Finite Group Databasebeta

← C2×#81113 #11667270

Subgroups — C2×#81113 #11667270

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×#5728C211yesmaximal, characteristic, complemented
2C2×#5580C211yesmaximal, characteristic
3C2×C3×#477C211yesmaximal, characteristic, complemented
4C2×#5084C211yesmaximal, characteristic, complemented
5C2×#5020C211yesmaximal, characteristic, complemented
6#81113C2116yesmaximal, direct product factor
7C2×#4878C212yesmaximal, complemented
8C2×#4885C212yesmaximal
9C2×#4846C211yesmaximal, characteristic, complemented
10C2×#4833C211yesmaximal, characteristic, complemented
11C2×#4800C212yesmaximal, complemented
12C2×#4814C212yesmaximal, complemented
13C2×#3233—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#947C2211yescharacteristic
17C2×#946—21no—
18C2×#946C2211yescharacteristic
19C2×Q8×Dic3—21no—
20#5728—24no—
21#5728C2218yescomplemented
22#5580—24no—
23#5580C2218yes—
24C2×#898—21no—
25C2×#897—21no—
26C2×#895C2211yescharacteristic
27C2×#894—21no—
28C2×#892C2212yes—
29C2×#890C2211yescharacteristic
30C2×#890C2212yes—
31C2×#887—21no—
32C2×C4×D12—21no—
33C2×C4×D12C2211yescharacteristic
34C2×#872—21no—
35C2×C42×D3C2211yescharacteristic
36C2×#869C2211yescharacteristic
37C3×#477C2218yescomplemented
38C2×C3×#107C2212yes—
39C2×C3×C4⋊C8C2211yescharacteristic
40C2×C3×#103C2212yes—
41C2×#835C2212yes—
42C2×#821C2212yes—
43C2×(C2×D3)⋊C8C2212yes—
44C2×#818C2212yes—
45C2×C8×Dic3C2212yes—
46#5084C2218yescomplemented
47#5020C2218yescomplemented
48C2×#809C2212yes—
49C2×#808C2212yes—
50C2×#808C2212yes—
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4885C22116yes—
53#4878C22116yescomplemented
54#4846C2218yescomplemented
55#4833C2218yescomplemented
56#4814C22116yescomplemented
57#4800C22116yescomplemented
58C2×#533—31no—
59C2×#529—31no—
60#3233—316no—
61C2×#499—32no—
62C2×#488—32no—
63C2×#485—32no—
64C2×#482—32no—
65C2×#477D311yescharacteristic, complemented
66C2×#475—31no—
67C2×#476—31no—
68C2×#474—31no—
69C2×#431—31no—
70C22×C3×Q8—21no—
71C22×C3×Q8C2311yescharacteristic
72C22×D12—21no—
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C22×C4×D3D411yescharacteristic
76C22×C4×D3D411yescharacteristic
77C22×C4×D3C2311yescharacteristic
78C22×Dic6—21no—
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8C2314yes—
82C3×C4×Q8—22no—
83C3×C4×Q8C2314yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90#947—22no—
91#947C2314yes—
92#946—28no—
93#946—22no—
94#946C2314yes—
95Q8×Dic3—28no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3C2312yes—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—22no—
110C2×C4×Dic3—22no—
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2312yes—
115#898—28no—
116#897—28no—
117#895—22no—
118#895C2314yes—
119#894—28no—
120#892—24no—
121#892C2318yes—
122#890—24no—
123#890—22no—
124#890C2314yes—
125#890C2318yes—
126#887—28no—
127C4×D12—28no—
128C4×D12—22no—
129C4×D12C2314yes—
130#872—28no—
131C42×D3D414yes—
132C42×D3C2314yes—
133#869—22no—
134#869C2314yes—
135C3×#107C2318yes—
136C3×C4⋊C8C2314yes—
137C3×#103C2318yes—
138#835C2318yes—
139#821C2318yes—
140(C2×D3)⋊C8C2318yes—
141#818C2318yes—
142C8×Dic3C2318yes—
143#809C2318yes—
144#808C2318yes—
145#808C2318yes—
146C4⋊(C3⋊C8)C2314yes—
147#533—38no—
148#533—64no—
149C2×C4⋊Q8—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151#529—38no—
152#529—64no—
153C2×#126—61no—
154C2×#125—32no—
155C2×#125—31no—
156C2×#124—31no—
157C2×#123—61no—
158C2×C22⋊Q8—31no—
159C2×C22⋊Q8—61no—
160C2×C22⋊Q8—61no—
161C2×#121—31no—
162C2×C4×Q8—61no—
163C2×C4×Q8—61no—
164C2×C4×Q8C2×D311yescharacteristic
165C2×C4×D4—31no—
166C2×C4×D4—61no—
167C2×#117—32no—
168C2×#117—61no—
169C22×C42—31no—
170#499—316no—
171#488—316no—
172#485—316no—
173#482—316no—
174#477C2×D318yescomplemented
175#476—38no—
176#475—38no—
177#474—38no—
178#431—38no—
179C2×#107—32no—
180C2×#107C2×D312yes—
181C2×#106—32no—
182C2×#106—32no—
183C2×C4⋊C8—31no—
184C2×C4⋊C8C2×D311yescharacteristic
185C2×#103—32no—
186C2×#103C2×D312yes—
187C2×#102—32no—
188C2×#102—32no—
189C2×C22⋊C8—32no—
190C2×C4×C8—32no—
191C23×D3—21no—
192C23×D3C2×D411yescharacteristic
193C2×C3×Q8—24no—
194C2×C3×Q8—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—44no—
198C2×C3×Q8C2412yes—
199C22×C3×C4—22no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—22no—
206C22×Dic3—21no—
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×D12—22no—
211C2×D12—22no—
212C2×D12—24no—
213C2×D12—21no—
214C2×D12—44no—
215C2×D12C2412yes—
216C2×C4×D3—24no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—24no—
220C2×C4×D3—42no—
221C2×C4×D3—44no—
222C2×C4×D3C2×D412yes—
223C2×C4×D3C2×D414yes—
224C2×C4×D3C2×D414yes—
225C2×C4×D3C2412yes—
226C2×Dic6—24no—
227C2×Dic6—44no—
228C2×C3×C8—24no—
229C2×C3×C8C2414yes—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4C2412yes—
235C3×C4⋊C4C2414yes—
236C3×C42—24no—
237C3×C42C2×D412yes—
238C3×C42C2412yes—
239(C2×D3)⋊C4—24no—
240(C2×D3)⋊C4—28no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—24no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245C4⋊Dic3—22no—
246C4⋊Dic3—24no—
247C4⋊Dic3—21no—
248C4⋊Dic3—24no—
249C4⋊Dic3—28no—
250C4⋊Dic3C2412yes—
251C4⋊Dic3C2414yes—
252Dic3⋊C4—24no—
253Dic3⋊C4—24no—
254Dic3⋊C4—28no—
255C4×Dic3—28no—
256C4×Dic3C2×D412yes—
257C4×Dic3C2×D414yes—
258C4×Dic3C2×D414yes—
259C4×Dic3C2412yes—
260C2×C3⋊C8—24no—
261C2×C3⋊C8C2414yes—
262C22×Q8—21no—
263C22×Q8—61no—
264C22×Q8C22×D311yescharacteristic
265C22×D4—31no—
266C22×D4—61no—
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—31no—
270C23×C4—61no—
271C22×C8—32no—
272C22×C8C22×D312yes—
273C4⋊Q8—22no—
274C4⋊Q8—34no—
275C4⋊Q8—62no—
276C4⋊Q8C22×D314yes—
277#126—68no—
278#125—34no—
279#125—38no—
280#125—62no—
281#125—64no—
282#124—34no—
283#124—62no—
284#123—68no—
285C22⋊Q8—34no—
286C22⋊Q8—68no—
287C22⋊Q8—62no—
288C22⋊Q8—68no—
289#121—34no—
290#121—62no—
291C4×Q8—22no—
292C4×Q8—68no—
293C4×Q8—68no—
294C4×Q8C22×D314yes—
295C4×D4—34no—
296C4×D4—68no—
297C4×D4—62no—
298#117—38no—
299#117—68no—
300#117—64no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4C22×D312yes—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C42—21no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—34no—
324C2×C42—34no—
325C2×C42—62no—
326C2×C42C22×D311yescharacteristic
327#107—38no—
328#107C22×D318yes—
329#106—38no—
330#106—38no—
331C4⋊C8—34no—
332C4⋊C8C22×D314yes—
333#103—38no—
334#103C22×D318yes—
335#102—38no—
336#102—38no—
337C22⋊C8—38no—
338C4×C8—38no—
339C23×C3C22×D411yescharacteristic
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3C22×D412yes—
348C3×Q8—22no—
349C3×Q8—22no—
350C3×Q8—48no—
351C3×Q8—42no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—44no—
359C2×C3×C4#13514yes—
360C2×C3×C4C22×D412yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D412yes—
363C2×C3×C42−1+412yes—
364C2×C3×C42−1+412yes—
365C2×C3×C4C2511yescharacteristic
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—24no—
370C2×Dic3—22no—
371C2×Dic3—44no—
372C2×Dic3—42no—
373C2×Dic3C22×D412yes—
374C2×Dic3C22×D412yes—
375C2×Dic3C22×D412yes—
376D12—22no—
377D12—22no—
378D12—48no—
379D12—42no—
380C4×D3—24no—
381C4×D3—28no—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—44no—
385C4×D3—48no—
386Dic6—48no—
387C3×C8—24no—
388C3⋊C8—24no—
389C24—31no—
390C24—61no—
391C2×Q8—21no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—64no—
397C2×Q8—124no—
398C2×Q8C23×D312yes—
399C2×D4—32no—
400C2×D4—64no—
401C2×D4—61no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—124no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—32no—
412C22×C4—34no—
413C22×C4—32no—
414C22×C4—34no—
415C22×C4—64no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—61no—
420C22×C4—64no—
421C22×C4—122no—
422C22×C4—124no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C23×D311yescharacteristic
426C2×C8—24no—
427C2×C8—34no—
428C2×C8—64no—
429C2×C8C23×D314yes—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—34no—
435C4⋊C4—32no—
436C4⋊C4—64no—
437C4⋊C4—64no—
438C4⋊C4—64no—
439C4⋊C4—68no—
440C4⋊C4—68no—
441C4⋊C4—62no—
442C4⋊C4—61no—
443C4⋊C4—64no—
444C4⋊C4C23×D314yes—
445C4⋊C4C23×D312yes—
446C22⋊C4—64no—
447C22⋊C4—64no—
448C22⋊C4—64no—
449C22⋊C4—68no—
450C22⋊C4—64no—
451C22⋊C4—64no—
452C42—24no—
453C42—34no—
454C42—34no—
455C42—32no—
456C42—32no—
457C42—68no—
458C42D3×D412yes—
459C42C23×D312yes—
460C22×C3#53612yes—
461C22×C3C2×#13512yes—
462C22×C3#57512yes—
463C22×C3C23×D411yescharacteristic
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—44no—
468C2×D3—42no—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—84no—
472C3×C4—22no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—44no—
476C3×C4#53612yes—
477C3×C4C2×#13514yes—
478C3×C4C2×2−1+411yescharacteristic
479C3×C4C2×2−1+411yescharacteristic
480Dic3—22no—
481Dic3—22no—
482Dic3—22no—
483Dic3—44no—
484Dic3—42no—
485C23—32no—
486C23—62no—
487C23—62no—
488C23—62no—
489C23—62no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23C2×D3×D411yescharacteristic
494Q8—22no—
495Q8—22no—
496Q8—48no—
497Q8—42no—
498Q8—128no—
499D4—62no—
500D4—62no—
501D4—128no—
502D4—122no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—68no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—62no—
522C2×C4—124no—
523C2×C4—122no—
524C2×C4—124no—
525C2×C4—128no—
526C2×C4#91914yes—
527C2×C4C2×D3×D412yes—
528C2×C4C2×D3×D412yes—
529C2×C4C2×D3×D411yescharacteristic
530C2×C4#100712yes—
531C2×C4#101012yes—
532C2×C4C24×D311yescharacteristic
533C8—24no—
534C8—64no—
535C2×C3#323314yescomplemented
536C2×C3C2×#53611yescharacteristic
537C2×C3C22×#13511yescharacteristic
538C2×C3C2×#57511yescharacteristic
539D3—42no—
540D3—82no—
541C22—62no—
542C22—62no—
543C22—62no—
544C22—124no—
545C22—122no—
546C22—122no—
547C22—122no—
548C22—244no—
549C22#562812yes—
550C22C2×#91912yes—
551C22#577312yes—
552C22C22×D3×D411yescharacteristic
553C4—22no—
554C4—42no—
555C4—44no—
556C4—42no—
557C4—62no—
558C4—62no—
559C4—62no—
560C4—122no—
561C4—124no—
562C4#562812yes—
563C4C2×#91914yes—
564C4C2×#100711yescharacteristic
565C4C2×#101011yescharacteristic
566C3C2×#323311yescharacteristic, complemented, Sylow
567C2—122no—
568C2—242no—
569C2#8111314yesdirect product factor
570C2C2×#562811yescharacteristic
571C2C22×#91911yescharacteristic
572C2C2×#577311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#477C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenterC2×#477D3
hypocenterC3C2×#3233
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
3. C2×C3×#477C2semidirect
4. C2×#5084C2semidirect
5. C2×#5020C2semidirect
6. #81113C2direct (N × H)
7. C2×#4878C2semidirect
9. C2×#4846C2semidirect
10. C2×#4833C2semidirect
11. C2×#4800C2semidirect
12. C2×#4814C2semidirect
21. #5728C22semidirect
37. C3×#477C22semidirect
46. #5084C22semidirect
47. #5020C22semidirect
53. #4878C22semidirect
54. #4846C22semidirect
55. #4833C22semidirect
56. #4814C22semidirect
57. #4800C22semidirect
65. C2×#477D3semidirect
174. #477C2×D3semidirect
535. C2×C3#3233semidirect
566. C3C2×#3233semidirect
569. C2#81113direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#55802
C2×C3×#4772
C2×#50842
C2×#50202
#811132
C2×#48782
C2×#48852
C2×#48462
C2×#48332
C2×#48002
C2×#48142
C2×#32333