharacteristic
483Q8—24no—
484Q8—42no—
485D4—64no—
486D4—128no—
487D4—122no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—32no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—32no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—68no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—128no—
510C2×C4—124no—
511C2×C4—128no—
512C2×C4—124no—
513C2×C4—122no—
514C2×C4C2×D3×D412yes—
515C2×C4C2×D3×D411yescharacteristic
516C2×C4#101014yes—
517C2×C4C24×D311yescharacteristic
518C8—24no—
519C8—64no—
520C2×C3—24no—
521C2×C3#323214yescomplemented
522C2×C3C2×#53611yescharacteristic
523C2×C3C22×#13511yescharacteristic
524C2×C3C2×#57511yescharacteristic
525D3—82no—
526C22—28no—
527C22—22no—
528C22—22no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22—244no—
533C22#564812yes—
534C22C2×#91914yes—
535C22C2×#91912yes—
536C22#577312yes—
537C22C22×D3×D411yescharacteristic
538C4—22no—
539C4—44no—
540C4—42no—
541C4—42no—
542C4—62no—
543C4—62no—
544C4—62no—
545C4—62no—
546C4—124no—
547C4—122no—
548C4#564812yes—
549C4C2×#101012yes—
550C3C2×#323211yescharacteristic, complemented, Sylow
551C2—24no—
552C2—242no—
553C2#8146214yesdirect product factor
554C2C2×#564811yescharacteristic
555C2C22×#91911yescharacteristic
556C2C2×#577311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#484C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#484D3
hypocenterC3C2×#3232
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C22×#890C2semidirect
3. C2×C3×#484C2semidirect
4. C2×#5043C2semidirect
6. #81462C2direct (N × H)
7. C2×#4891C2semidirect
8. C2×#4878C2semidirect
9. C2×#4800C2semidirect
10. C2×#4796C2semidirect
12. C2×#4718C2semidirect
34. C2×#890C22semidirect
37. C3×#484C22semidirect
47. #5043C22semidirect
52. #4891C22semidirect
53. #4878C22semidirect
54. #4800C22semidirect
55. #4796C22semidirect
57. #4718C22semidirect
63. C2×#484D3semidirect
172. #484C2×D3semidirect
521. C2×C3#3232semidirect
550. C3C2×#3232semidirect
553. C2#81462direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56242
C22×#8902
C2×C3×#4842
C2×#50432
C2×#49682
#814622
C2×#48912
C2×#48782
C2×#48002
C2×#47962
C2×#47362
C2×#47182
C2×#32323