Finite Group Databasebeta

← C2×#76285 #11001057

Subgroups — C2×#76285 #11001057

Order 768.

3182 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#890C211yesmaximal, characteristic, complemented
2C22×C42×D3C211yesmaximal, characteristic
3C2×C3×#389C211yesmaximal, characteristic, complemented
4C2×#4989C212yesmaximal
5C2×#4987C211yesmaximal, characteristic
6C2×#4987C211yesmaximal, characteristic
7C2×#4980C212yesmaximal, complemented
8C2×#4977C211yesmaximal, characteristic, complemented
9C2×#4928C211yesmaximal, characteristic, complemented
10#76285C2116yesmaximal, direct product factor
11C2×#4668C212yesmaximal
12C2×#4652C212yesmaximal, complemented
13C2×#2851—31nomaximal, Sylow
14C23×C4×D3C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C42C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2212yes—
26C22×Dic3⋊C4C2212yes—
27C22×Dic3⋊C4C2212yes—
28C22×C4×Dic3C2211yescharacteristic
29C22×C4×Dic3C2211yescharacteristic
30C22×C4×Dic3C2211yescharacteristic
31C2×#890—28no—
32C2×#890C2218yescomplemented
33C2×C42×D3—28no—
34C2×C42×D3C2218yes—
35C3×#389C2218yescomplemented
36C2×C3×#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42#4989C22116yes—
43#4987C2218yes—
44#4987C2218yes—
45#4980C22116yescomplemented
46#4977C2218yescomplemented
47#4928C2218yescomplemented
48#4668C22116yes—
49#4652C22116yescomplemented
50C22×#125—31no—
51C23×C42—31no—
52#2851—316no—
53C2×#389—31no—
54C2×#389D311yescharacteristic, complemented
55C2×#387—32no—
56C2×#387—32no—
57C2×#385—32no—
58C2×#385—32no—
59C2×#382—31no—
60C2×#380—31no—
61C2×#380—31no—
62C24×D3C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2312yes—
69C23×Dic3C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C22×C4×D3—24no—
74C22×C4×D3—24no—
75C22×C4×D3—24no—
76C22×C4×D3—24no—
77C22×C4×D3—24no—
78C22×C4×D3—24no—
79C22×C4×D3—24no—
80C22×C4×D3—24no—
81C22×C4×D3Q818yes—
82C22×C4×D3C2314yes—
83C22×C4×D3C2314yes—
84C22×C4×D3C2314yes—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4—28no—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C4⋊C4C2318yes—
90C2×C3×C4⋊C4C2314yes—
91C2×C3×C42—24no—
92C2×C3×C42C2314yes—
93C2×(C2×D3)⋊C4—28no—
94C2×(C2×D3)⋊C4—28no—
95C2×(C2×D3)⋊C4C2318yes—
96C2×(C2×D3)⋊C4C2318yes—
97C2×C4⋊Dic3—24no—
98C2×C4⋊Dic3—24no—
99C2×C4⋊Dic3—28no—
100C2×C4⋊Dic3C2318yes—
101C2×C4⋊Dic3C2314yes—
102C2×C4⋊Dic3C2314yes—
103C2×Dic3⋊C4—28no—
104C2×Dic3⋊C4—28no—
105C2×Dic3⋊C4C2318yes—
106C2×Dic3⋊C4C2318yes—
107C2×C4×Dic3—24no—
108C2×C4×Dic3—24no—
109C2×C4×Dic3D418yes—
110C2×C4×Dic3C2314yes—
111C2×C4×Dic3C2314yes—
112C2×C4×Dic3C2314yes—
113#890—232no—
114C42×D3—232no—
115C3×#95C2318yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C3⋊#95C2318yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C24×C4—31no—
122C24×C4—31no—
123C24×C4—31no—
124C2×#125—38no—
125C2×#125—68no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4C2×D311yescharacteristic
132C22×C4⋊C4C2×D311yescharacteristic
133C22×C4⋊C4C2×D312yes—
134C22×C22⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C42—31no—
137C22×C42—31no—
138C22×C42—31no—
139C22×C42—38no—
140C22×C42—68no—
141C22×C42C2×D311yescharacteristic
142#389—38no—
143#389C2×D318yescomplemented
144#387—316no—
145#387—316no—
146#385—316no—
147#385—316no—
148#382—38no—
149#380—38no—
150#380—38no—
151C2×#95—32no—
152C2×#95—32no—
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95C2×D312yes—
157C24×C3C2411yescharacteristic
158C23×D3—22no—
159C23×D3—22no—
160C23×D3—22no—
161C23×D3—22no—
162C23×D3C2×Q814yes—
163C23×D3#4114yes—
164C23×D3#4114yes—
165C23×D3C2412yes—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×Q814yes—
178C22×C3×C4#4114yes—
179C22×C3×C4#4114yes—
180C22×C3×C4#4114yes—
181C22×C3×C4C2414yes—
182C22×C3×C4C2414yes—
183C22×C3×C4C2412yes—
184C22×C3×C4C2412yes—
185C22×C3×C4C2412yes—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3C2×D414yes—
199C22×Dic3C2×D414yes—
200C22×Dic3C2×Q814yes—
201C22×Dic3C2414yes—
202C22×Dic3C2414yes—
203C22×Dic3C2412yes—
204C22×Dic3C2412yes—
205C22×Dic3C2412yes—
206C2×C4×D3—232no—
207C2×C4×D3—232no—
208C2×C4×D3—28no—
209C2×C4×D3—28no—
210C2×C4×D3—232no—
211C2×C4×D3—28no—
212C2×C4×D3—28no—
213C2×C4×D3—28no—
214C2×C4×D3—28no—
215C2×C4×D3—28no—
216C2×C4×D3—28no—
217C2×C4×D3C2×Q8116yes—
218C3×C4⋊C4—28no—
219C3×C4⋊C4—28no—
220C3×C4⋊C4—216no—
221C3×C42—28no—
222(C2×D3)⋊C4—216no—
223(C2×D3)⋊C4—216no—
224C4⋊Dic3—28no—
225C4⋊Dic3—28no—
226C4⋊Dic3—216no—
227Dic3⋊C4—216no—
228Dic3⋊C4—216no—
229C4×Dic3—28no—
230C4×Dic3—28no—
231C4×Dic3C2×D4116yes—
232C25—31no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—38no—
237C23×C4—34no—
238C23×C4—32no—
239C23×C4—32no—
240C23×C4—34no—
241C23×C4—34no—
242C23×C4—64no—
243C23×C4—64no—
244C23×C4—64no—
245C23×C4—64no—
246C23×C4—64no—
247C23×C4—64no—
248C23×C4—64no—
249C23×C4—64no—
250C23×C4C22×D312yes—
251C23×C4C22×D312yes—
252C23×C4C22×D311yescharacteristic
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255#125—632no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—34no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—38no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—38no—
264C2×C4⋊C4—68no—
265C2×C4⋊C4—68no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—64no—
268C2×C4⋊C4—68no—
269C2×C4⋊C4C22×D314yes—
270C2×C4⋊C4C22×D314yes—
271C2×C4⋊C4C22×D318yes—
272C2×C22⋊C4—38no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—68no—
275C2×C22⋊C4—68no—
276C2×C42—24no—
277C2×C42—38no—
278C2×C42—34no—
279C2×C42—34no—
280C2×C42—34no—
281C2×C42—632no—
282C2×C42—64no—
283C2×C42—64no—
284C2×C42C22×D314yes—
285#95—38no—
286#95—38no—
287#95—38no—
288#95—38no—
289#95—38no—
290#95C22×D318yes—
291C23×C3C22×D412yes—
292C23×C3C22×Q812yes—
293C23×C3C2×#4112yes—
294C23×C3C2×#4112yes—
295C23×C3C2×#4112yes—
296C23×C3C2×#4112yes—
297C23×C3C2×#4112yes—
298C23×C3C2511yescharacteristic
299C22×D3—28no—
300C22×D3—28no—
301C22×D3—28no—
302C22×D3—28no—
303C22×D3—28no—
304C22×D3—28no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—22no—
309C22×D3#125116yescomplemented
310C22×D3C22×Q814yes—
311C22×D3C2×#4114yes—
312C22×D3C2×#4114yes—
313C2×C3×C4—28no—
314C2×C3×C4—28no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—48no—
326C2×C3×C4—48no—
327C2×C3×C4C22⋊Q8116yes—
328C2×C3×C4C22×D414yes—
329C2×C3×C4C22×Q814yes—
330C2×C3×C4C2×#4114yes—
331C2×C3×C4C2×#4114yes—
332C2×C3×C4C2×#4114yes—
333C2×Dic3—28no—
334C2×Dic3—28no—
335C2×Dic3—28no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—48no—
349C2×Dic3—48no—
350C2×Dic3C22×D414yes—
351C2×Dic3C22×D414yes—
352C2×Dic3C22×Q814yes—
353C4×D3—232no—
354C4×D3—232no—
355C4×D3—232no—
356C24—34no—
357C24—34no—
358C24—34no—
359C24—32no—
360C24—62no—
361C24—62no—
362C24—62no—
363C24—62no—
364C24C23×D311yescharacteristic
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—34no—
376C22×C4—34no—
377C22×C4—34no—
378C22×C4—32no—
379C22×C4—32no—
380C22×C4—32no—
381C22×C4—316no—
382C22×C4—34no—
383C22×C4—34no—
384C22×C4—68no—
385C22×C4—68no—
386C22×C4—68no—
387C22×C4—68no—
388C22×C4—632no—
389C22×C4—632no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—632no—
397C22×C4—68no—
398C22×C4—68no—
399C22×C4—68no—
400C22×C4—68no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4#23514yes—
408C22×C4D3×D414yes—
409C22×C4#23714yes—
410C22×C4D3×Q814yes—
411C22×C4#23914yes—
412C22×C4C23×D314yes—
413C22×C4C23×D314yes—
414C22×C4C23×D312yes—
415C22×C4C23×D312yes—
416C22×C4C23×D312yes—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—216no—
420C4⋊C4—616no—
421C4⋊C4—616no—
422C4⋊C4—68no—
423C4⋊C4—68no—
424C4⋊C4—616no—
425C22⋊C4—616no—
426C22⋊C4—616no—
427C42—28no—
428C42—316no—
429C42—68no—
430C42—68no—
431C22×C3C2×C22⋊Q814yes—
432C22×C3#52514yes—
433C22×C3#52514yes—
434C22×C3C2×#12514yes—
435C22×C3#52914yes—
436C22×C3#52914yes—
437C22×C3#53314yes—
438C22×C3C23×D411yescharacteristic
439C22×C3C23×Q811yescharacteristic
440C22×C3C22×#4111yescharacteristic
441C22×C3C22×#4111yescharacteristic
442C22×C3C22×#4111yescharacteristic
443C22×C3C22×#4111yescharacteristic
444C22×C3C22×#4111yescharacteristic
445C2×D3—232no—
446C2×D3—24no—
447C2×D3—24no—
448C2×D3—24no—
449C2×D3—24no—
450C2×D3—24no—
451C2×D3—24no—
452C2×D3C2×#12518yescomplemented
453C3×C4—24no—
454C3×C4—24no—
455C3×C4—44no—
456C3×C4—44no—
457C3×C4C2×C22⋊Q818yes—
458Dic3—24no—
459Dic3—24no—
460Dic3—24no—
461Dic3—44no—
462Dic3—44no—
463C23—316no—
464C23—34no—
465C23—34no—
466C23—34no—
467C23—62no—
468C23—62no—
469C23—62no—
470C23—62no—
471C23—68no—
472C23—68no—
473C23—68no—
474C23—68no—
475C23—68no—
476C23—68no—
477C23C2×#23512yes—
478C23C2×D3×D412yes—
479C23C2×#23712yes—
480C23C2×D3×Q812yes—
481C23C2×#23912yes—
482C23D3×#4112yes—
483C23D3×#4112yes—
484C23C24×D311yescharacteristic
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—34no—
498C2×C4—34no—
499C2×C4—34no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—632no—
503C2×C4—632no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—68no—
511C2×C4—68no—
512C2×C4—68no—
513C2×C4—632no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—128no—
521C2×C4—128no—
522C2×C4#896116yes—
523C2×C4C2×#23514yes—
524C2×C4C2×D3×D414yes—
525C2×C4C2×#23714yes—
526C2×C4C2×D3×Q814yes—
527C2×C4C2×#23914yes—
528C2×C3#285118yescomplemented
529C2×C3C22×C22⋊Q811yescharacteristic
530C2×C3C2×#52511yescharacteristic
531C2×C3C2×#52511yescharacteristic
532C2×C3C22×#12511yescharacteristic
533C2×C3C2×#52911yescharacteristic
534C2×C3C2×#52911yescharacteristic
535C2×C3C2×#53311yescharacteristic
536D3—28no—
537C22—38no—
538C22—64no—
539C22—64no—
540C22—64no—
541C22—64no—
542C22—64no—
543C22—64no—
544C22—632no—
545C22C2×#89614yes—
546C22#555714yes—
547C22#558014yes—
548C22#560614yes—
549C22#562714yes—
550C22D3×#12514yes—
551C22#572814yes—
552C22C22×#23511yescharacteristic
553C22C22×D3×D411yescharacteristic
554C22C22×#23711yescharacteristic
555C22C22×D3×Q811yescharacteristic
556C22C22×#23911yescharacteristic
557C22C2×D3×#4111yescharacteristic
558C22C2×D3×#4111yescharacteristic
559C4—24no—
560C4—24no—
561C4—44no—
562C4—44no—
563C4—64no—
564C4—64no—
565C4—64no—
566C4—124no—
567C4—124no—
568C4C2×#89618yes—
569C3C2×#285111yescharacteristic, complemented, Sylow
570C2—68no—
571C2#7628518yesdirect product factor
572C2C22×#89611yescharacteristic
573C2C2×#555711yescharacteristic
574C2C2×#558011yescharacteristic
575C2C2×#560611yescharacteristic
576C2C2×#562711yescharacteristic
577C2C2×D3×#12511yescharacteristic
578C2C2×#572811yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#389C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×#389D3
hypocenterC3C2×#2851
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#890C2semidirect
3. C2×C3×#389C2semidirect
7. C2×#4980C2semidirect
8. C2×#4977C2semidirect
9. C2×#4928C2semidirect
10. #76285C2direct (N × H)
12. C2×#4652C2semidirect
32. C2×#890C22semidirect
35. C3×#389C22semidirect
45. #4980C22semidirect
46. #4977C22semidirect
47. #4928C22semidirect
49. #4652C22semidirect
54. C2×#389D3semidirect
143. #389C2×D3semidirect
309. C22×D3#125semidirect
452. C2×D3C2×#125semidirect
528. C2×C3#2851semidirect
569. C3C2×#2851semidirect
571. C2#76285direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8902
C22×C42×D32
C2×C3×#3892
C2×#49892
C2×#49872
C2×#49802
C2×#49772
C2×#49282
#762852
C2×#46682
C2×#46522
C2×#28513